Exponential, Logarithmic, Sine, and Cosine Integrals - 6.14 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
6.14.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\expintE@{t}\diff{t} = \frac{1}{a}\ln@{1+a}}
\int_{0}^{\infty}e^{-at}\expintE@{t}\diff{t} = \frac{1}{a}\ln@{1+a}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > -1} 
int(exp(- a*t)*Ei(t), t = 0..infinity) = (1)/(a)*ln(1 + a)

Integrate[Exp[- a*t]*ExpIntegralE[1, t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a]*Log[1 + a]
Failure Successful
Failed [4 / 4]
Result: -.1487623676-2.094395103*I
Test Values: {a = 1.5}


Result: -.5753641448
Test Values: {a = -.5}

... skip entries to safe data
 Successful [Tested: 4]
6.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\cosint@{t}\diff{t} = -\frac{1}{2a}\ln@{1+a^{2}}}
\int_{0}^{\infty}e^{-at}\cosint@{t}\diff{t} = -\frac{1}{2a}\ln@{1+a^{2}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*Ci(t), t = 0..infinity) = -(1)/(2*a)*ln(1 + (a)^(2))

Integrate[Exp[- a*t]*CosIntegral[t], {t, 0, Infinity}, GenerateConditions->None] == -Divide[1,2*a]*Log[1 + (a)^(2)]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
6.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\shiftsinint@{t}\diff{t} = -\frac{1}{a}\atan@@{a}}
\int_{0}^{\infty}e^{-at}\shiftsinint@{t}\diff{t} = -\frac{1}{a}\atan@@{a}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*Ssi(t), t = 0..infinity) = -(1)/(a)*arctan(a)

Integrate[Exp[- a*t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}, GenerateConditions->None] == -Divide[1,a]*ArcTan[a]
Successful Failure -
Failed [3 / 3]
Result: -902994.0050351195
Test Values: {Rule[a, 1.5]}


Result: -902991.9106400171
Test Values: {Rule[a, 0.5]}

... skip entries to safe data
6.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\expintE^{2}@{t}\diff{t} = 2\ln@@{2}}
\int_{0}^{\infty}\expintE^{2}@{t}\diff{t} = 2\ln@@{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((Ei(t))^(2), t = 0..infinity) = 2*ln(2)

Integrate[(ExpIntegralE[1, t])^(2), {t, 0, Infinity}, GenerateConditions->None] == 2*Log[2]
Failure Successful Error Successful [Tested: 1]
6.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\cos@@{t}\cosint@{t}\diff{t} = \int_{0}^{\infty}\sin@@{t}\shiftsinint@{t}\diff{t}}
\int_{0}^{\infty}\cos@@{t}\cosint@{t}\diff{t} = \int_{0}^{\infty}\sin@@{t}\shiftsinint@{t}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(cos(t)*Ci(t), t = 0..infinity) = int(sin(t)*Ssi(t), t = 0..infinity)

Integrate[Cos[t]*CosIntegral[t], {t, 0, Infinity}, GenerateConditions->None] == Integrate[Sin[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [1 / 1]
Result: 902989.9925173485
Test Values: {}

6.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\sin@@{t}\shiftsinint@{t}\diff{t} = -\tfrac{1}{4}\pi}
\int_{0}^{\infty}\sin@@{t}\shiftsinint@{t}\diff{t} = -\tfrac{1}{4}\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(sin(t)*Ssi(t), t = 0..infinity) = -(1)/(4)*Pi

Integrate[Sin[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}, GenerateConditions->None] == -Divide[1,4]*Pi
Successful Failure Skip - symbolical successful subtest
Failed [1 / 1]
Result: -902989.9925173485
Test Values: {}

6.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\cosint^{2}@{t}\diff{t} = \int_{0}^{\infty}\shiftsinint^{2}@{t}\diff{t}}
\int_{0}^{\infty}\cosint^{2}@{t}\diff{t} = \int_{0}^{\infty}\shiftsinint^{2}@{t}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((Ci(t))^(2), t = 0..infinity) = int((Ssi(t))^(2), t = 0..infinity)

Integrate[(CosIntegral[t])^(2), {t, 0, Infinity}, GenerateConditions->None] == Integrate[(SinIntegral[t] - Pi/2)^(2), {t, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
6.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\shiftsinint^{2}@{t}\diff{t} = \tfrac{1}{2}\pi}
\int_{0}^{\infty}\shiftsinint^{2}@{t}\diff{t} = \tfrac{1}{2}\pi		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int((Ssi(t))^(2), t = 0..infinity) = (1)/(2)*Pi

Integrate[(SinIntegral[t] - Pi/2)^(2), {t, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*Pi
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
6.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\cosint@{t}\shiftsinint@{t}\diff{t} = \ln@@{2}}
\int_{0}^{\infty}\cosint@{t}\shiftsinint@{t}\diff{t} = \ln@@{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
int(Ci(t)*Ssi(t), t = 0..infinity) = ln(2)

Integrate[CosIntegral[t]*SinIntegral[t] - Pi/2, {t, 0, Infinity}, GenerateConditions->None] == Log[2]
Failure Failure Successful [Tested: 0]
Failed [1 / 1]
Result: -902996.3337464853
Test Values: {}

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