Optimal. Leaf size=34 \[ -\frac {1}{2} \tanh ^{-1}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2320, 3853,
3855} \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \tan \left (1-e^x\right ) \sec \left (1-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 3853
Rule 3855
Rubi steps
\begin {align*} \int e^x \sec ^3\left (1-e^x\right ) \, dx &=\text {Subst}\left (\int \sec ^3(1-x) \, dx,x,e^x\right )\\ &=-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right )+\frac {1}{2} \text {Subst}\left (\int \sec (1-x) \, dx,x,e^x\right )\\ &=-\frac {1}{2} \tanh ^{-1}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tanh ^{-1}\left (\sin \left (1-e^x\right )\right )-\frac {1}{2} \sec \left (1-e^x\right ) \tan \left (1-e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 28, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\sec \left (-1+{\mathrm e}^{x}\right ) \tan \left (-1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (\sec \left (-1+{\mathrm e}^{x}\right )+\tan \left (-1+{\mathrm e}^{x}\right )\right )}{2}\) | \(28\) |
default | \(\frac {\sec \left (-1+{\mathrm e}^{x}\right ) \tan \left (-1+{\mathrm e}^{x}\right )}{2}+\frac {\ln \left (\sec \left (-1+{\mathrm e}^{x}\right )+\tan \left (-1+{\mathrm e}^{x}\right )\right )}{2}\) | \(28\) |
norman | \(\frac {\tan ^{3}\left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )+\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )}{\left (\tan ^{2}\left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )-1\right )^{2}}-\frac {\ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )-1\right )}{2}+\frac {\ln \left (\tan \left (-\frac {1}{2}+\frac {{\mathrm e}^{x}}{2}\right )+1\right )}{2}\) | \(57\) |
risch | \(-\frac {i \left ({\mathrm e}^{3 i \left (-1+{\mathrm e}^{x}\right )}-{\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}\right )}{\left ({\mathrm e}^{2 i \left (-1+{\mathrm e}^{x}\right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}-i\right )}{2}+\frac {\ln \left ({\mathrm e}^{i \left (-1+{\mathrm e}^{x}\right )}+i\right )}{2}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 39, normalized size = 1.15 \begin {gather*} -\frac {\sin \left (e^{x} - 1\right )}{2 \, {\left (\sin \left (e^{x} - 1\right )^{2} - 1\right )}} + \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs.
\(2 (21) = 42\).
time = 0.36, size = 52, normalized size = 1.53 \begin {gather*} \frac {\cos \left (e^{x} - 1\right )^{2} \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \cos \left (e^{x} - 1\right )^{2} \log \left (-\sin \left (e^{x} - 1\right ) + 1\right ) + 2 \, \sin \left (e^{x} - 1\right )}{4 \, \cos \left (e^{x} - 1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{x} \sec ^{3}{\left (e^{x} - 1 \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.56, size = 41, normalized size = 1.21 \begin {gather*} -\frac {\sin \left (e^{x} - 1\right )}{2 \, {\left (\sin \left (e^{x} - 1\right )^{2} - 1\right )}} + \frac {1}{4} \, \log \left (\sin \left (e^{x} - 1\right ) + 1\right ) - \frac {1}{4} \, \log \left (-\sin \left (e^{x} - 1\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.53, size = 78, normalized size = 2.29 \begin {gather*} -\mathrm {atan}\left ({\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\right )\,1{}\mathrm {i}-\frac {{\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{{\mathrm {e}}^{-2{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}+1}+\frac {{\mathrm {e}}^{-\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{2\,{\mathrm {e}}^{-2{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,2{}\mathrm {i}}+{\mathrm {e}}^{-4{}\mathrm {i}}\,{\mathrm {e}}^{{\mathrm {e}}^x\,4{}\mathrm {i}}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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