Optimal. Leaf size=39 \[ i \text {Li}_2\left (-i e^x\right )-i \text {Li}_2\left (i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+x \tan ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {5203, 4180, 2279, 2391} \[ i \text {PolyLog}\left (2,-i e^x\right )-i \text {PolyLog}\left (2,i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+x \tan ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 5203
Rubi steps
\begin {align*} \int \tan ^{-1}(\sinh (x)) \, dx &=x \tan ^{-1}(\sinh (x))-\int x \text {sech}(x) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )+x \tan ^{-1}(\sinh (x))+i \int \log \left (1-i e^x\right ) \, dx-i \int \log \left (1+i e^x\right ) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )+x \tan ^{-1}(\sinh (x))+i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^x\right )-i \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^x\right )\\ &=-2 x \tan ^{-1}\left (e^x\right )+x \tan ^{-1}(\sinh (x))+i \text {Li}_2\left (-i e^x\right )-i \text {Li}_2\left (i e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 64, normalized size = 1.64 \[ x \tan ^{-1}(\sinh (x))+i \left (\text {Li}_2\left (-i e^{-x}\right )-\text {Li}_2\left (i e^{-x}\right )+x \left (\log \left (1-i e^{-x}\right )-\log \left (1+i e^{-x}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 58, normalized size = 1.49 \[ x \arctan \left (\sinh \relax (x)\right ) + i \, x \log \left (i \, \cosh \relax (x) + i \, \sinh \relax (x) + 1\right ) - i \, x \log \left (-i \, \cosh \relax (x) - i \, \sinh \relax (x) + 1\right ) - i \, {\rm Li}_2\left (i \, \cosh \relax (x) + i \, \sinh \relax (x)\right ) + i \, {\rm Li}_2\left (-i \, \cosh \relax (x) - i \, \sinh \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \arctan \left (\sinh \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 52, normalized size = 1.33 \[ x \arctan \left (\sinh \relax (x )\right )-i x \left (\ln \left (1-i {\mathrm e}^{x}\right )-\ln \left (1+i {\mathrm e}^{x}\right )\right )+i \dilog \left (1+i {\mathrm e}^{x}\right )-i \dilog \left (1-i {\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right ) - 2 \, \int \frac {x e^{x}}{e^{\left (2 \, x\right )} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {atan}\left (\mathrm {sinh}\relax (x)\right ) \,d x \]
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 \[ \int \operatorname {atan}{\left (\sinh {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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