Optimal. Leaf size=26 \[ \frac {1}{2} i \text {sech}^2(x)+\frac {1}{2} \tan ^{-1}(\sinh (x))-\frac {1}{2} \tanh (x) \text {sech}(x) \]
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Rubi [A] time = 0.06, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2706, 2606, 30, 2611, 3770} \[ \frac {1}{2} i \text {sech}^2(x)+\frac {1}{2} \tan ^{-1}(\sinh (x))-\frac {1}{2} \tanh (x) \text {sech}(x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 2611
Rule 2706
Rule 3770
Rubi steps
\begin {align*} \int \frac {\tanh (x)}{i+\sinh (x)} \, dx &=-\left (i \int \text {sech}^2(x) \tanh (x) \, dx\right )+\int \text {sech}(x) \tanh ^2(x) \, dx\\ &=-\frac {1}{2} \text {sech}(x) \tanh (x)+i \operatorname {Subst}(\int x \, dx,x,\text {sech}(x))+\frac {1}{2} \int \text {sech}(x) \, dx\\ &=\frac {1}{2} \tan ^{-1}(\sinh (x))+\frac {1}{2} i \text {sech}^2(x)-\frac {1}{2} \text {sech}(x) \tanh (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 0.77 \[ \frac {1}{2} \tan ^{-1}(\sinh (x))-\frac {1}{2 (\sinh (x)+i)} \]
Antiderivative was successfully verified.
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fricas [B] time = 3.17, size = 56, normalized size = 2.15 \[ \frac {{\left (i \, e^{\left (2 \, x\right )} - 2 \, e^{x} - i\right )} \log \left (e^{x} + i\right ) + {\left (-i \, e^{\left (2 \, x\right )} + 2 \, e^{x} + i\right )} \log \left (e^{x} - i\right ) - 2 \, e^{x}}{2 \, e^{\left (2 \, x\right )} + 4 i \, e^{x} - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 53, normalized size = 2.04 \[ \frac {-i \, e^{\left (-x\right )} + i \, e^{x} + 2}{4 \, {\left (e^{\left (-x\right )} - e^{x} - 2 i\right )}} + \frac {1}{4} i \, \log \left (-e^{\left (-x\right )} + e^{x} + 2 i\right ) - \frac {1}{4} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 45, normalized size = 1.73 \[ -\frac {i}{\left (\tanh \left (\frac {x}{2}\right )+i\right )^{2}}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )}{2}+\frac {1}{\tanh \left (\frac {x}{2}\right )+i}-\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )-i\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 42, normalized size = 1.62 \[ \frac {e^{\left (-x\right )}}{-2 i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} + \frac {1}{2} i \, \log \left (i \, e^{\left (-x\right )} + 1\right ) - \frac {1}{2} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 29, normalized size = 1.12 \[ \mathrm {atan}\left ({\mathrm {e}}^x\right )+\frac {1{}\mathrm {i}}{{\mathrm {e}}^{2\,x}-1+{\mathrm {e}}^x\,2{}\mathrm {i}}-\frac {1}{{\mathrm {e}}^x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 32, normalized size = 1.23 \[ \frac {\log {\left (e^{x} - i \right )}}{2} - \frac {\log {\left (e^{x} + i \right )}}{2} + \frac {e^{x}}{- e^{2 x} - 2 i e^{x} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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