Optimal. Leaf size=45 \[ -\frac{i}{2 (1+i \sinh (x))}-\frac{3}{4} i \log (-\sinh (x)+i)-\frac{1}{4} i \log (\sinh (x)+i) \]
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Rubi [A] time = 0.0475287, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 88} \[ -\frac{i}{2 (1+i \sinh (x))}-\frac{3}{4} i \log (-\sinh (x)+i)-\frac{1}{4} i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tanh (x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{x^2}{(i-i x) (i+i x)^2} \, dx,x,i \sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{i}{4 (-1+x)}+\frac{i}{2 (1+x)^2}-\frac{3 i}{4 (1+x)}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\frac{3}{4} i \log (i-\sinh (x))-\frac{1}{4} i \log (i+\sinh (x))-\frac{i}{2 (1+i \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.0374018, size = 39, normalized size = 0.87 \[ \frac{1}{4} \left (-\frac{2}{\sinh (x)-i}-3 i \log (-\sinh (x)+i)-i \log (\sinh (x)+i)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 65, normalized size = 1.4 \begin{align*} i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{3\,i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) +{i \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}+i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{\frac{i}{2}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03818, size = 61, normalized size = 1.36 \begin{align*} -i \, x + \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{3}{2} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6308, size = 207, normalized size = 4.6 \begin{align*} \frac{2 i \, x e^{\left (2 \, x\right )} + 2 \,{\left (2 \, x - 1\right )} e^{x} +{\left (-i \, e^{\left (2 \, x\right )} - 2 \, e^{x} + i\right )} \log \left (e^{x} + i\right ) +{\left (-3 i \, e^{\left (2 \, x\right )} - 6 \, e^{x} + 3 i\right )} \log \left (e^{x} - i\right ) - 2 i \, x}{2 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} - 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1512, size = 74, normalized size = 1.64 \begin{align*} \frac{3 i \, e^{\left (-x\right )} - 3 i \, e^{x} - 2}{4 \,{\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}} - \frac{1}{4} i \, \log \left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right ) - \frac{3}{4} i \, \log \left (-e^{\left (-x\right )} + e^{x} - 2 i\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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