Optimal. Leaf size=29 \[ \frac{\text{csch}^2(x)}{2}+2 i \text{csch}(x)+2 \log (\sinh (x))-2 \log (\sinh (x)+i) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0480786, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2707, 77} \[ \frac{\text{csch}^2(x)}{2}+2 i \text{csch}(x)+2 \log (\sinh (x))-2 \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2707
Rule 77
Rubi steps
\begin{align*} \int \frac{\coth ^3(x)}{(i+\sinh (x))^2} \, dx &=-\operatorname{Subst}\left (\int \frac{i-x}{x^3 (i+x)} \, dx,x,\sinh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{x^3}+\frac{2 i}{x^2}-\frac{2}{x}+\frac{2}{i+x}\right ) \, dx,x,\sinh (x)\right )\\ &=2 i \text{csch}(x)+\frac{\text{csch}^2(x)}{2}+2 \log (\sinh (x))-2 \log (i+\sinh (x))\\ \end{align*}
Mathematica [A] time = 0.016163, size = 29, normalized size = 1. \[ \frac{\text{csch}^2(x)}{2}+2 i \text{csch}(x)+2 \log (\sinh (x))-2 \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 51, normalized size = 1.8 \begin{align*} -i\tanh \left ({\frac{x}{2}} \right ) +{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{i \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{1}{8} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+2\,\ln \left ( \tanh \left ( x/2 \right ) \right ) -4\,\ln \left ( \tanh \left ( x/2 \right ) +i \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.14238, size = 86, normalized size = 2.97 \begin{align*} \frac{-4 i \, e^{\left (-x\right )} - 2 \, e^{\left (-2 \, x\right )} + 4 i \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) - 4 \, \log \left (e^{\left (-x\right )} - i\right ) + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.06189, size = 207, normalized size = 7.14 \begin{align*} \frac{2 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{\left (2 \, x\right )} - 1\right ) - 4 \,{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \log \left (e^{x} + i\right ) + 4 i \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - 4 i \, e^{x}}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.688085, size = 53, normalized size = 1.83 \begin{align*} \frac{4 i e^{3 x} + 2 e^{2 x} - 4 i e^{x}}{e^{4 x} - 2 e^{2 x} + 1} - 4 \log{\left (e^{x} + i \right )} + 2 \log{\left (e^{2 x} - 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.12132, size = 72, normalized size = 2.48 \begin{align*} \frac{4 i \, e^{\left (3 \, x\right )} + 2 \, e^{\left (2 \, x\right )} - 4 i \, e^{x}}{{\left (e^{x} + 1\right )}^{2}{\left (e^{x} - 1\right )}^{2}} + 2 \, \log \left (e^{x} + 1\right ) - 4 \, \log \left (e^{x} + i\right ) + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]