Optimal. Leaf size=77 \[ -\frac{i}{8 (1-i \sinh (x))}-\frac{3 i}{4 (1+i \sinh (x))}+\frac{i}{8 (1+i \sinh (x))^2}-\frac{11}{16} i \log (-\sinh (x)+i)-\frac{5}{16} i \log (\sinh (x)+i) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0672505, antiderivative size = 77, 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 = {3879, 88} \[ -\frac{i}{8 (1-i \sinh (x))}-\frac{3 i}{4 (1+i \sinh (x))}+\frac{i}{8 (1+i \sinh (x))^2}-\frac{11}{16} i \log (-\sinh (x)+i)-\frac{5}{16} i \log (\sinh (x)+i) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{i+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{x^4}{(i-i x)^2 (i+i x)^3} \, dx,x,i \sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \left (-\frac{i}{8 (-1+x)^2}-\frac{5 i}{16 (-1+x)}-\frac{i}{4 (1+x)^3}+\frac{3 i}{4 (1+x)^2}-\frac{11 i}{16 (1+x)}\right ) \, dx,x,i \sinh (x)\right )\\ &=-\frac{11}{16} i \log (i-\sinh (x))-\frac{5}{16} i \log (i+\sinh (x))-\frac{i}{8 (1-i \sinh (x))}+\frac{i}{8 (1+i \sinh (x))^2}-\frac{3 i}{4 (1+i \sinh (x))}\\ \end{align*}
Mathematica [A] time = 0.115431, size = 61, normalized size = 0.79 \[ \frac{1}{16} \left (-\frac{2 \left (5 \sinh ^2(x)+3 i \sinh (x)+6\right )}{(\sinh (x)-i)^2 (\sinh (x)+i)}-11 i \log (-\sinh (x)+i)-5 i \log (\sinh (x)+i)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 109, normalized size = 1.4 \begin{align*} i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) -{\frac{11\,i}{8}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) +{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-4}}+{{\frac{i}{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-2}}+ \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-3}+ \left ( \tanh \left ({\frac{x}{2}} \right ) -i \right ) ^{-1}+i\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) -{\frac{5\,i}{8}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) +{{\frac{i}{4}} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-2}}-{\frac{1}{4} \left ( \tanh \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.04781, size = 130, normalized size = 1.69 \begin{align*} -i \, x + \frac{5 \, e^{\left (-x\right )} + 6 i \, e^{\left (-2 \, x\right )} + 14 \, e^{\left (-3 \, x\right )} - 6 i \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{8 i \, e^{\left (-x\right )} - 4 \, e^{\left (-2 \, x\right )} + 16 i \, e^{\left (-3 \, x\right )} + 4 \, e^{\left (-4 \, x\right )} + 8 i \, e^{\left (-5 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - 4} - \frac{5}{8} i \, \log \left (e^{\left (-x\right )} - i\right ) - \frac{11}{8} i \, \log \left (i \, e^{\left (-x\right )} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.58237, size = 568, normalized size = 7.38 \begin{align*} \frac{8 i \, x e^{\left (6 \, x\right )} + 2 \,{\left (8 \, x - 5\right )} e^{\left (5 \, x\right )} +{\left (8 i \, x - 12 i\right )} e^{\left (4 \, x\right )} + 4 \,{\left (8 \, x - 7\right )} e^{\left (3 \, x\right )} +{\left (-8 i \, x + 12 i\right )} e^{\left (2 \, x\right )} + 2 \,{\left (8 \, x - 5\right )} e^{x} +{\left (-5 i \, e^{\left (6 \, x\right )} - 10 \, e^{\left (5 \, x\right )} - 5 i \, e^{\left (4 \, x\right )} - 20 \, e^{\left (3 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - 10 \, e^{x} + 5 i\right )} \log \left (e^{x} + i\right ) +{\left (-11 i \, e^{\left (6 \, x\right )} - 22 \, e^{\left (5 \, x\right )} - 11 i \, e^{\left (4 \, x\right )} - 44 \, e^{\left (3 \, x\right )} + 11 i \, e^{\left (2 \, x\right )} - 22 \, e^{x} + 11 i\right )} \log \left (e^{x} - i\right ) - 8 i \, x}{8 \, e^{\left (6 \, x\right )} - 16 i \, e^{\left (5 \, x\right )} + 8 \, e^{\left (4 \, x\right )} - 32 i \, e^{\left (3 \, x\right )} - 8 \, e^{\left (2 \, x\right )} - 16 i \, e^{x} - 8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{3}{\left (x \right )}}{\operatorname{csch}{\left (x \right )} + i}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13793, size = 132, normalized size = 1.71 \begin{align*} \frac{5 \, e^{\left (-x\right )} - 5 \, e^{x} - 6 i}{16 \,{\left (-i \, e^{\left (-x\right )} + i \, e^{x} - 2\right )}} + \frac{33 i \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 84 \, e^{\left (-x\right )} + 84 \, e^{x} - 52 i}{32 \,{\left (e^{\left (-x\right )} - e^{x} + 2 i\right )}^{2}} - \frac{5}{16} i \, \log \left (i \, e^{\left (-x\right )} - i \, e^{x} + 2\right ) - \frac{11}{16} i \, \log \left (i \, e^{\left (-x\right )} - i \, e^{x} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]