Optimal. Leaf size=16 \[ -\frac{2}{\cosh (x)+1}-\log (\cosh (x)+1) \]
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Rubi [A] time = 0.0581354, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4392, 2667, 43} \[ -\frac{2}{\cosh (x)+1}-\log (\cosh (x)+1) \]
Antiderivative was successfully verified.
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Rule 4392
Rule 2667
Rule 43
Rubi steps
\begin{align*} \int (-\coth (x)+\text{csch}(x))^3 \, dx &=i \int (i-i \cosh (x))^3 \text{csch}^3(x) \, dx\\ &=-\operatorname{Subst}\left (\int \frac{i+x}{(i-x)^2} \, dx,x,-i \cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{2 i}{(-i+x)^2}+\frac{1}{-i+x}\right ) \, dx,x,-i \cosh (x)\right )\\ &=-\frac{2 i}{i+i \cosh (x)}-\log (1+\cosh (x))\\ \end{align*}
Mathematica [B] time = 0.0519006, size = 43, normalized size = 2.69 \[ -\text{sech}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right )-\log (\sinh (x))+3 \log \left (\tanh \left (\frac{x}{2}\right )\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 41, normalized size = 2.6 \begin{align*} -\ln \left ( \sinh \left ( x \right ) \right ) +{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{2}}{2}}-3\,{\frac{\cosh \left ( x \right ) }{ \left ( \sinh \left ( x \right ) \right ) ^{2}}}+{\rm csch} \left (x\right ){\rm coth} \left (x\right )-2\,{\it Artanh} \left ({{\rm e}^{x}} \right ) +{\frac{3\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{2\, \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.19223, size = 92, normalized size = 5.75 \begin{align*} \frac{3}{2} \, \coth \left (x\right )^{2} - x + \frac{4 \,{\left (e^{\left (-x\right )} + e^{\left (-3 \, x\right )}\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - 2 \, \log \left (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 = 2.07745, size = 335, normalized size = 20.94 \begin{align*} \frac{x \cosh \left (x\right )^{2} + x \sinh \left (x\right )^{2} + 2 \,{\left (x - 2\right )} \cosh \left (x\right ) - 2 \,{\left (\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \,{\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1} \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 3 \coth{\left (x \right )} \operatorname{csch}^{2}{\left (x \right )}\, dx - \int - 3 \coth ^{2}{\left (x \right )} \operatorname{csch}{\left (x \right )}\, dx - \int \coth ^{3}{\left (x \right )}\, dx - \int - \operatorname{csch}^{3}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11969, size = 26, normalized size = 1.62 \begin{align*} x - \frac{4 \, e^{x}}{{\left (e^{x} + 1\right )}^{2}} - 2 \, \log \left (e^{x} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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