Optimal. Leaf size=18 \[ \frac{2}{1-\cosh (x)}+\log (1-\cosh (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.056183, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {4392, 2667, 43} \[ \frac{2}{1-\cosh (x)}+\log (1-\cosh (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
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 = 0.0506899, size = 41, normalized size = 2.28 \[ -\text{csch}^2\left (\frac{x}{2}\right )-2 \log \left (\sinh \left (\frac{x}{2}\right )\right )+\log (\sinh (x))+3 \log \left (\tanh \left (\frac{x}{2}\right )\right )+2 \log \left (\cosh \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.018, size = 39, normalized size = 2.2 \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.05576, size = 89, normalized size = 4.94 \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.02317, size = 336, normalized size = 18.67 \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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\coth{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.1337, size = 30, normalized size = 1.67 \begin{align*} -x - \frac{4 \, e^{x}}{{\left (e^{x} - 1\right )}^{2}} + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]