Optimal. Leaf size=34 \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0331728, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2611, 3770} \[ -\frac{\tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2611
Rule 3770
Rubi steps
\begin{align*} \int \coth ^2(a+b x) \text{csch}(a+b x) \, dx &=-\frac{\coth (a+b x) \text{csch}(a+b x)}{2 b}+\frac{1}{2} \int \text{csch}(a+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cosh (a+b x))}{2 b}-\frac{\coth (a+b x) \text{csch}(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 0.032511, size = 57, normalized size = 1.68 \[ -\frac{\text{csch}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}-\frac{\text{sech}^2\left (\frac{1}{2} (a+b x)\right )}{8 b}+\frac{\log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.017, size = 45, normalized size = 1.3 \begin{align*}{\frac{1}{b} \left ( -{\frac{\cosh \left ( bx+a \right ) }{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}}+{\frac{{\rm csch} \left (bx+a\right ){\rm coth} \left (bx+a\right )}{2}}-{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.23242, size = 113, normalized size = 3.32 \begin{align*} -\frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac{e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{b{\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.34596, size = 1088, normalized size = 32. \begin{align*} -\frac{2 \, \cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, \sinh \left (b x + a\right )^{3} +{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) -{\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \,{\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \,{\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 2 \, \cosh \left (b x + a\right )}{2 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16513, size = 122, normalized size = 3.59 \begin{align*} -\frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right )}{4 \, b} + \frac{\log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} - \frac{e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}}{{\left ({\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]