Optimal. Leaf size=39 \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+b x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0626266, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3220, 3768, 3770} \[ \frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}+b x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3220
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \text{csch}^3(c+d x) \left (a+b \sinh ^3(c+d x)\right ) \, dx &=-\left (i \int \left (i b+i a \text{csch}^3(c+d x)\right ) \, dx\right )\\ &=b x+a \int \text{csch}^3(c+d x) \, dx\\ &=b x-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}-\frac{1}{2} a \int \text{csch}(c+d x) \, dx\\ &=b x+\frac{a \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac{a \coth (c+d x) \text{csch}(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0129128, size = 63, normalized size = 1.62 \[ -\frac{a \text{csch}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \text{sech}^2\left (\frac{1}{2} (c+d x)\right )}{8 d}-\frac{a \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+b x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.04, size = 37, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( a \left ( -{\frac{{\rm csch} \left (dx+c\right ){\rm coth} \left (dx+c\right )}{2}}+{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) \right ) + \left ( dx+c \right ) b \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.09883, size = 123, normalized size = 3.15 \begin{align*} b x + \frac{1}{2} \, a{\left (\frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac{2 \,{\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d{\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.12639, size = 1413, normalized size = 36.23 \begin{align*} \frac{2 \, b d x \cosh \left (d x + c\right )^{4} + 2 \, b d x \sinh \left (d x + c\right )^{4} - 4 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right )^{3} + 2 \,{\left (4 \, b d x \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right )^{3} + 2 \, b d x + 2 \,{\left (6 \, b d x \cosh \left (d x + c\right )^{2} - 2 \, b d x - 3 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 2 \, a \cosh \left (d x + c\right ) +{\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) -{\left (a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, a \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (a \cosh \left (d x + c\right )^{3} - a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \,{\left (4 \, b d x \cosh \left (d x + c\right )^{3} - 4 \, b d x \cosh \left (d x + c\right ) - 3 \, a \cosh \left (d x + c\right )^{2} - a\right )} \sinh \left (d x + c\right )}{2 \,{\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \,{\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \,{\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\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 = 1.16116, size = 108, normalized size = 2.77 \begin{align*} \frac{{\left (d x + c\right )} b}{d} + \frac{a \log \left (e^{\left (d x + c\right )} + 1\right )}{2 \, d} - \frac{a \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{2 \, d} - \frac{a e^{\left (3 \, d x + 3 \, c\right )} + a e^{\left (d x + c\right )}}{d{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]