Optimal. Leaf size=19 \[ \frac{\text{sech}^2(x)}{2 a}-\frac{\text{sech}(x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0659567, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2706, 2606, 30, 8} \[ \frac{\text{sech}^2(x)}{2 a}-\frac{\text{sech}(x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2706
Rule 2606
Rule 30
Rule 8
Rubi steps
\begin{align*} \int \frac{\tanh ^3(x)}{a+a \cosh (x)} \, dx &=\frac{\int \text{sech}(x) \tanh (x) \, dx}{a}-\frac{\int \text{sech}^2(x) \tanh (x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}(\int 1 \, dx,x,\text{sech}(x))}{a}+\frac{\operatorname{Subst}(\int x \, dx,x,\text{sech}(x))}{a}\\ &=-\frac{\text{sech}(x)}{a}+\frac{\text{sech}^2(x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0213752, size = 17, normalized size = 0.89 \[ \frac{2 \sinh ^4\left (\frac{x}{2}\right ) \text{sech}^2(x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 18, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( - \left ( \cosh \left ( x \right ) \right ) ^{-1}+{\frac{1}{2\, \left ( \cosh \left ( x \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.07107, size = 95, normalized size = 5. \begin{align*} -\frac{2 \, e^{\left (-x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} + \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} - \frac{2 \, e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.82161, size = 221, normalized size = 11.63 \begin{align*} -\frac{2 \,{\left (\cosh \left (x\right )^{2} +{\left (2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + \sinh \left (x\right )^{2} - \cosh \left (x\right ) + 1\right )}}{a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) \sinh \left (x\right )^{2} + a \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right ) +{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )} \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*} \frac{\int \frac{\tanh ^{3}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15578, size = 30, normalized size = 1.58 \begin{align*} -\frac{2 \,{\left (e^{\left (-x\right )} + e^{x} - 1\right )}}{a{\left (e^{\left (-x\right )} + e^{x}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]