Optimal. Leaf size=30 \[ -\frac{\tanh ^4(x)}{4 a}+\frac{\text{sech}^3(x)}{3 a}-\frac{\text{sech}(x)}{a} \]
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Rubi [A] time = 0.0858699, antiderivative size = 30, 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, 2607, 30, 2606} \[ -\frac{\tanh ^4(x)}{4 a}+\frac{\text{sech}^3(x)}{3 a}-\frac{\text{sech}(x)}{a} \]
Antiderivative was successfully verified.
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Rule 2706
Rule 2607
Rule 30
Rule 2606
Rubi steps
\begin{align*} \int \frac{\tanh ^5(x)}{a+a \cosh (x)} \, dx &=\frac{\int \text{sech}(x) \tanh ^3(x) \, dx}{a}-\frac{\int \text{sech}^2(x) \tanh ^3(x) \, dx}{a}\\ &=-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,i \tanh (x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\text{sech}(x)\right )}{a}\\ &=-\frac{\text{sech}(x)}{a}+\frac{\text{sech}^3(x)}{3 a}-\frac{\tanh ^4(x)}{4 a}\\ \end{align*}
Mathematica [A] time = 0.0273653, size = 25, normalized size = 0.83 \[ \frac{2 \sinh ^6\left (\frac{x}{2}\right ) (5 \cosh (x)+3) \text{sech}^4(x)}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 30, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( - \left ( \cosh \left ( x \right ) \right ) ^{-1}+{\frac{1}{3\, \left ( \cosh \left ( x \right ) \right ) ^{3}}}-{\frac{1}{4\, \left ( \cosh \left ( x \right ) \right ) ^{4}}}+{\frac{1}{2\, \left ( \cosh \left ( x \right ) \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08009, size = 301, normalized size = 10.03 \begin{align*} -\frac{2 \, e^{\left (-x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} + \frac{2 \, e^{\left (-2 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac{10 \, e^{\left (-3 \, x\right )}}{3 \,{\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} - \frac{10 \, e^{\left (-5 \, x\right )}}{3 \,{\left (4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a\right )}} + \frac{2 \, e^{\left (-6 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} - \frac{2 \, e^{\left (-7 \, x\right )}}{4 \, a e^{\left (-2 \, x\right )} + 6 \, a e^{\left (-4 \, x\right )} + 4 \, a e^{\left (-6 \, x\right )} + a e^{\left (-8 \, x\right )} + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8375, size = 562, normalized size = 18.73 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (x\right )^{4} + 3 \,{\left (4 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + 3 \, \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{3} +{\left (18 \, \cosh \left (x\right )^{2} - 9 \, \cosh \left (x\right ) + 8\right )} \sinh \left (x\right )^{2} + 8 \, \cosh \left (x\right )^{2} +{\left (12 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) + 3\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right ) + 5\right )}}{3 \,{\left (a \cosh \left (x\right )^{5} + 5 \, a \cosh \left (x\right ) \sinh \left (x\right )^{4} + a \sinh \left (x\right )^{5} + 5 \, a \cosh \left (x\right )^{3} +{\left (10 \, a \cosh \left (x\right )^{2} + 3 \, a\right )} \sinh \left (x\right )^{3} + 5 \,{\left (2 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 10 \, a \cosh \left (x\right ) +{\left (5 \, a \cosh \left (x\right )^{4} + 9 \, a \cosh \left (x\right )^{2} + 2 \, a\right )} \sinh \left (x\right )\right )}} \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*} \frac{\int \frac{\tanh ^{5}{\left (x \right )}}{\cosh{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19375, size = 65, normalized size = 2.17 \begin{align*} -\frac{2 \,{\left (3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 3 \,{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4 \, e^{\left (-x\right )} - 4 \, e^{x} + 6\right )}}{3 \, a{\left (e^{\left (-x\right )} + e^{x}\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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