Optimal. Leaf size=33 \[ \frac{(a-a \cosh (x))^4}{4 a^5}-\frac{2 (a-a \cosh (x))^3}{3 a^4} \]
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Rubi [A] time = 0.0556596, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2667, 43} \[ \frac{(a-a \cosh (x))^4}{4 a^5}-\frac{2 (a-a \cosh (x))^3}{3 a^4} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh ^5(x)}{a+a \cosh (x)} \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^2 (a+x) \, dx,x,a \cosh (x)\right )}{a^5}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a-x)^2-(a-x)^3\right ) \, dx,x,a \cosh (x)\right )}{a^5}\\ &=-\frac{2 (a-a \cosh (x))^3}{3 a^4}+\frac{(a-a \cosh (x))^4}{4 a^5}\\ \end{align*}
Mathematica [A] time = 0.0203131, size = 21, normalized size = 0.64 \[ \frac{2 \sinh ^6\left (\frac{x}{2}\right ) (3 \cosh (x)+5)}{3 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 87, normalized size = 2.6 \begin{align*} 32\,{\frac{1}{a} \left ({\frac{1}{128\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{4}}}-{\frac{5}{192\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{3}}}+{\frac{5}{256\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{2}}}+{\frac{5}{256\,\tanh \left ( x/2 \right ) +256}}+{\frac{1}{128\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{4}}}+{\frac{5}{192\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{3}}}+{\frac{5}{256\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{2}}}-{\frac{5}{256\,\tanh \left ( x/2 \right ) -256}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14559, size = 81, normalized size = 2.45 \begin{align*} -\frac{{\left (8 \, e^{\left (-x\right )} + 12 \, e^{\left (-2 \, x\right )} - 72 \, e^{\left (-3 \, x\right )} - 3\right )} e^{\left (4 \, x\right )}}{192 \, a} + \frac{72 \, e^{\left (-x\right )} - 12 \, e^{\left (-2 \, x\right )} - 8 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-4 \, x\right )}}{192 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79853, size = 165, normalized size = 5. \begin{align*} \frac{3 \, \cosh \left (x\right )^{4} + 3 \, \sinh \left (x\right )^{4} - 8 \, \cosh \left (x\right )^{3} + 6 \,{\left (3 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{2} - 12 \, \cosh \left (x\right )^{2} + 72 \, \cosh \left (x\right )}{96 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.05777, size = 104, normalized size = 3.15 \begin{align*} - \frac{4 \tanh ^{8}{\left (\frac{x}{2} \right )}}{3 a \tanh ^{8}{\left (\frac{x}{2} \right )} - 12 a \tanh ^{6}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 12 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 3 a} + \frac{16 \tanh ^{6}{\left (\frac{x}{2} \right )}}{3 a \tanh ^{8}{\left (\frac{x}{2} \right )} - 12 a \tanh ^{6}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 12 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 3 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20462, size = 69, normalized size = 2.09 \begin{align*} \frac{{\left (72 \, e^{\left (3 \, x\right )} - 12 \, e^{\left (2 \, x\right )} - 8 \, e^{x} + 3\right )} e^{\left (-4 \, x\right )} + 3 \, e^{\left (4 \, x\right )} - 8 \, e^{\left (3 \, x\right )} - 12 \, e^{\left (2 \, x\right )} + 72 \, e^{x}}{192 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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