Optimal. Leaf size=46 \[ \frac{(a-a \cosh (x))^6}{6 a^7}-\frac{4 (a-a \cosh (x))^5}{5 a^6}+\frac{(a-a \cosh (x))^4}{a^5} \]
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Rubi [A] time = 0.0649232, antiderivative size = 46, 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} \[ \frac{(a-a \cosh (x))^6}{6 a^7}-\frac{4 (a-a \cosh (x))^5}{5 a^6}+\frac{(a-a \cosh (x))^4}{a^5} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \frac{\sinh ^7(x)}{a+a \cosh (x)} \, dx &=-\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^2 \, dx,x,a \cosh (x)\right )}{a^7}\\ &=-\frac{\operatorname{Subst}\left (\int \left (4 a^2 (a-x)^3-4 a (a-x)^4+(a-x)^5\right ) \, dx,x,a \cosh (x)\right )}{a^7}\\ &=\frac{(a-a \cosh (x))^4}{a^5}-\frac{4 (a-a \cosh (x))^5}{5 a^6}+\frac{(a-a \cosh (x))^6}{6 a^7}\\ \end{align*}
Mathematica [A] time = 0.0302479, size = 27, normalized size = 0.59 \[ \frac{4 \sinh ^8\left (\frac{x}{2}\right ) (28 \cosh (x)+5 \cosh (2 x)+27)}{15 a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.034, size = 107, normalized size = 2.3 \begin{align*} 128\,{\frac{1}{a} \left ({\frac{1}{768\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{6}}}-{\frac{7}{1280\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{5}}}+{\frac{7}{1024\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{4}}}-{\frac{7}{2048\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{2}}}-{\frac{7}{2048\,\tanh \left ( x/2 \right ) +2048}}+{\frac{1}{768\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{6}}}+{\frac{7}{1280\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{5}}}+{\frac{7}{1024\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{4}}}-{\frac{7}{2048\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{2}}}+{\frac{7}{2048\,\tanh \left ( x/2 \right ) -2048}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09906, size = 113, normalized size = 2.46 \begin{align*} -\frac{{\left (12 \, e^{\left (-x\right )} + 30 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} - 75 \, e^{\left (-4 \, x\right )} + 600 \, e^{\left (-5 \, x\right )} - 5\right )} e^{\left (6 \, x\right )}}{1920 \, a} - \frac{600 \, e^{\left (-x\right )} - 75 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} + 30 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-6 \, x\right )}}{1920 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81454, size = 313, normalized size = 6.8 \begin{align*} \frac{5 \, \cosh \left (x\right )^{6} + 5 \, \sinh \left (x\right )^{6} - 12 \, \cosh \left (x\right )^{5} + 15 \,{\left (5 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) - 2\right )} \sinh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{4} + 100 \, \cosh \left (x\right )^{3} + 15 \,{\left (5 \, \cosh \left (x\right )^{4} - 8 \, \cosh \left (x\right )^{3} - 12 \, \cosh \left (x\right )^{2} + 20 \, \cosh \left (x\right ) + 5\right )} \sinh \left (x\right )^{2} + 75 \, \cosh \left (x\right )^{2} - 600 \, \cosh \left (x\right )}{960 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 12.7959, size = 218, normalized size = 4.74 \begin{align*} \frac{16 \tanh ^{12}{\left (\frac{x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac{x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 15 a} - \frac{96 \tanh ^{10}{\left (\frac{x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac{x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 15 a} + \frac{240 \tanh ^{8}{\left (\frac{x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac{x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac{x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 15 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20752, size = 101, normalized size = 2.2 \begin{align*} -\frac{{\left (600 \, e^{\left (5 \, x\right )} - 75 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} + 30 \, e^{\left (2 \, x\right )} + 12 \, e^{x} - 5\right )} e^{\left (-6 \, x\right )} - 5 \, e^{\left (6 \, x\right )} + 12 \, e^{\left (5 \, x\right )} + 30 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 600 \, e^{x}}{1920 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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