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 = 0.06, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 43
Rule 2667
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*}
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Mathematica [A] time = 0.03, 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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fricas [B] time = 1.38, size = 94, normalized size = 2.04 \[ \frac {5 \, \cosh \relax (x)^{6} + 5 \, \sinh \relax (x)^{6} - 12 \, \cosh \relax (x)^{5} + 15 \, {\left (5 \, \cosh \relax (x)^{2} - 4 \, \cosh \relax (x) - 2\right )} \sinh \relax (x)^{4} - 30 \, \cosh \relax (x)^{4} + 100 \, \cosh \relax (x)^{3} + 15 \, {\left (5 \, \cosh \relax (x)^{4} - 8 \, \cosh \relax (x)^{3} - 12 \, \cosh \relax (x)^{2} + 20 \, \cosh \relax (x) + 5\right )} \sinh \relax (x)^{2} + 75 \, \cosh \relax (x)^{2} - 600 \, \cosh \relax (x)}{960 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 75, normalized size = 1.63 \[ -\frac {{\left (600 \, e^{\left (5 \, x\right )} - 75 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} + 30 \, e^{\left (2 \, x\right )} + 12 \, e^{x} - 5\right )} e^{\left (-6 \, x\right )} - 5 \, e^{\left (6 \, x\right )} + 12 \, e^{\left (5 \, x\right )} + 30 \, e^{\left (4 \, x\right )} - 100 \, e^{\left (3 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 600 \, e^{x}}{1920 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 107, normalized size = 2.33 \[ \frac {\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}+\frac {7}{10 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}+\frac {7}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {7}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {7}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {1}{6 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {7}{10 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {7}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {7}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {7}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 84, normalized size = 1.83 \[ -\frac {{\left (12 \, e^{\left (-x\right )} + 30 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} - 75 \, e^{\left (-4 \, x\right )} + 600 \, e^{\left (-5 \, x\right )} - 5\right )} e^{\left (6 \, x\right )}}{1920 \, a} - \frac {600 \, e^{\left (-x\right )} - 75 \, e^{\left (-2 \, x\right )} - 100 \, e^{\left (-3 \, x\right )} + 30 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-5 \, x\right )} - 5 \, e^{\left (-6 \, x\right )}}{1920 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 107, normalized size = 2.33 \[ \frac {5\,{\mathrm {e}}^{-2\,x}}{128\,a}-\frac {5\,{\mathrm {e}}^{-x}}{16\,a}+\frac {5\,{\mathrm {e}}^{2\,x}}{128\,a}+\frac {5\,{\mathrm {e}}^{-3\,x}}{96\,a}+\frac {5\,{\mathrm {e}}^{3\,x}}{96\,a}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a}-\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}-\frac {{\mathrm {e}}^{5\,x}}{160\,a}+\frac {{\mathrm {e}}^{-6\,x}}{384\,a}+\frac {{\mathrm {e}}^{6\,x}}{384\,a}-\frac {5\,{\mathrm {e}}^x}{16\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.62, size = 284, normalized size = 6.17 \[ \frac {320 \tanh ^{6}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} - \frac {240 \tanh ^{4}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} + \frac {96 \tanh ^{2}{\left (\frac {x}{2} \right )}}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} - \frac {16}{15 a \tanh ^{12}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{10}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{8}{\left (\frac {x}{2} \right )} - 300 a \tanh ^{6}{\left (\frac {x}{2} \right )} + 225 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 90 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 15 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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