Optimal. Leaf size=31 \[ \frac {x}{2 a}+\frac {\sinh ^3(x)}{3 a}-\frac {\sinh (x) \cosh (x)}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ \frac {x}{2 a}+\frac {\sinh ^3(x)}{3 a}-\frac {\sinh (x) \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 2682
Rubi steps
\begin {align*} \int \frac {\sinh ^4(x)}{a+a \cosh (x)} \, dx &=\frac {\sinh ^3(x)}{3 a}-\frac {\int \sinh ^2(x) \, dx}{a}\\ &=-\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\sinh ^3(x)}{3 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}-\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\sinh ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 25, normalized size = 0.81 \[ \frac {6 x-3 \sinh (x)-3 \sinh (2 x)+\sinh (3 x)}{12 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.02, size = 27, normalized size = 0.87 \[ \frac {\sinh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) - 1\right )} \sinh \relax (x) + 6 \, x}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 40, normalized size = 1.29 \[ \frac {{\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + 12 \, x + e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - 3 \, e^{x}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 103, normalized size = 3.32 \[ -\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 a}-\frac {1}{3 a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{a \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 a \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 54, normalized size = 1.74 \[ -\frac {{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )}}{24 \, a} + \frac {x}{2 \, a} + \frac {3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.94, size = 59, normalized size = 1.90 \[ \frac {{\mathrm {e}}^{-x}}{8\,a}+\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}+\frac {{\mathrm {e}}^{3\,x}}{24\,a}+\frac {x}{2\,a}-\frac {{\mathrm {e}}^x}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.30, size = 294, normalized size = 9.48 \[ \frac {3 x \tanh ^{6}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {9 x \tanh ^{4}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} + \frac {9 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {3 x}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {6 \tanh ^{5}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} - \frac {16 \tanh ^{3}{\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} + \frac {6 \tanh {\left (\frac {x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 6 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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