Optimal. Leaf size=18 \[ \frac {\sinh ^3(x)}{3 a}+\frac {\sinh (x)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3175, 2633} \[ \frac {\sinh ^3(x)}{3 a}+\frac {\sinh (x)}{a} \]
Antiderivative was successfully verified.
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Rule 2633
Rule 3175
Rubi steps
\begin {align*} \int \frac {\cosh ^5(x)}{a+a \sinh ^2(x)} \, dx &=\frac {\int \cosh ^3(x) \, dx}{a}\\ &=\frac {i \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{a}\\ &=\frac {\sinh (x)}{a}+\frac {\sinh ^3(x)}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.06 \[ \frac {\frac {3 \sinh (x)}{4}+\frac {1}{12} \sinh (3 x)}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.57, size = 20, normalized size = 1.11 \[ \frac {\sinh \relax (x)^{3} + 3 \, {\left (\cosh \relax (x)^{2} + 3\right )} \sinh \relax (x)}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.02, size = 29, normalized size = 1.61 \[ -\frac {{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 67, normalized size = 3.72 \[ \frac {-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 34, normalized size = 1.89 \[ \frac {{\left (9 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )}}{24 \, a} - \frac {9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 35, normalized size = 1.94 \[ \frac {{\mathrm {e}}^{3\,x}}{24\,a}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a}-\frac {3\,{\mathrm {e}}^{-x}}{8\,a}+\frac {3\,{\mathrm {e}}^x}{8\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.09, size = 124, normalized size = 6.89 \[ - \frac {6 \tanh ^{5}{\left (\frac {x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 3 a} + \frac {4 \tanh ^{3}{\left (\frac {x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 3 a} - \frac {6 \tanh {\left (\frac {x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac {x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac {x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac {x}{2} \right )} - 3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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