Optimal. Leaf size=18 \[ \frac{\sinh ^3(x)}{3 a}+\frac{\sinh (x)}{a} \]
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Rubi [A] time = 0.0516122, antiderivative size = 18, normalized size of antiderivative = 1., 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 3175
Rule 2633
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*}
Mathematica [A] time = 0.0034921, 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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Maple [B] time = 0.025, size = 67, normalized size = 3.7 \begin{align*} 2\,{\frac{1}{a} \left ( -1/6\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-3}+1/4\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-2}-1/2\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}-1/6\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-3}-1/4\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-2}-1/2\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.03111, size = 46, normalized size = 2.56 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47582, size = 65, normalized size = 3.61 \begin{align*} \frac{\sinh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )}{12 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 10.0482, size = 124, normalized size = 6.89 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14123, size = 39, normalized size = 2.17 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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