Optimal. Leaf size=20 \[ \frac{x}{2 a}-\frac{\sinh (x) \cosh (x)}{2 a} \]
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Rubi [A] time = 0.0460336, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3175, 2635, 8} \[ \frac{x}{2 a}-\frac{\sinh (x) \cosh (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3175
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sinh ^4(x)}{a-a \cosh ^2(x)} \, dx &=-\frac{\int \sinh ^2(x) \, dx}{a}\\ &=-\frac{\cosh (x) \sinh (x)}{2 a}+\frac{\int 1 \, dx}{2 a}\\ &=\frac{x}{2 a}-\frac{\cosh (x) \sinh (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0041697, size = 19, normalized size = 0.95 \[ -\frac{\frac{1}{4} \sinh (2 x)-\frac{x}{2}}{a} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 78, normalized size = 3.9 \begin{align*}{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07239, size = 34, normalized size = 1.7 \begin{align*} \frac{x}{2 \, a} - \frac{e^{\left (2 \, x\right )}}{8 \, a} + \frac{e^{\left (-2 \, x\right )}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73236, size = 41, normalized size = 2.05 \begin{align*} -\frac{\cosh \left (x\right ) \sinh \left (x\right ) - x}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.66779, size = 153, normalized size = 7.65 \begin{align*} \frac{x \tanh ^{4}{\left (\frac{x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} - \frac{2 x \tanh ^{2}{\left (\frac{x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} + \frac{x}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} - \frac{2 \tanh ^{3}{\left (\frac{x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} - \frac{2 \tanh{\left (\frac{x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28488, size = 35, normalized size = 1.75 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 4 \, x + e^{\left (2 \, x\right )}}{8 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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