Optimal. Leaf size=29 \[ \frac{1}{3} \sinh ^2(x)^{3/2} \coth (x)-\frac{2}{3} \sqrt{\sinh ^2(x)} \coth (x) \]
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Rubi [A] time = 0.0255441, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3176, 3203, 3207, 2638} \[ \frac{1}{3} \sinh ^2(x)^{3/2} \coth (x)-\frac{2}{3} \sqrt{\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3203
Rule 3207
Rule 2638
Rubi steps
\begin{align*} \int \left (-1+\cosh ^2(x)\right )^{3/2} \, dx &=\int \sinh ^2(x)^{3/2} \, dx\\ &=\frac{1}{3} \coth (x) \sinh ^2(x)^{3/2}-\frac{2}{3} \int \sqrt{\sinh ^2(x)} \, dx\\ &=\frac{1}{3} \coth (x) \sinh ^2(x)^{3/2}-\frac{1}{3} \left (2 \text{csch}(x) \sqrt{\sinh ^2(x)}\right ) \int \sinh (x) \, dx\\ &=-\frac{2}{3} \coth (x) \sqrt{\sinh ^2(x)}+\frac{1}{3} \coth (x) \sinh ^2(x)^{3/2}\\ \end{align*}
Mathematica [A] time = 0.021668, size = 23, normalized size = 0.79 \[ \frac{1}{12} \sqrt{\sinh ^2(x)} (\cosh (3 x)-9 \cosh (x)) \text{csch}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 21, normalized size = 0.7 \begin{align*}{\frac{\cosh \left ( x \right ) \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}-3 \right ) }{3\,\sinh \left ( x \right ) }\sqrt{ \left ( \sinh \left ( x \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73285, size = 31, normalized size = 1.07 \begin{align*} -\frac{1}{24} \, e^{\left (3 \, x\right )} + \frac{3}{8} \, e^{\left (-x\right )} - \frac{1}{24} \, e^{\left (-3 \, x\right )} + \frac{3}{8} \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06496, size = 73, normalized size = 2.52 \begin{align*} \frac{1}{12} \, \cosh \left (x\right )^{3} + \frac{1}{4} \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \frac{3}{4} \, \cosh \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27537, size = 89, normalized size = 3.07 \begin{align*} -\frac{1}{24} \,{\left (9 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )\right )} e^{\left (-3 \, x\right )} + \frac{1}{24} \, e^{\left (3 \, x\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \frac{3}{8} \, e^{x} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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