Optimal. Leaf size=33 \[ \frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac{2}{3} \sqrt{-\cosh ^2(x)} \tanh (x) \]
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Rubi [A] time = 0.0293963, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3176, 3203, 3207, 2637} \[ \frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac{2}{3} \sqrt{-\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
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Rule 3176
Rule 3203
Rule 3207
Rule 2637
Rubi steps
\begin{align*} \int \left (-1-\sinh ^2(x)\right )^{3/2} \, dx &=\int \left (-\cosh ^2(x)\right )^{3/2} \, dx\\ &=\frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac{2}{3} \int \sqrt{-\cosh ^2(x)} \, dx\\ &=\frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)-\frac{1}{3} \left (2 \sqrt{-\cosh ^2(x)} \text{sech}(x)\right ) \int \cosh (x) \, dx\\ &=-\frac{2}{3} \sqrt{-\cosh ^2(x)} \tanh (x)+\frac{1}{3} \left (-\cosh ^2(x)\right )^{3/2} \tanh (x)\\ \end{align*}
Mathematica [A] time = 0.0063351, size = 25, normalized size = 0.76 \[ -\frac{1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt{-\cosh ^2(x)} \text{sech}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 21, normalized size = 0.6 \begin{align*}{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) \left ( \left ( \cosh \left ( x \right ) \right ) ^{2}+2 \right ) }{3}{\frac{1}{\sqrt{- \left ( \cosh \left ( x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.54951, size = 72, normalized size = 2.18 \begin{align*} \frac{3 \, e^{\left (-2 \, x\right )}}{8 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac{3}{2}}} - \frac{3 \, e^{\left (-4 \, x\right )}}{8 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac{3}{2}}} - \frac{e^{\left (-6 \, x\right )}}{24 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac{3}{2}}} + \frac{1}{24 \, \left (-e^{\left (-2 \, x\right )}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.80619, size = 81, normalized size = 2.45 \begin{align*} \frac{1}{24} \,{\left (-i \, e^{\left (6 \, x\right )} - 9 i \, e^{\left (4 \, x\right )} + 9 i \, e^{\left (2 \, x\right )} + i\right )} e^{\left (-3 \, 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 [C] time = 1.19878, size = 34, normalized size = 1.03 \begin{align*} \frac{1}{24} i \,{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - \frac{1}{24} i \, e^{\left (3 \, x\right )} - \frac{3}{8} i \, e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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