Optimal. Leaf size=33 \[ \frac {1}{3} \left (-\sinh ^2(x)\right )^{3/2} \coth (x)+\frac {2}{3} \sqrt {-\sinh ^2(x)} \coth (x) \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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, 2638} \[ \frac {1}{3} \left (-\sinh ^2(x)\right )^{3/2} \coth (x)+\frac {2}{3} \sqrt {-\sinh ^2(x)} \coth (x) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3176
Rule 3203
Rule 3207
Rubi steps
\begin {align*} \int \left (1-\cosh ^2(x)\right )^{3/2} \, dx &=\int \left (-\sinh ^2(x)\right )^{3/2} \, dx\\ &=\frac {1}{3} \coth (x) \left (-\sinh ^2(x)\right )^{3/2}+\frac {2}{3} \int \sqrt {-\sinh ^2(x)} \, dx\\ &=\frac {1}{3} \coth (x) \left (-\sinh ^2(x)\right )^{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) \left (-\sinh ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.76 \[ -\frac {1}{12} \sqrt {-\sinh ^2(x)} (\cosh (3 x)-9 \cosh (x)) \text {csch}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 1, normalized size = 0.03 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.15, size = 66, normalized size = 2.00 \[ -\frac {1}{24} i \, {\left (9 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )\right )} e^{\left (-3 \, x\right )} + \frac {1}{24} i \, e^{\left (3 \, x\right )} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) - \frac {3}{8} i \, e^{x} \mathrm {sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 21, normalized size = 0.64 \[ \frac {\sinh \relax (x ) \cosh \relax (x ) \left (\sinh ^{2}\relax (x )-2\right )}{3 \sqrt {-\left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.88, size = 23, normalized size = 0.70 \[ \frac {1}{24} i \, e^{\left (3 \, x\right )} - \frac {3}{8} i \, e^{\left (-x\right )} + \frac {1}{24} i \, e^{\left (-3 \, x\right )} - \frac {3}{8} i \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (1-{\mathrm {cosh}\relax (x)}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1 - \cosh ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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