Optimal. Leaf size=45 \[ \frac {2}{3} i F(i x|-1)-2 i E(i x|-1)-\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x) \]
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Rubi [A] time = 0.06, antiderivative size = 45, 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 = {3180, 3172, 3177, 3182} \[ \frac {2}{3} i F(i x|-1)-2 i E(i x|-1)-\frac {1}{3} \sinh (x) \sqrt {1-\sinh ^2(x)} \cosh (x) \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3177
Rule 3180
Rule 3182
Rubi steps
\begin {align*} \int \left (1-\sinh ^2(x)\right )^{3/2} \, dx &=-\frac {1}{3} \cosh (x) \sinh (x) \sqrt {1-\sinh ^2(x)}+\frac {1}{3} \int \frac {4-6 \sinh ^2(x)}{\sqrt {1-\sinh ^2(x)}} \, dx\\ &=-\frac {1}{3} \cosh (x) \sinh (x) \sqrt {1-\sinh ^2(x)}-\frac {2}{3} \int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx+2 \int \sqrt {1-\sinh ^2(x)} \, dx\\ &=-2 i E(i x|-1)+\frac {2}{3} i F(i x|-1)-\frac {1}{3} \cosh (x) \sinh (x) \sqrt {1-\sinh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 45, normalized size = 1.00 \[ \frac {1}{12} \left (8 i F(i x|-1)-24 i E(i x|-1)-\sinh (2 x) \sqrt {6-2 \cosh (2 x)}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-\sinh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-\sinh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 103, normalized size = 2.29 \[ \frac {\sqrt {-\left (-1+\sinh ^{2}\relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \left (\sinh \relax (x ) \left (\cosh ^{4}\relax (x )\right )+10 \sqrt {-\left (\cosh ^{2}\relax (x )\right )+2}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \relax (x ), i\right )-6 \sqrt {-\left (\cosh ^{2}\relax (x )\right )+2}\, \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \relax (x ), i\right )-2 \left (\cosh ^{2}\relax (x )\right ) \sinh \relax (x )\right )}{3 \sqrt {1-\left (\sinh ^{4}\relax (x )\right )}\, \cosh \relax (x ) \sqrt {1-\left (\sinh ^{2}\relax (x )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-\sinh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (1-{\mathrm {sinh}\relax (x)}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (1 - \sinh ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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