Optimal. Leaf size=87 \[ \frac {1}{3} \sinh (x) \sqrt {\sinh ^2(x)-1} \cosh (x)+\frac {2 i \sqrt {1-\sinh ^2(x)} F(i x|-1)}{3 \sqrt {\sinh ^2(x)-1}}+\frac {2 i \sqrt {\sinh ^2(x)-1} E(i x|-1)}{\sqrt {1-\sinh ^2(x)}} \]
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Rubi [A] time = 0.09, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ \frac {1}{3} \sinh (x) \sqrt {\sinh ^2(x)-1} \cosh (x)+\frac {2 i \sqrt {1-\sinh ^2(x)} F(i x|-1)}{3 \sqrt {\sinh ^2(x)-1}}+\frac {2 i \sqrt {\sinh ^2(x)-1} E(i x|-1)}{\sqrt {1-\sinh ^2(x)}} \]
Antiderivative was successfully verified.
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Rule 3172
Rule 3177
Rule 3178
Rule 3180
Rule 3182
Rule 3183
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\\ &=\frac {1}{3} \cosh (x) \sinh (x) \sqrt {-1+\sinh ^2(x)}-\frac {\left (2 \sqrt {1-\sinh ^2(x)}\right ) \int \frac {1}{\sqrt {1-\sinh ^2(x)}} \, dx}{3 \sqrt {-1+\sinh ^2(x)}}-\frac {\left (2 \sqrt {-1+\sinh ^2(x)}\right ) \int \sqrt {1-\sinh ^2(x)} \, dx}{\sqrt {1-\sinh ^2(x)}}\\ &=\frac {2 i F(i x|-1) \sqrt {1-\sinh ^2(x)}}{3 \sqrt {-1+\sinh ^2(x)}}+\frac {1}{3} \cosh (x) \sinh (x) \sqrt {-1+\sinh ^2(x)}+\frac {2 i E(i x|-1) \sqrt {-1+\sinh ^2(x)}}{\sqrt {1-\sinh ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 78, normalized size = 0.90 \[ \frac {\frac {\sinh (4 x)-6 \sinh (2 x)}{\sqrt {2}}+8 i \sqrt {3-\cosh (2 x)} F(i x|-1)-24 i \sqrt {3-\cosh (2 x)} E(i x|-1)}{12 \sqrt {\cosh (2 x)-3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, 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 = 106, normalized size = 1.22 \[ \frac {\sqrt {\left (-1+\sinh ^{2}\relax (x )\right ) \left (\cosh ^{2}\relax (x )\right )}\, \left (\sinh \relax (x ) \left (\cosh ^{4}\relax (x )\right )+2 i \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\left (\cosh ^{2}\relax (x )\right )+2}\, \EllipticF \left (i \sinh \relax (x ), i\right )-6 i \sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\left (\cosh ^{2}\relax (x )\right )+2}\, \EllipticE \left (i \sinh \relax (x ), i\right )-2 \left (\cosh ^{2}\relax (x )\right ) \sinh \relax (x )\right )}{3 \sqrt {\sinh ^{4}\relax (x )-1}\, \cosh \relax (x ) \sqrt {-1+\sinh ^{2}\relax (x )}} \]
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 = -0.01 \[ \int {\left ({\mathrm {sinh}\relax (x)}^2-1\right )}^{3/2} \,d x \]
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 \[ \int \left (\sinh ^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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