Optimal. Leaf size=87 \[ \frac {2 i F(i x|-1) \sqrt {1-\sinh ^2(x)}}{3 \sqrt {-1+\sinh ^2(x)}}+\frac {1}{3} \cosh (x) \sinh (x) \sqrt {-1+\sinh ^2(x)}+\frac {2 i E(i x|-1) \sqrt {-1+\sinh ^2(x)}}{\sqrt {1-\sinh ^2(x)}} \]
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Rubi [A]
time = 0.06, 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 = {3259, 3251,
3257, 3256, 3262, 3261} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3251
Rule 3256
Rule 3257
Rule 3259
Rule 3261
Rule 3262
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 = 0.09, size = 78, normalized size = 0.90 \begin {gather*} \frac {-24 i \sqrt {3-\cosh (2 x)} E(i x|-1)+8 i \sqrt {3-\cosh (2 x)} F(i x|-1)+\frac {-6 \sinh (2 x)+\sinh (4 x)}{\sqrt {2}}}{12 \sqrt {-3+\cosh (2 x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 106, normalized size = 1.22
method | result | size |
default | \(\frac {\sqrt {\left (-1+\sinh ^{2}\left (x \right )\right ) \left (\cosh ^{2}\left (x \right )\right )}\, \left (\left (\cosh ^{4}\left (x \right )\right ) \sinh \left (x \right )+2 i \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \sqrt {-\left (\cosh ^{2}\left (x \right )\right )+2}\, \EllipticF \left (i \sinh \left (x \right ), i\right )-6 i \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \sqrt {-\left (\cosh ^{2}\left (x \right )\right )+2}\, \EllipticE \left (i \sinh \left (x \right ), i\right )-2 \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right )\right )}{3 \sqrt {\sinh ^{4}\left (x \right )-1}\, \cosh \left (x \right ) \sqrt {-1+\sinh ^{2}\left (x \right )}}\) | \(106\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.09, size = 10, normalized size = 0.11 \begin {gather*} {\rm integral}\left ({\left (\sinh \left (x\right )^{2} - 1\right )}^{\frac {3}{2}}, x\right ) \end {gather*}
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 \begin {gather*} \int \left (\sinh ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left ({\mathrm {sinh}\left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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