Optimal. Leaf size=45 \[ -2 i E(i x|-1)+\frac {2}{3} i F(i x|-1)-\frac {1}{3} \cosh (x) \sinh (x) \sqrt {1-\sinh ^2(x)} \]
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Rubi [A]
time = 0.05, 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 = {3259, 3251,
3256, 3261} \begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 3251
Rule 3256
Rule 3259
Rule 3261
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 = 0.05, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{12} \left (-24 i E(i x|-1)+8 i F(i x|-1)-\sqrt {6-2 \cosh (2 x)} \sinh (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.98, size = 103, normalized size = 2.29
method | result | size |
default | \(\frac {\sqrt {-\left (-1+\sinh ^{2}\left (x \right )\right ) \left (\cosh ^{2}\left (x \right )\right )}\, \left (\left (\cosh ^{4}\left (x \right )\right ) \sinh \left (x \right )+10 \sqrt {-\left (\cosh ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticF \left (\sinh \left (x \right ), i\right )-6 \sqrt {-\left (\cosh ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \EllipticE \left (\sinh \left (x \right ), i\right )-2 \left (\cosh ^{2}\left (x \right )\right ) \sinh \left (x \right )\right )}{3 \sqrt {1-\left (\sinh ^{4}\left (x \right )\right )}\, \cosh \left (x \right ) \sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}\) | \(103\) |
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 = 0.08, size = 12, normalized size = 0.27 \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 (1 - \sinh ^{2}{\left (x \right )}\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 = -0.02 \begin {gather*} \int {\left (1-{\mathrm {sinh}\left (x\right )}^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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