Optimal. Leaf size=29 \[ -\frac {2}{3} \coth (x) \sqrt {\sinh ^2(x)}+\frac {1}{3} \coth (x) \sinh ^2(x)^{3/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3255, 3282,
3286, 2718} \begin {gather*} \frac {1}{3} \sinh ^2(x)^{3/2} \coth (x)-\frac {2}{3} \sqrt {\sinh ^2(x)} \coth (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 3255
Rule 3282
Rule 3286
Rubi steps
\begin {align*} \int \left (-1+\cosh ^2(x)\right )^{3/2} \, dx &=\int \sinh ^2(x)^{3/2} \, dx\\ &=\frac {1}{3} \coth (x) \sinh ^2(x)^{3/2}-\frac {2}{3} \int \sqrt {\sinh ^2(x)} \, dx\\ &=\frac {1}{3} \coth (x) \sinh ^2(x)^{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) \sinh ^2(x)^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{12} (-9 \cosh (x)+\cosh (3 x)) \text {csch}(x) \sqrt {\sinh ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.94, size = 21, normalized size = 0.72
method | result | size |
default | \(\frac {\sqrt {-\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \cosh \left (x \right ) \left (\cosh ^{2}\left (x \right )-3\right )}{3 \sinh \left (x \right )}\) | \(21\) |
risch | \(\frac {{\mathrm e}^{4 x} \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{24 \,{\mathrm e}^{2 x}-24}-\frac {3 \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{8 \left ({\mathrm e}^{2 x}-1\right )}-\frac {3 \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{8 \left ({\mathrm e}^{2 x}-1\right )}+\frac {{\mathrm e}^{-2 x} \sqrt {\left ({\mathrm e}^{2 x}-1\right )^{2} {\mathrm e}^{-2 x}}}{24 \,{\mathrm e}^{2 x}-24}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 23, normalized size = 0.79 \begin {gather*} -\frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {3}{8} \, e^{\left (-x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {3}{8} \, e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 19, normalized size = 0.66 \begin {gather*} \frac {1}{12} \, \cosh \left (x\right )^{3} + \frac {1}{4} \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \frac {3}{4} \, \cosh \left (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 (\cosh ^{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 [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (21) = 42\).
time = 0.42, size = 66, normalized size = 2.28 \begin {gather*} -\frac {1}{24} \, {\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} \, e^{\left (3 \, x\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) - \frac {3}{8} \, e^{x} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \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 = -0.03 \begin {gather*} \int {\left ({\mathrm {cosh}\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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