Optimal. Leaf size=29 \[ \frac {2}{3} \sqrt {\cosh ^2(x)} \tanh (x)+\frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x) \]
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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, 2717} \begin {gather*} \frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac {2}{3} \sqrt {\cosh ^2(x)} \tanh (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3255
Rule 3282
Rule 3286
Rubi steps
\begin {align*} \int \left (1+\sinh ^2(x)\right )^{3/2} \, dx &=\int \cosh ^2(x)^{3/2} \, dx\\ &=\frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac {2}{3} \int \sqrt {\cosh ^2(x)} \, dx\\ &=\frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac {1}{3} \left (2 \sqrt {\cosh ^2(x)} \text {sech}(x)\right ) \int \cosh (x) \, dx\\ &=\frac {2}{3} \sqrt {\cosh ^2(x)} \tanh (x)+\frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{12} \sqrt {\cosh ^2(x)} \text {sech}(x) (9 \sinh (x)+\sinh (3 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 21, normalized size = 0.72
method | result | size |
default | \(\frac {\sqrt {\frac {1}{2}+\frac {\cosh \left (2 x \right )}{2}}\, \sinh \left (x \right ) \left (\sinh ^{2}\left (x \right )+3\right )}{3 \cosh \left (x \right )}\) | \(21\) |
risch | \(\frac {{\mathrm e}^{4 x} \sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{24+24 \,{\mathrm e}^{2 x}}+\frac {3 \sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}\, {\mathrm e}^{2 x}}{8 \left (1+{\mathrm e}^{2 x}\right )}-\frac {3 \sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{8 \left (1+{\mathrm e}^{2 x}\right )}-\frac {{\mathrm e}^{-2 x} \sqrt {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-2 x}}}{24 \left (1+{\mathrm e}^{2 x}\right )}\) | \(114\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, 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 = 0.43, size = 17, normalized size = 0.59 \begin {gather*} \frac {1}{12} \, \sinh \left (x\right )^{3} + \frac {1}{4} \, {\left (\cosh \left (x\right )^{2} + 3\right )} \sinh \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 (\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 [A]
time = 0.42, size = 25, normalized size = 0.86 \begin {gather*} -\frac {1}{24} \, {\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, 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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \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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