Optimal. Leaf size=29 \[ \frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac {2}{3} \sqrt {\cosh ^2(x)} \tanh (x) \]
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Rubi [A] time = 0.03, 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 = {3176, 3203, 3207, 2637} \[ \frac {1}{3} \cosh ^2(x)^{3/2} \tanh (x)+\frac {2}{3} \sqrt {\cosh ^2(x)} \tanh (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3176
Rule 3203
Rule 3207
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 \[ \frac {1}{12} (9 \sinh (x)+\sinh (3 x)) \sqrt {\cosh ^2(x)} \text {sech}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 2.91, size = 17, normalized size = 0.59 \[ \frac {1}{12} \, \sinh \relax (x)^{3} + \frac {1}{4} \, {\left (\cosh \relax (x)^{2} + 3\right )} \sinh \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 25, normalized size = 0.86 \[ -\frac {1}{24} \, {\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {3}{8} \, e^{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 21, normalized size = 0.72 \[ \frac {\sqrt {\frac {\cosh \left (2 x \right )}{2}+\frac {1}{2}}\, \sinh \relax (x ) \left (\sinh ^{2}\relax (x )+3\right )}{3 \cosh \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 23, normalized size = 0.79 \[ \frac {1}{24} \, e^{\left (3 \, x\right )} - \frac {3}{8} \, e^{\left (-x\right )} - \frac {1}{24} \, e^{\left (-3 \, x\right )} + \frac {3}{8} \, e^{x} \]
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 \[ \int {\left ({\mathrm {sinh}\relax (x)}^2+1\right )}^{3/2} \,d x \]
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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