Optimal. Leaf size=60 \[ \frac {3 \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{5/3}}{5 b d \sqrt {\cosh ^2(c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722}
\begin {gather*} \frac {3 \cosh (c+d x) (b \sinh (c+d x))^{5/3} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\sinh ^2(c+d x)\right )}{5 b d \sqrt {\cosh ^2(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rubi steps
\begin {align*} \int (b \sinh (c+d x))^{2/3} \, dx &=\frac {3 \cosh (c+d x) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{5/3}}{5 b d \sqrt {\cosh ^2(c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.95 \begin {gather*} \frac {3 \sqrt {\cosh ^2(c+d x)} \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{2/3} \tanh (c+d x)}{5 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \left (b \sinh \left (d x +c \right )\right )^{\frac {2}{3}}\, dx\]
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 = 0.00, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sinh {\left (c + d x \right )}\right )^{\frac {2}{3}}\, 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 (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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