Optimal. Leaf size=60 \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{7/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};-\sinh ^2(c+d x)\right )}{7 b d \sqrt{\cosh ^2(c+d x)}} \]
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Rubi [A] time = 0.0154059, antiderivative size = 60, normalized size of antiderivative = 1., 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 = {2643} \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{7/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};-\sinh ^2(c+d x)\right )}{7 b d \sqrt{\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int (b \sinh (c+d x))^{4/3} \, dx &=\frac{3 \cosh (c+d x) \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{7/3}}{7 b d \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0525996, size = 57, normalized size = 0.95 \[ \frac{3 \sqrt{\cosh ^2(c+d x)} \tanh (c+d x) (b \sinh (c+d x))^{4/3} \, _2F_1\left (\frac{1}{2},\frac{7}{6};\frac{13}{6};-\sinh ^2(c+d x)\right )}{7 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int \left ( b\sinh \left ( dx+c \right ) \right ) ^{{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sinh \left (d x + c\right )\right )^{\frac{1}{3}} b \sinh \left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sinh \left (d x + c\right )\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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