Optimal. Leaf size=60 \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\sinh ^2(c+d x)\right )}{2 b d \sqrt{\cosh ^2(c+d x)}} \]
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Rubi [A] time = 0.0149039, 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} \[ \frac{3 \cosh (c+d x) (b \sinh (c+d x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\sinh ^2(c+d x)\right )}{2 b d \sqrt{\cosh ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{b \sinh (c+d x)}} \, dx &=\frac{3 \cosh (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\sinh ^2(c+d x)\right ) (b \sinh (c+d x))^{2/3}}{2 b d \sqrt{\cosh ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0397989, size = 57, normalized size = 0.95 \[ \frac{3 \sqrt{\cosh ^2(c+d x)} \tanh (c+d x) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-\sinh ^2(c+d x)\right )}{2 d \sqrt [3]{b \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{b\sinh \left ( dx+c \right ) }}}\, 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 \frac{1}{\left (b \sinh \left (d x + c\right )\right )^{\frac{1}{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 (\frac{\left (b \sinh \left (d x + c\right )\right )^{\frac{2}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{b \sinh{\left (c + d x \right )}}}\, dx \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 \frac{1}{\left (b \sinh \left (d x + c\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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