Integrand size = 12, antiderivative size = 58 \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \sqrt [3]{b \sinh (c+d x)}}{b d \sqrt {\cosh ^2(c+d x)}} \]
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Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2722} \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \cosh (c+d x) \sqrt [3]{b \sinh (c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right )}{b d \sqrt {\cosh ^2(c+d x)}} \]
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Rule 2722
Rubi steps \begin{align*} \text {integral}& = \frac {3 \cosh (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \sqrt [3]{b \sinh (c+d x)}}{b d \sqrt {\cosh ^2(c+d x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\frac {3 \sqrt {\cosh ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\sinh ^2(c+d x)\right ) \tanh (c+d x)}{d (b \sinh (c+d x))^{2/3}} \]
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\[\int \frac {1}{\left (b \sinh \left (d x +c \right )\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (b \sinh {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \sinh \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b \sinh (c+d x))^{2/3}} \, dx=\int \frac {1}{{\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{2/3}} \,d x \]
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