Integrand size = 14, antiderivative size = 60 \[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3+2 m),\frac {1}{6} (9+2 m),\coth ^2(c+d x)\right )}{d (3+2 m)} \]
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Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3740, 3557, 371} \[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (2 m+3),\frac {1}{6} (2 m+9),\coth ^2(c+d x)\right )}{d (2 m+3)} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \int \coth ^{\frac {2 m}{3}}(c+d x) \, dx \\ & = -\frac {\left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^{2 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3+2 m),\frac {1}{6} (9+2 m),\coth ^2(c+d x)\right )}{d (3+2 m)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02 \[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\frac {\coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3+2 m),\frac {1}{6} (9+2 m),\coth ^2(c+d x)\right )}{d \left (1+\frac {2 m}{3}\right )} \]
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\[\int \left (b \coth \left (d x +c \right )^{m}\right )^{\frac {2}{3}}d x\]
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Exception generated. \[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\int \left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}} \,d x } \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{2/3} \,d x \]
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