Integrand size = 14, antiderivative size = 60 \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\frac {3 \coth (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3-2 m),\frac {1}{6} (9-2 m),\coth ^2(c+d x)\right )}{d (3-2 m) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
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Time = 0.03 (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} \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\frac {3 \coth (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3-2 m),\frac {1}{6} (9-2 m),\coth ^2(c+d x)\right )}{d (3-2 m) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \frac {\coth ^{\frac {2 m}{3}}(c+d x) \int \coth ^{-\frac {2 m}{3}}(c+d x) \, dx}{\left (b \coth ^m(c+d x)\right )^{2/3}} \\ & = -\frac {\coth ^{\frac {2 m}{3}}(c+d x) \text {Subst}\left (\int \frac {x^{-2 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \left (b \coth ^m(c+d x)\right )^{2/3}} \\ & = \frac {3 \coth (c+d x) \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) \left (b \coth ^m(c+d x)\right )^{2/3}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\frac {\coth (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{6} (3-2 m),\frac {1}{6} (9-2 m),\coth ^2(c+d x)\right )}{d \left (1-\frac {2 m}{3}\right ) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
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\[\int \frac {1}{\left (b \coth \left (d x +c \right )^{m}\right )^{\frac {2}{3}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{\left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{2/3}} \,d x \]
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