Integrand size = 14, antiderivative size = 69 \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\frac {2 \coth ^{1-m}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\coth ^2(c+d x)\right )}{b d (2-3 m) \sqrt {b \coth ^m(c+d x)}} \]
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Time = 0.04 (sec) , antiderivative size = 69, 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} \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\frac {2 \coth ^{1-m}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\coth ^2(c+d x)\right )}{b d (2-3 m) \sqrt {b \coth ^m(c+d x)}} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \frac {\coth ^{\frac {m}{2}}(c+d x) \int \coth ^{-\frac {3 m}{2}}(c+d x) \, dx}{b \sqrt {b \coth ^m(c+d x)}} \\ & = -\frac {\coth ^{\frac {m}{2}}(c+d x) \text {Subst}\left (\int \frac {x^{-3 m/2}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{b d \sqrt {b \coth ^m(c+d x)}} \\ & = \frac {2 \coth ^{1-m}(c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2-3 m),\frac {3 (2-m)}{4},\coth ^2(c+d x)\right )}{b d (2-3 m) \sqrt {b \coth ^m(c+d x)}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\frac {\coth (c+d x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2-3 m),-\frac {3}{4} (-2+m),\coth ^2(c+d x)\right )}{d \left (1-\frac {3 m}{2}\right ) \left (b \coth ^m(c+d x)\right )^{3/2}} \]
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\[\int \frac {1}{\left (b \coth \left (d x +c \right )^{m}\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{3/2}} \,d x \]
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