Integrand size = 14, antiderivative size = 63 \[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\frac {2 b \coth ^{1+m}(c+d x) \sqrt {b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2+3 m),\frac {3 (2+m)}{4},\coth ^2(c+d x)\right )}{d (2+3 m)} \]
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Time = 0.04 (sec) , antiderivative size = 63, 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} \[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\frac {2 b \coth ^{m+1}(c+d x) \sqrt {b \coth ^m(c+d x)} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (3 m+2),\frac {3 (m+2)}{4},\coth ^2(c+d x)\right )}{d (3 m+2)} \]
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Rule 371
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left (b \coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \int \coth ^{\frac {3 m}{2}}(c+d x) \, dx \\ & = -\frac {\left (b \coth ^{-\frac {m}{2}}(c+d x) \sqrt {b \coth ^m(c+d x)}\right ) \text {Subst}\left (\int \frac {x^{3 m/2}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {2 b \coth ^{1+m}(c+d x) \sqrt {b \coth ^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 )}{d (2+3 m)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\frac {\coth (c+d x) \left (b \coth ^m(c+d x)\right )^{3/2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{4} (2+3 m),\frac {3 (2+m)}{4},\coth ^2(c+d x)\right )}{d \left (1+\frac {3 m}{2}\right )} \]
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\[\int \left (b \coth \left (d x +c \right )^{m}\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\int \left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {3}{2}} \,d x } \]
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\[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\int { \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (b \coth ^m(c+d x)\right )^{3/2} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{3/2} \,d x \]
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