Optimal. Leaf size=69 \[ \frac{2 \coth ^{1-m}(c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\coth ^2(c+d x)\right )}{b d (2-3 m) \sqrt{b \coth ^m(c+d x)}} \]
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Rubi [A] time = 0.0481878, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3659, 3476, 364} \[ \frac{2 \coth ^{1-m}(c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 m);\frac{3 (2-m)}{4};\coth ^2(c+d x)\right )}{b d (2-3 m) \sqrt{b \coth ^m(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\left (b \coth ^m(c+d x)\right )^{3/2}} \, dx &=\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) \operatorname{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) \, _2F_1\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*}
Mathematica [A] time = 0.0661005, size = 58, normalized size = 0.84 \[ -\frac{2 \coth (c+d x) \, _2F_1\left (1,\frac{1}{4} (2-3 m);-\frac{3}{4} (m-2);\coth ^2(c+d x)\right )}{d (3 m-2) \left (b \coth ^m(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{m} \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )^{m}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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