Optimal. Leaf size=60 \[ \frac{3 \coth (c+d x) \, _2F_1\left (1,\frac{1}{6} (3-2 m);\frac{1}{6} (9-2 m);\coth ^2(c+d x)\right )}{d (3-2 m) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
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Rubi [A] time = 0.0445252, antiderivative size = 60, 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} \[ \frac{3 \coth (c+d x) \, _2F_1\left (1,\frac{1}{6} (3-2 m);\frac{1}{6} (9-2 m);\coth ^2(c+d x)\right )}{d (3-2 m) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
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 )^{2/3}} \, dx &=\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) \operatorname{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) \, _2F_1\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*}
Mathematica [A] time = 0.0398727, size = 60, normalized size = 1. \[ -\frac{3 \coth (c+d x) \, _2F_1\left (1,\frac{1}{6} (3-2 m);\frac{1}{6} (9-2 m);\coth ^2(c+d x)\right )}{d (2 m-3) \left (b \coth ^m(c+d x)\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{m} \right ) ^{-{\frac{2}{3}}}\, 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{2}{3}}}\,{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \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{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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