Optimal. Leaf size=54 \[ \frac{3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{m+3}{6};\frac{m+9}{6};\coth ^2(c+d x)\right )}{d (m+3)} \]
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Rubi [A] time = 0.0437366, antiderivative size = 54, 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} \[ \frac{3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{m+3}{6};\frac{m+9}{6};\coth ^2(c+d x)\right )}{d (m+3)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \sqrt [3]{b \coth ^m(c+d x)} \, dx &=\left (\coth ^{-\frac{m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \int \coth ^{\frac{m}{3}}(c+d x) \, dx\\ &=-\frac{\left (\coth ^{-\frac{m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^{m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{3+m}{6};\frac{9+m}{6};\coth ^2(c+d x)\right )}{d (3+m)}\\ \end{align*}
Mathematica [A] time = 0.0386344, size = 54, normalized size = 1. \[ \frac{3 \coth (c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{m+3}{6};\frac{m+9}{6};\coth ^2(c+d x)\right )}{d (m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.114, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{b \left ({\rm coth} \left (dx+c\right ) \right ) ^{m}}\, 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 \left (b \coth \left (d x + c\right )^{m}\right )^{\frac{1}{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 \sqrt [3]{b \coth ^{m}{\left (c + d x \right )}}\, 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 \left (b \coth \left (d x + c\right )^{m}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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