Optimal. Leaf size=65 \[ \frac{3 b \coth ^{m+1}(c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{1}{6} (4 m+3);\frac{1}{6} (4 m+9);\coth ^2(c+d x)\right )}{d (4 m+3)} \]
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Rubi [A] time = 0.0453596, antiderivative size = 65, 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} \[ \frac{3 b \coth ^{m+1}(c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{1}{6} (4 m+3);\frac{1}{6} (4 m+9);\coth ^2(c+d x)\right )}{d (4 m+3)} \]
Antiderivative was successfully verified.
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Rule 3659
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \left (b \coth ^m(c+d x)\right )^{4/3} \, dx &=\left (b \coth ^{-\frac{m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \int \coth ^{\frac{4 m}{3}}(c+d x) \, dx\\ &=-\frac{\left (b \coth ^{-\frac{m}{3}}(c+d x) \sqrt [3]{b \coth ^m(c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^{4 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac{3 b \coth ^{1+m}(c+d x) \sqrt [3]{b \coth ^m(c+d x)} \, _2F_1\left (1,\frac{1}{6} (3+4 m);\frac{1}{6} (9+4 m);\coth ^2(c+d x)\right )}{d (3+4 m)}\\ \end{align*}
Mathematica [A] time = 0.0703297, size = 60, normalized size = 0.92 \[ \frac{3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{4/3} \, _2F_1\left (1,\frac{1}{6} (4 m+3);\frac{1}{6} (4 m+9);\coth ^2(c+d x)\right )}{d (4 m+3)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int \left ( b \left ({\rm coth} \left (dx+c\right ) \right ) ^{m} \right ) ^{{\frac{4}{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 \left (b \coth \left (d x + c\right )^{m}\right )^{\frac{4}{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(-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 \left (b \coth \left (d x + c\right )^{m}\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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