Optimal. Leaf size=60 \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, 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} \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3659
Rubi steps
\begin {align*} \int \left (b \coth ^m(c+d x)\right )^{2/3} \, dx &=\left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \int \coth ^{\frac {2 m}{3}}(c+d x) \, dx\\ &=-\frac {\left (\coth ^{-\frac {2 m}{3}}(c+d x) \left (b \coth ^m(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^{2 m/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _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)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 1.00 \[ \frac {3 \coth (c+d x) \left (b \coth ^m(c+d x)\right )^{2/3} \, _2F_1\left (1,\frac {1}{6} (2 m+3);\frac {1}{6} (2 m+9);\coth ^2(c+d x)\right )}{d (2 m+3)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{m}\left (d x +c \right )\right )\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth \left (d x + c\right )^{m}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^m\right )}^{2/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \coth ^{m}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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