Optimal. Leaf size=289 \[ -\frac {3 \tanh (c+d x) \left (b \coth ^2(c+d x)\right )^{2/3}}{d}-\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\sqrt {3} \left (b \coth ^2(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} \left (b \coth ^2(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)} \]
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Rubi [A] time = 0.18, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3658, 3473, 3476, 329, 210, 634, 618, 204, 628, 206} \[ -\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\sqrt {3} \left (b \coth ^2(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} \left (b \coth ^2(c+d x)\right )^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 \tanh (c+d x) \left (b \coth ^2(c+d x)\right )^{2/3}}{d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3473
Rule 3476
Rule 3658
Rubi steps
\begin {align*} \int \left (b \coth ^2(c+d x)\right )^{2/3} \, dx &=\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \int \coth ^{\frac {4}{3}}(c+d x) \, dx}{\coth ^{\frac {4}{3}}(c+d x)}\\ &=-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)} \, dx}{\coth ^{\frac {4}{3}}(c+d x)}\\ &=-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}-\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{x^{2/3} \left (-1+x^2\right )} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {4}{3}}(c+d x)}\\ &=-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}-\frac {\left (3 \left (b \coth ^2(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}\\ &=-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {4}{3}}(c+d x)}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^2(c+d x)\right )^{2/3}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}-\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (3 \left (b \coth ^2(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (3 \left (b \coth ^2(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}\\ &=\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^2(c+d x)\right )^{2/3}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}-\frac {\left (3 \left (b \coth ^2(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\left (3 \left (b \coth ^2(c+d x)\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {4}{3}}(c+d x)}\\ &=-\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^2(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^2(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\tanh ^{-1}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^2(c+d x)\right )^{2/3}}{d \coth ^{\frac {4}{3}}(c+d x)}-\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}+\frac {\left (b \coth ^2(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {4}{3}}(c+d x)}-\frac {3 \left (b \coth ^2(c+d x)\right )^{2/3} \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 43, normalized size = 0.15 \[ \frac {3 \tanh (c+d x) \left (b \coth ^2(c+d x)\right )^{2/3} \left (\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};\coth ^2(c+d x)\right )-1\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 2037, normalized size = 7.05 \[ \text {result too large to display} \]
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 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \left (b \left (\coth ^{2}\left (d x +c \right )\right )\right )^{\frac {2}{3}}\, dx \]
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 \[ \int \left (b \coth \left (d x + c\right )^{2}\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{2/3} \,d x \]
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 \[ \int \left (b \coth ^{2}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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