Integrand size = 14, antiderivative size = 291 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d} \]
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Time = 0.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3554, 3557, 335, 302, 648, 632, 210, 642, 212} \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {3 \tanh (c+d x) \left (b \coth ^4(c+d x)\right )^{2/3}}{5 d}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)} \]
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Rule 210
Rule 212
Rule 302
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3554
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b \coth ^4(c+d x)\right )^{2/3} \int \coth ^{\frac {8}{3}}(c+d x) \, dx}{\coth ^{\frac {8}{3}}(c+d x)} \\ & = -\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{\coth ^{\frac {8}{3}}(c+d x)} \\ & = -\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{d \coth ^{\frac {8}{3}}(c+d x)} \\ & = -\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac {\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {8}{3}}(c+d x)} \\ & = -\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \coth ^{\frac {8}{3}}(c+d x)} \\ & = \frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}-\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 d \coth ^{\frac {8}{3}}(c+d x)} \\ & = \frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (b \coth ^4(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 {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(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 {8}{3}}(c+d x)}-\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d}+\frac {\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\left (3 \left (b \coth ^4(c+d x)\right )^{2/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 d \coth ^{\frac {8}{3}}(c+d x)} \\ & = \frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{2 d \coth ^{\frac {8}{3}}(c+d x)}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \left (b \coth ^4(c+d x)\right )^{2/3}}{d \coth ^{\frac {8}{3}}(c+d x)}-\frac {\left (b \coth ^4(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 {8}{3}}(c+d x)}+\frac {\left (b \coth ^4(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 {8}{3}}(c+d x)}-\frac {3 \left (b \coth ^4(c+d x)\right )^{2/3} \tanh (c+d x)}{5 d} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\frac {\left (b \coth ^4(c+d x)\right )^{2/3} \left (20 \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )-12 \coth ^{\frac {5}{3}}(c+d x)+5 \left (2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-2 \sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )-\log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )+\log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )\right )\right )}{20 d \coth ^{\frac {8}{3}}(c+d x)} \]
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\[\int \left (b \coth \left (d x +c \right )^{4}\right )^{\frac {2}{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (239) = 478\).
Time = 0.28 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.12 \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=-\frac {10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 10 \, {\left (\sqrt {3} \cosh \left (d x + c\right )^{2} + 2 \, \sqrt {3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sqrt {3} \sinh \left (d x + c\right )^{2} - \sqrt {3}\right )} {\left (b^{2}\right )}^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b - 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}}{3 \, b}\right ) + 5 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) + 5 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 10 \, \left (-b^{2}\right )^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) - 10 \, {\left (b^{2}\right )}^{\frac {1}{3}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right ) + 12 \, {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1\right )} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}}}{20 \, {\left (d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} - d\right )}} \]
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\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int \left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {2}{3}}\, dx \]
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\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}} \,d x } \]
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\[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int { \left (b \coth \left (d x + c\right )^{4}\right )^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int \left (b \coth ^4(c+d x)\right )^{2/3} \, dx=\int {\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{2/3} \,d x \]
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