Integrand size = 14, antiderivative size = 369 \[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=-\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}} \]
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Time = 0.19 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3739, 3555, 3557, 335, 302, 648, 632, 210, 642, 212} \[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\sqrt {3} \coth ^{\frac {4}{3}}(c+d x) \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}} \]
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Rule 210
Rule 212
Rule 302
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3555
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {16}{3}}(c+d x)} \, dx}{b \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {10}{3}}(c+d x)} \, dx}{b \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {4}{3}}(c+d x)} \, dx}{b \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \int \coth ^{\frac {2}{3}}(c+d x) \, dx}{b \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {x^{2/3}}{-1+x^2} \, dx,x,\coth (c+d x)\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-\frac {1}{2}-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-\frac {1}{2}+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{b d \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\left (3 \coth ^{\frac {4}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\coth (c+d x)}\right )}{2 b d \sqrt [3]{b \coth ^4(c+d x)}} \\ & = -\frac {3 \coth (c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {4}{3}}(c+d x)}{2 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {4}{3}}(c+d x)}{b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}+\frac {\coth ^{\frac {4}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh (c+d x)}{7 b d \sqrt [3]{b \coth ^4(c+d x)}}-\frac {3 \tanh ^3(c+d x)}{13 b d \sqrt [3]{b \coth ^4(c+d x)}} \\ \end{align*}
Time = 1.75 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\frac {-91 \coth (c+d x) \left (6+\sqrt [6]{\coth ^2(c+d x)} \log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [6]{\coth ^2(c+d x)} \log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)} \log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)} \log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )-6 \tanh (c+d x) \left (13+7 \tanh ^2(c+d x)\right )}{182 b d \sqrt [3]{b \coth ^4(c+d x)}} \]
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\[\int \frac {1}{\left (b \coth \left (d x +c \right )^{4}\right )^{\frac {4}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 3650 vs. \(2 (311) = 622\).
Time = 0.54 (sec) , antiderivative size = 15579, normalized size of antiderivative = 42.22 \[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\int \frac {1}{\left (b \coth ^{4}{\left (c + d x \right )}\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{4}\right )^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{4}\right )^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (b \coth ^4(c+d x)\right )^{4/3}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^4\right )}^{4/3}} \,d x \]
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