Integrand size = 14, antiderivative size = 264 \[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {3739, 3557, 335, 216, 648, 632, 210, 642, 212} \[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=-\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right )}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \coth ^{\frac {2}{3}}(c+d x) \arctan \left (\frac {2 \sqrt [3]{\coth (c+d x)}+1}{\sqrt {3}}\right )}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right )}{d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)-\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (\coth ^{\frac {2}{3}}(c+d x)+\sqrt [3]{\coth (c+d x)}+1\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}} \]
[In]
[Out]
Rule 210
Rule 212
Rule 216
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3557
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \frac {\coth ^{\frac {2}{3}}(c+d x) \int \frac {1}{\coth ^{\frac {2}{3}}(c+d x)} \, dx}{\sqrt [3]{b \coth ^2(c+d x)}} \\ & = -\frac {\coth ^{\frac {2}{3}}(c+d x) \text {Subst}\left (\int \frac {1}{x^{2/3} \left (-1+x^2\right )} \, dx,x,\coth (c+d x)\right )}{d \sqrt [3]{b \coth ^2(c+d x)}} \\ & = -\frac {\left (3 \coth ^{\frac {2}{3}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \sqrt [3]{b \coth ^2(c+d x)}} \\ & = \frac {\coth ^{\frac {2}{3}}(c+d x) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \text {Subst}\left (\int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \text {Subst}\left (\int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx,x,\sqrt [3]{\coth (c+d x)}\right )}{d \sqrt [3]{b \coth ^2(c+d x)}} \\ & = \frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\left (3 \coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\left (3 \coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}} \\ & = \frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\left (3 \coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\left (3 \coth ^{\frac {2}{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 d \sqrt [3]{b \coth ^2(c+d x)}} \\ & = -\frac {\sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\sqrt {3} \arctan \left (\frac {1+2 \sqrt [3]{\coth (c+d x)}}{\sqrt {3}}\right ) \coth ^{\frac {2}{3}}(c+d x)}{2 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\text {arctanh}\left (\sqrt [3]{\coth (c+d x)}\right ) \coth ^{\frac {2}{3}}(c+d x)}{d \sqrt [3]{b \coth ^2(c+d x)}}-\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1-\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}}+\frac {\coth ^{\frac {2}{3}}(c+d x) \log \left (1+\sqrt [3]{\coth (c+d x)}+\coth ^{\frac {2}{3}}(c+d x)\right )}{4 d \sqrt [3]{b \coth ^2(c+d x)}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.69 \[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=-\frac {\coth (c+d x) \left (\log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \left (-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+\log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )\right )}{2 d \sqrt [6]{\coth ^2(c+d x)} \sqrt [3]{b \coth ^2(c+d x)}} \]
[In]
[Out]
\[\int \frac {1}{\left (\coth \left (d x +c \right )^{2} b \right )^{\frac {1}{3}}}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1594 vs. \(2 (216) = 432\).
Time = 0.41 (sec) , antiderivative size = 8338, normalized size of antiderivative = 31.58 \[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=\int \frac {1}{\sqrt [3]{b \coth ^{2}{\left (c + d x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )^{2}\right )^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt [3]{b \coth ^2(c+d x)}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {coth}\left (c+d\,x\right )}^2\right )}^{1/3}} \,d x \]
[In]
[Out]