Integrand size = 12, antiderivative size = 218 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d} \]
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Time = 0.17 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335, 216, 648, 632, 210, 642, 212} \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d} \]
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Rule 210
Rule 212
Rule 216
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {1}{x^{2/3} \left (-b^2+x^2\right )} \, dx,x,b \coth (c+d x)\right )}{d} \\ & = -\frac {(3 b) \text {Subst}\left (\int \frac {1}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {\sqrt [3]{b}-\frac {x}{2}}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{2/3} d}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{b}+\frac {x}{2}}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{b^{2/3} d}+\frac {\text {Subst}\left (\int \frac {1}{b^{2/3}-x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{\sqrt [3]{b} d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{b}+2 x}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{2/3} d}+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{b}+2 x}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 b^{2/3} d}+\frac {3 \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 \sqrt [3]{b} d}+\frac {3 \text {Subst}\left (\int \frac {1}{b^{2/3}+\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{4 \sqrt [3]{b} d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{2/3} d}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{2 b^{2/3} d} \\ & = -\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{2 b^{2/3} d}+\frac {\text {arctanh}\left (\frac {\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac {\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac {\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=-\frac {\sqrt [3]{b \coth (c+d x)} \left (\log \left (1-\sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1+\sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \left (-\sqrt [3]{-1} \log \left (1-\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )+\sqrt [3]{-1} \log \left (1+\sqrt [3]{-1} \sqrt [6]{\coth ^2(c+d x)}\right )-\log \left (1-(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )+\log \left (1+(-1)^{2/3} \sqrt [6]{\coth ^2(c+d x)}\right )\right )\right )}{2 b d \sqrt [6]{\coth ^2(c+d x)}} \]
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Time = 0.10 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \,b^{\frac {2}{3}}}\) | \(193\) |
default | \(-\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}-b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}+\frac {\ln \left (b^{\frac {2}{3}}+b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\arctan \left (\frac {\left (1+\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{2 b^{\frac {2}{3}} d}+\frac {\ln \left (\left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+b^{\frac {1}{3}}\right )}{2 d \,b^{\frac {2}{3}}}-\frac {\ln \left (b^{\frac {2}{3}}-b^{\frac {1}{3}} \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}+\left (b \coth \left (d x +c \right )\right )^{\frac {2}{3}}\right )}{4 b^{\frac {2}{3}} d}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \coth \left (d x +c \right )\right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}-1\right )}{3}\right )}{2 d \,b^{\frac {2}{3}}}\) | \(193\) |
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Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (166) = 332\).
Time = 0.26 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.63 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\frac {2 \, \sqrt {3} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} b \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}} - 2 \, \sqrt {3} \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} \sqrt {-\left (-b^{2}\right )^{\frac {1}{3}}}}{3 \, b^{2}}\right ) + 2 \, \sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (-\frac {\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{6}} {\left ({\left (b^{2}\right )}^{\frac {1}{3}} b - 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right )}}{3 \, b^{2}}\right ) + \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} - \left (-b^{2}\right )^{\frac {1}{3}} b + \left (-b^{2}\right )^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {2}{3}} + {\left (b^{2}\right )}^{\frac {1}{3}} b - {\left (b^{2}\right )}^{\frac {2}{3}} \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}}\right ) - 2 \, \left (-b^{2}\right )^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} - \left (-b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (b \left (\frac {b \cosh \left (d x + c\right )}{\sinh \left (d x + c\right )}\right )^{\frac {1}{3}} + {\left (b^{2}\right )}^{\frac {2}{3}}\right )}{4 \, b^{2} d} \]
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\[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\int \frac {1}{\left (b \coth {\left (c + d x \right )}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\int { \frac {1}{\left (b \coth \left (d x + c\right )\right )^{\frac {2}{3}}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\text {Exception raised: TypeError} \]
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Time = 2.07 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(b \coth (c+d x))^{2/3}} \, dx=\frac {\mathrm {atanh}\left (\frac {{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{b^{1/3}}\right )}{b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}-\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{-243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d}-\frac {\mathrm {atan}\left (\frac {b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}\,243{}\mathrm {i}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}+\frac {243\,\sqrt {3}\,b^{10/3}\,{\left (b\,\mathrm {coth}\left (c+d\,x\right )\right )}^{1/3}}{243\,b^{11/3}+\sqrt {3}\,b^{11/3}\,243{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,b^{2/3}\,d} \]
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