Optimal. Leaf size=218 \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{2/3} d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 b^{2/3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d} \]
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Rubi [A] time = 0.23531, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {3476, 329, 210, 634, 618, 204, 628, 206} \[ -\frac{\log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}+\frac{\log \left (b^{2/3}+\sqrt [3]{b} \sqrt [3]{b \coth (c+d x)}+(b \coth (c+d x))^{2/3}\right )}{4 b^{2/3} d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{2/3} d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}+1}{\sqrt{3}}\right )}{2 b^{2/3} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b \coth (c+d x))^{2/3}} \, dx &=-\frac{b \operatorname{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) \operatorname{Subst}\left (\int \frac{1}{-b^2+x^6} \, dx,x,\sqrt [3]{b \coth (c+d x)}\right )}{d}\\ &=\frac{\operatorname{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{\operatorname{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{\operatorname{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{\tanh ^{-1}\left (\frac{\sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}\right )}{b^{2/3} d}-\frac{\operatorname{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{\operatorname{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 \operatorname{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 \operatorname{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{\tanh ^{-1}\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 \operatorname{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 \operatorname{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} \tan ^{-1}\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} \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b \coth (c+d x)}}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{2 b^{2/3} d}+\frac{\tanh ^{-1}\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*}
Mathematica [C] time = 0.0286256, size = 36, normalized size = 0.17 \[ \frac{3 \sqrt [3]{b \coth (c+d x)} \, _2F_1\left (\frac{1}{6},1;\frac{7}{6};\coth ^2(c+d x)\right )}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 193, normalized size = 0.9 \begin{align*}{\frac{1}{2\,d}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}+\sqrt [3]{b} \right ){b}^{-{\frac{2}{3}}}}-{\frac{1}{4\,d}\ln \left ({b}^{{\frac{2}{3}}}-\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{2\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}}}-{\frac{1}{2\,d}\ln \left ( \sqrt [3]{b{\rm coth} \left (dx+c\right )}-\sqrt [3]{b} \right ){b}^{-{\frac{2}{3}}}}+{\frac{1}{4\,d}\ln \left ({b}^{{\frac{2}{3}}}+\sqrt [3]{b}\sqrt [3]{b{\rm coth} \left (dx+c\right )}+ \left ( b{\rm coth} \left (dx+c\right ) \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{2\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,{\frac{\sqrt [3]{b{\rm coth} \left (dx+c\right )}}{\sqrt [3]{b}}} \right ) } \right ){b}^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth \left (d x + c\right )\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01017, size = 991, normalized size = 4.55 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b \coth{\left (c + d x \right )}\right )^{\frac{2}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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