Optimal. Leaf size=244 \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b c^{4/3}}+\frac{3}{b c \sqrt [3]{c \cot (a+b x)}} \]
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Rubi [A] time = 0.424563, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3474, 3476, 329, 295, 634, 618, 204, 628, 203} \[ \frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}-\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b c^{4/3}}+\frac{3}{b c \sqrt [3]{c \cot (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3474
Rule 3476
Rule 329
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{(c \cot (a+b x))^{4/3}} \, dx &=\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}-\frac{\int (c \cot (a+b x))^{2/3} \, dx}{c^2}\\ &=\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{\operatorname{Subst}\left (\int \frac{x^{2/3}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b c}\\ &=\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{3 \operatorname{Subst}\left (\int \frac{x^4}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c}\\ &=\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{c}}{2}+\frac{\sqrt{3} x}{2}}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{\sqrt [3]{c}}{2}-\frac{\sqrt{3} x}{2}}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{-\sqrt{3} \sqrt [3]{c}+2 x}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}-\frac{\sqrt{3} \operatorname{Subst}\left (\int \frac{\sqrt{3} \sqrt [3]{c}+2 x}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}-\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c}+\frac{\operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt{3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{\sqrt{3} \log \left (c^{2/3}-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac{\sqrt{3} \log \left (c^{2/3}+\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt{3} \sqrt [3]{c}}\right )}{2 \sqrt{3} b c^{4/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt{3} \sqrt [3]{c}}\right )}{2 \sqrt{3} b c^{4/3}}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}-\frac{6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac{\tan ^{-1}\left (\frac{1}{3} \left (3 \sqrt{3}+\frac{6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac{3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac{\sqrt{3} \log \left (c^{2/3}-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac{\sqrt{3} \log \left (c^{2/3}+\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0569193, size = 38, normalized size = 0.16 \[ \frac{3 \text{Hypergeometric2F1}\left (-\frac{1}{6},1,\frac{5}{6},-\cot ^2(a+b x)\right )}{b c \sqrt [3]{c \cot (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 229, normalized size = 0.9 \begin{align*} 3\,{\frac{1}{bc\sqrt [3]{c\cot \left ( bx+a \right ) }}}+{\frac{\sqrt{3}}{4\,b{c}^{3}} \left ({c}^{2} \right ) ^{{\frac{5}{6}}}\ln \left ( \sqrt{3}\sqrt [6]{{c}^{2}}\sqrt [3]{c\cot \left ( bx+a \right ) }- \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}-\sqrt [3]{{c}^{2}} \right ) }+{\frac{1}{2\,bc}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{2}}}}}+{\frac{1}{bc}\arctan \left ({\sqrt [3]{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [6]{{c}^{2}}}}} \right ){\frac{1}{\sqrt [6]{{c}^{2}}}}}-{\frac{\sqrt{3}}{4\,b{c}^{3}} \left ({c}^{2} \right ) ^{{\frac{5}{6}}}\ln \left ( \left ( c\cot \left ( bx+a \right ) \right ) ^{{\frac{2}{3}}}+\sqrt{3}\sqrt [6]{{c}^{2}}\sqrt [3]{c\cot \left ( bx+a \right ) }+\sqrt [3]{{c}^{2}} \right ) }+{\frac{1}{2\,bc}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{c}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 (c \cot{\left (a + b x \right )}\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c \cot \left (b x + a\right )\right )^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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