Optimal. Leaf size=242 \[ -\frac{\sqrt{3} c^{4/3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac{\sqrt{3} c^{4/3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac{c^{4/3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}+\frac{c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b}-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b} \]
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Rubi [A] time = 0.469561, antiderivative size = 242, 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 = {3473, 3476, 329, 209, 634, 618, 204, 628, 203} \[ -\frac{\sqrt{3} c^{4/3} \log \left (-\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac{\sqrt{3} c^{4/3} \log \left (\sqrt{3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac{c^{4/3} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}+\frac{c^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt{3}\right )}{2 b}-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3476
Rule 329
Rule 209
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int (c \cot (a+b x))^{4/3} \, dx &=-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}-c^2 \int \frac{1}{(c \cot (a+b x))^{2/3}} \, dx\\ &=-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac{c^3 \operatorname{Subst}\left (\int \frac{1}{x^{2/3} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac{\left (3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}+\frac{c^{4/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{c}-\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}+\frac{c^{4/3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{c}+\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}+\frac{c^{5/3} \operatorname{Subst}\left (\int \frac{1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac{\left (\sqrt{3} c^{4/3}\right ) \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}+\frac{\left (\sqrt{3} c^{4/3}\right ) \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}+\frac{c^{5/3} \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}+\frac{c^{5/3} \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}\\ &=\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac{\sqrt{3} c^{4/3} \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}+\frac{\sqrt{3} c^{4/3} \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}+\frac{c^{4/3} \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}-\frac{c^{4/3} \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}\\ &=\frac{c^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac{c^{4/3} \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}+\frac{c^{4/3} \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}-\frac{3 c \sqrt [3]{c \cot (a+b x)}}{b}-\frac{\sqrt{3} c^{4/3} \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}+\frac{\sqrt{3} c^{4/3} \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}\\ \end{align*}
Mathematica [C] time = 0.0301896, size = 38, normalized size = 0.16 \[ \frac{3 c \sqrt [3]{c \cot (a+b x)} \left (\text{Hypergeometric2F1}\left (\frac{1}{6},1,\frac{7}{6},-\cot ^2(a+b x)\right )-1\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 214, normalized size = 0.9 \begin{align*} -3\,{\frac{c\sqrt [3]{c\cot \left ( bx+a \right ) }}{b}}-{\frac{c\sqrt{3}}{4\,b}\sqrt [6]{{c}^{2}}\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{c}{2\,b}\sqrt [6]{{c}^{2}}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}-\sqrt{3} \right ) }+{\frac{c}{b}\sqrt [6]{{c}^{2}}\arctan \left ({\sqrt [3]{c\cot \left ( bx+a \right ) }{\frac{1}{\sqrt [6]{{c}^{2}}}}} \right ) }+{\frac{c\sqrt{3}}{4\,b}\sqrt [6]{{c}^{2}}\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{c}{2\,b}\sqrt [6]{{c}^{2}}\arctan \left ( 2\,{\frac{\sqrt [3]{c\cot \left ( bx+a \right ) }}{\sqrt [6]{{c}^{2}}}}+\sqrt{3} \right ) } \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 \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 \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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