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 = 0.47, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, 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 203
Rule 204
Rule 209
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3473
Rule 3476
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*}
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Mathematica [C] time = 0.03, size = 38, normalized size = 0.16 \[ \frac {3 c \sqrt [3]{c \cot (a+b x)} \left (\, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-\cot ^2(a+b x)\right )-1\right )}{b} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cot \left (b x + a\right )\right )^{\frac {4}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 214, normalized size = 0.88 \[ -\frac {3 c \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{b}-\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b}+\frac {c \sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b}+\frac {c \left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 196, normalized size = 0.81 \[ \frac {{\left (\sqrt {3} c^{\frac {1}{3}} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) - \sqrt {3} c^{\frac {1}{3}} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 2 \, c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) + 4 \, c^{\frac {1}{3}} \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right ) - 12 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}\right )} c}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 246, normalized size = 1.02 \[ -\frac {3\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{5/6}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left ({\left (-1\right )}^{1/6}\,c^{1/3}-2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}-{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{4/3}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}-{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \cot {\left (a + b x \right )}\right )^{\frac {4}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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