Optimal. Leaf size=225 \[ -\frac {\sqrt {3} c^{2/3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{2/3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}-\frac {c^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b} \]
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Rubi [A] time = 0.38, antiderivative size = 225, normalized size of antiderivative = 1.00, 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, 295, 634, 618, 204, 628, 203} \[ -\frac {\sqrt {3} c^{2/3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}+\frac {\sqrt {3} c^{2/3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b}-\frac {c^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b}-\frac {c^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 295
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int (c \cot (a+b x))^{2/3} \, dx &=-\frac {c \operatorname {Subst}\left (\int \frac {x^{2/3}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac {(3 c) \operatorname {Subst}\left (\int \frac {x^4}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac {c^{2/3} \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}-\frac {c^{2/3} \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}-\frac {c \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac {c^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {\left (\sqrt {3} c^{2/3}\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^{2/3}\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 \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 \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^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}-\frac {\sqrt {3} c^{2/3} \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^{2/3} \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^{2/3} \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^{2/3} \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^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b}+\frac {c^{2/3} \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^{2/3} \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 {\sqrt {3} c^{2/3} \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^{2/3} \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 = 0.05, size = 40, normalized size = 0.18 \[ -\frac {3 (c \cot (a+b x))^{5/3} \, _2F_1\left (\frac {5}{6},1;\frac {11}{6};-\cot ^2(a+b x)\right )}{5 b c} \]
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 \[ \int \left (c \cot \left (b x + a\right )\right )^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 203, normalized size = 0.90 \[ -\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b c}-\frac {c \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b \left (c^{2}\right )^{\frac {1}{6}}}-\frac {c \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b c}-\frac {c \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b \left (c^{2}\right )^{\frac {1}{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 182, normalized size = 0.81 \[ \frac {{\left (\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.82, size = 260, normalized size = 1.16 \[ -\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}-\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}-\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b}+\frac {{\left (-1\right )}^{1/6}\,c^{2/3}\,\ln \left (\frac {972\,c^9}{b^3}+\frac {486\,{\left (-1\right )}^{1/6}\,c^{26/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{b^3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]
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 \[ \int \left (c \cot {\left (a + b x \right )}\right )^{\frac {2}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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