Optimal. Leaf size=225 \[ \frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b c^{2/3}} \]
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Rubi [A] time = 0.31, 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, 209, 634, 618, 204, 628, 203} \[ \frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{2/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{2/3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b c^{2/3}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 209
Rule 329
Rule 618
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{(c \cot (a+b x))^{2/3}} \, dx &=-\frac {c \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) \operatorname {Subst}\left (\int \frac {1}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac {\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 c^{2/3}}-\frac {\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 c^{2/3}}-\frac {\operatorname {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b \sqrt [3]{c}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\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^{2/3}}-\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^{2/3}}-\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 \sqrt [3]{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 \sqrt [3]{c}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\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^{2/3}}-\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^{2/3}}-\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^{2/3}}+\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^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{2/3}}+\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^{2/3}}-\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^{2/3}}+\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^{2/3}}-\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^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 38, normalized size = 0.17 \[ -\frac {3 \sqrt [3]{c \cot (a+b x)} \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-\cot ^2(a+b x)\right )}{b c} \]
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 \[ \int \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 209, normalized size = 0.93 \[ \frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}-\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b c}-\frac {\left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{2 b c}-\frac {\left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{b c}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{4 b c}-\frac {\left (c^{2}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{2 b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 182, normalized size = 0.81 \[ -\frac {c {\left (\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.55, size = 231, normalized size = 1.03 \[ -\frac {{\left (-1\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{5/6}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\,1{}\mathrm {i}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b\,c^{2/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{2/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}-{\left (-1\right )}^{1/6}\,c^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{2/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left ({\left (-1\right )}^{1/6}\,c^{1/3}-2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{2/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (2\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}+{\left (-1\right )}^{1/6}\,c^{1/3}-{\left (-1\right )}^{2/3}\,\sqrt {3}\,c^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{2/3}} \]
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 \[ \int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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