Optimal. Leaf size=131 \[ \frac {\sqrt [3]{c} \log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b}+\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b} \]
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Rubi [A] time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3476, 329, 275, 292, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{c} \log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b}+\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 617
Rule 628
Rule 634
Rule 3476
Rubi steps
\begin {align*} \int \sqrt [3]{c \cot (a+b x)} \, dx &=-\frac {c \operatorname {Subst}\left (\int \frac {\sqrt [3]{x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b}\\ &=-\frac {(3 c) \operatorname {Subst}\left (\int \frac {x^3}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b}\\ &=-\frac {(3 c) \operatorname {Subst}\left (\int \frac {x}{c^2+x^3} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}\\ &=\frac {\sqrt [3]{c} \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+x} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \operatorname {Subst}\left (\int \frac {c^{2/3}+x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b}\\ &=\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \operatorname {Subst}\left (\int \frac {-c^{2/3}+2 x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b}-\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b}\\ &=\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b}-\frac {\left (3 \sqrt [3]{c}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}\right )}{2 b}\\ &=\frac {\sqrt {3} \sqrt [3]{c} \tan ^{-1}\left (\frac {1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}}{\sqrt {3}}\right )}{2 b}+\frac {\sqrt [3]{c} \log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b}-\frac {\sqrt [3]{c} \log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 40, normalized size = 0.31 \[ -\frac {3 (c \cot (a+b x))^{4/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\cot ^2(a+b x)\right )}{4 b c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 211, normalized size = 1.61 \[ -\frac {2 \, \sqrt {3} c^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} c - 2 \, \sqrt {3} c^{\frac {1}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}}{3 \, c}\right ) - 2 \, c^{\frac {1}{3}} \log \left (c^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) + c^{\frac {1}{3}} \log \left (\frac {c^{\frac {4}{3}} \sin \left (2 \, b x + 2 \, a\right ) - c^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \]
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 \[ \int \left (c \cot \left (b x + a\right )\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 114, normalized size = 0.87 \[ \frac {c \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{2 b \left (c^{2}\right )^{\frac {1}{3}}}-\frac {c \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c^{2}\right )^{\frac {1}{3}} \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{4 b \left (c^{2}\right )^{\frac {1}{3}}}-\frac {c \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 b \left (c^{2}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 102, normalized size = 0.78 \[ -\frac {c {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (c^{\frac {2}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {2}{3}}} + \frac {\log \left (c^{\frac {4}{3}} - c^{\frac {2}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {4}{3}}\right )}{c^{\frac {2}{3}}} - \frac {2 \, \log \left (c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {2}{3}}}\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 134, normalized size = 1.02 \[ \frac {c^{1/3}\,\ln \left (81\,c^{16/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}+81\,c^6\right )}{2\,b}-\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}-\frac {81\,c^{16/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b}+\frac {c^{1/3}\,\ln \left (\frac {81\,c^6}{b^4}+\frac {162\,c^{16/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^4}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b} \]
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 \[ \int \sqrt [3]{c \cot {\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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