Integrand size = 26, antiderivative size = 327 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {8 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d} \]
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Time = 0.68 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3642, 3679, 3681, 3562, 59, 631, 210, 31, 3680} \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {8 \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} d}-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}+\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}} \]
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Rule 31
Rule 59
Rule 210
Rule 631
Rule 3562
Rule 3642
Rule 3679
Rule 3680
Rule 3681
Rubi steps \begin{align*} \text {integral}& = -\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac {\int \cot ^2(c+d x) \left (\frac {i a}{3}-\frac {5}{3} a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{2 a} \\ & = -\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac {\int \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)} \left (-\frac {16 a^2}{9}-\frac {2}{9} i a^2 \tan (c+d x)\right ) \, dx}{2 a^2} \\ & = -\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-i \int \sqrt [3]{a+i a \tan (c+d x)} \, dx-\frac {8 \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{9 a} \\ & = -\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac {(8 a) \text {Subst}\left (\int \frac {1}{x (a+i a x)^{2/3}} \, dx,x,\tan (c+d x)\right )}{9 d}-\frac {a \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = \frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}+\frac {\left (4 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}-\frac {\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}+\frac {\left (4 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}-\frac {\left (3 a^{2/3}\right ) \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+i a \tan (c+d x)}\right )}{2 \sqrt [3]{2} d} \\ & = \frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d}-\frac {\left (8 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{3 d}+\frac {\left (3 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}\right )}{2^{2/3} d} \\ & = \frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {8 \sqrt [3]{a} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} d}-\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{2^{2/3} d}+\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}+\frac {4 \sqrt [3]{a} \log (\tan (c+d x))}{9 d}-\frac {4 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{3 d}+\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{6 d}-\frac {\cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{2 d} \\ \end{align*}
Time = 1.63 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.12 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {-32 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )+18 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+32 \sqrt [3]{a} \log \left (\sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-18 \sqrt [3]{2} \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )-16 \sqrt [3]{a} \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+9 \sqrt [3]{2} \sqrt [3]{a} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )+6 i \cot (c+d x) \sqrt [3]{a+i a \tan (c+d x)}+18 \cot ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{36 d} \]
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\[\int \left (\cot ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 678 vs. \(2 (239) = 478\).
Time = 0.26 (sec) , antiderivative size = 678, normalized size of antiderivative = 2.07 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\int \sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.94 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {a^{2} {\left (\frac {18 \, \sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} - \frac {6 \, {\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} + a^{3}} - \frac {32 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {5}{3}}} + \frac {9 \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} - \frac {18 \cdot 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}{a^{\frac {5}{3}}} - \frac {16 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{a^{\frac {5}{3}}} + \frac {32 \, \log \left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{a^{\frac {5}{3}}}\right )}}{36 \, d} \]
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Exception generated. \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \]
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Time = 5.15 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.28 \[ \int \cot ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {8\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}+\frac {\frac {a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{6}+\frac {a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{3}}{d\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2+a^2\,d-2\,a\,d\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}-\frac {4\,\ln \left (d\,{\left (-\frac {a}{d^3}\right )}^{1/3}-2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\frac {4\,\ln \left (2\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-d\,{\left (-\frac {a}{d^3}\right )}^{1/3}+\sqrt {3}\,d\,{\left (-\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (-\frac {a}{d^3}\right )}^{1/3}}{9}+\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}-\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3}-\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+\frac {2^{1/3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,d\,{\left (\frac {a}{d^3}\right )}^{1/3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {a}{4\,d^3}\right )}^{1/3} \]
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