Integrand size = 26, antiderivative size = 234 \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}}+\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}-\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a d} \]
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Time = 0.38 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3641, 3673, 3608, 3562, 59, 631, 210, 31} \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {\sqrt {3} \sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{2^{2/3} d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2\ 2^{2/3} d}-\frac {\sqrt [3]{a} \log (\cos (c+d x))}{2\ 2^{2/3} d}-\frac {i \sqrt [3]{a} x}{2\ 2^{2/3}} \]
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Rule 31
Rule 59
Rule 210
Rule 631
Rule 3562
Rule 3608
Rule 3641
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 \int \tan (c+d x) \left (2 a+\frac {1}{3} i a \tan (c+d x)\right ) \sqrt [3]{a+i a \tan (c+d x)} \, dx}{7 a} \\ & = \frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a d}-\frac {3 \int \sqrt [3]{a+i a \tan (c+d x)} \left (-\frac {i a}{3}+2 a \tan (c+d x)\right ) \, dx}{7 a} \\ & = -\frac {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a d}+i \int \sqrt [3]{a+i a \tan (c+d x)} \, dx \\ & = -\frac {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a 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 {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a 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 (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 {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 {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a 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 {\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 {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 {18 \sqrt [3]{a+i a \tan (c+d x)}}{7 d}+\frac {3 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{7 d}-\frac {3 (a+i a \tan (c+d x))^{4/3}}{28 a d} \\ \end{align*}
Time = 1.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.06 \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {14 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-14 \sqrt [3]{2} \sqrt [3]{a} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+7 \sqrt [3]{2} \sqrt [3]{a} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-75 \sqrt [3]{a+i a \tan (c+d x)}-3 i \tan (c+d x) \sqrt [3]{a+i a \tan (c+d x)}+12 \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{28 d} \]
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Time = 0.57 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{3}}}{7 d \,a^{2}}+\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4 a d}-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}-\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{2 d}+\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{4 d}+\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{2 d}\) | \(198\) |
default | \(-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{3}}}{7 d \,a^{2}}+\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4 a d}-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}-\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{2 d}+\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{4 d}+\frac {a^{\frac {1}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{2 d}\) | \(198\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (172) = 344\).
Time = 0.27 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.68 \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {3 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} {\left (15 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 21 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 14\right )} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )} + 7 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (-i \, \sqrt {3} d + d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, \sqrt {3} d + d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (\left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (i \, \sqrt {3} d - d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) + 7 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (i \, \sqrt {3} d + d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, \sqrt {3} d + d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (\left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (-i \, \sqrt {3} d - d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right ) - 14 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} \log \left (2 \, \left (\frac {1}{4}\right )^{\frac {1}{3}} d \left (-\frac {a}{d^{3}}\right )^{\frac {1}{3}} + 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {1}{3}} e^{\left (\frac {2}{3} i \, d x + \frac {2}{3} i \, c\right )}\right )}{14 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\int \sqrt [3]{i a \left (\tan {\left (c + d x \right )} - i\right )} \tan ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.81 \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\frac {14 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {13}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) + 7 \cdot 2^{\frac {1}{3}} a^{\frac {13}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 14 \cdot 2^{\frac {1}{3}} a^{\frac {13}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{3}} a^{2} + 21 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{3} - 84 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{4}}{28 \, a^{4} d} \]
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\[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} \tan \left (d x + c\right )^{3} \,d x } \]
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Time = 1.01 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00 \[ \int \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)} \, dx=-\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}+\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{4\,a\,d}-\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/3}}{7\,a^2\,d}+\frac {2^{1/3}\,{\left (-a\right )}^{1/3}\,\ln \left (9\,a\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-9\,2^{1/3}\,{\left (-a\right )}^{4/3}\right )}{2\,d}+\frac {4^{2/3}\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}-\frac {9\,2^{1/3}\,{\left (-a\right )}^{4/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,d}-\frac {4^{2/3}\,{\left (-a\right )}^{1/3}\,\ln \left (\frac {9\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}+\frac {9\,2^{1/3}\,{\left (-a\right )}^{4/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,d}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,d} \]
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