Integrand size = 26, antiderivative size = 251 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {i a^{4/3} x}{2^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d} \]
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Time = 0.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3641, 3673, 3608, 3559, 3562, 59, 631, 210, 31} \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {i a^{4/3} x}{2^{2/3}}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}-\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d} \]
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Rule 31
Rule 59
Rule 210
Rule 631
Rule 3559
Rule 3562
Rule 3608
Rule 3641
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {3 \int \tan (c+d x) (a+i a \tan (c+d x))^{4/3} \left (2 a+\frac {4}{3} i a \tan (c+d x)\right ) \, dx}{10 a} \\ & = \frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac {3 \int (a+i a \tan (c+d x))^{4/3} \left (-\frac {4 i a}{3}+2 a \tan (c+d x)\right ) \, dx}{10 a} \\ & = -\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+i \int (a+i a \tan (c+d x))^{4/3} \, dx \\ & = -\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+(2 i a) \int \sqrt [3]{a+i a \tan (c+d x)} \, dx \\ & = -\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{2/3}} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^{4/3} x}{2^{2/3}}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}+\frac {\left (3 a^{4/3}\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^{2/3} d}+\frac {\left (3 a^{5/3}\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 )}{\sqrt [3]{2} d} \\ & = -\frac {i a^{4/3} x}{2^{2/3}}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d}-\frac {\left (3 \sqrt [3]{2} a^{4/3}\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 )}{d} \\ & = -\frac {i a^{4/3} x}{2^{2/3}}+\frac {\sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{d}-\frac {a^{4/3} \log (\cos (c+d x))}{2^{2/3} d}-\frac {3 a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{2^{2/3} d}-\frac {3 a \sqrt [3]{a+i a \tan (c+d x)}}{d}-\frac {9 (a+i a \tan (c+d x))^{4/3}}{20 d}+\frac {3 \tan ^2(c+d x) (a+i a \tan (c+d x))^{4/3}}{10 d}-\frac {6 (a+i a \tan (c+d x))^{7/3}}{35 a d} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.12 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {140 \sqrt [3]{2} \sqrt {3} a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-140 \sqrt [3]{2} a^{4/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )+70 \sqrt [3]{2} a^{4/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+i a \tan (c+d x)}+(a+i a \tan (c+d x))^{2/3}\right )-507 a \sqrt [3]{a+i a \tan (c+d x)}-111 i a \tan (c+d x) \sqrt [3]{a+i a \tan (c+d x)}+66 a \tan ^2(c+d x) \sqrt [3]{a+i a \tan (c+d x)}+42 i a \tan ^3(c+d x) \sqrt [3]{a+i a \tan (c+d x)}}{140 d} \]
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Time = 0.55 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {10}{3}}}{10 d \,a^{2}}+\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{3}}}{7 a d}-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4 d}-\frac {3 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\) | \(217\) |
default | \(-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {10}{3}}}{10 d \,a^{2}}+\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{3}}}{7 a d}-\frac {3 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}{4 d}-\frac {3 a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{d}-\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{2 d}+\frac {a^{\frac {4}{3}} 2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{d}\) | \(217\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (190) = 380\).
Time = 0.27 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.00 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {3 \cdot 2^{\frac {1}{3}} {\left (121 \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 240 \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 245 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 70 \, a\right )} \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 )} + 35 \cdot 2^{\frac {1}{3}} {\left ({\left (-i \, \sqrt {3} d + d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\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^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2 \cdot 2^{\frac {1}{3}} a \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 )} + 2^{\frac {1}{3}} {\left (i \, \sqrt {3} d - d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) + 35 \cdot 2^{\frac {1}{3}} {\left ({\left (i \, \sqrt {3} d + d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (i \, \sqrt {3} d + d\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, {\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^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2 \cdot 2^{\frac {1}{3}} a \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 )} + 2^{\frac {1}{3}} {\left (-i \, \sqrt {3} d - d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}}}{2 \, a}\right ) - 70 \cdot 2^{\frac {1}{3}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} \log \left (\frac {2^{\frac {1}{3}} a \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 )} + 2^{\frac {1}{3}} \left (-\frac {a^{4}}{d^{3}}\right )^{\frac {1}{3}} d}{a}\right )}{70 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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\[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}} \tan ^{3}{\left (c + d x \right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.83 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\frac {140 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {16}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) + 70 \cdot 2^{\frac {1}{3}} a^{\frac {16}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) - 140 \cdot 2^{\frac {1}{3}} a^{\frac {16}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) - 42 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {10}{3}} a^{2} + 60 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{3}} a^{3} - 105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} a^{4} - 420 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{5}}{140 \, a^{4} d} \]
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\[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}} \tan \left (d x + c\right )^{3} \,d x } \]
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Time = 5.30 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.97 \[ \int \tan ^3(c+d x) (a+i a \tan (c+d x))^{4/3} \, dx=-\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}{4\,d}+\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/3}}{7\,a\,d}-\frac {3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{10/3}}{10\,a^2\,d}-\frac {3\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}-\frac {2^{1/3}\,a^{4/3}\,\ln \left ({\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{d}-\frac {2^{1/3}\,a^{4/3}\,\ln \left (\frac {18\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}-\frac {9\,2^{1/3}\,a^{7/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d}+\frac {2^{1/3}\,a^{4/3}\,\ln \left (\frac {18\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{d}+\frac {9\,2^{1/3}\,a^{7/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{d}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{d} \]
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