Integrand size = 26, antiderivative size = 282 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {x}{8 \sqrt [3]{2} a^{4/3}}+\frac {i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d} \]
[Out]
Time = 0.55 (sec) , antiderivative size = 282, 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, 3676, 3673, 3607, 3562, 57, 631, 210, 31} \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {i \sqrt {3} \arctan \left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt {3} \sqrt [3]{a}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}-\frac {x}{8 \sqrt [3]{2} a^{4/3}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}} \]
[In]
[Out]
Rule 31
Rule 57
Rule 210
Rule 631
Rule 3562
Rule 3607
Rule 3641
Rule 3673
Rule 3676
Rubi steps \begin{align*} \text {integral}& = \frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {3 \int \frac {\tan ^2(c+d x) \left (3 a-\frac {4}{3} i a \tan (c+d x)\right )}{(a+i a \tan (c+d x))^{4/3}} \, dx}{5 a} \\ & = -\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}+\frac {9 \int \frac {\tan (c+d x) \left (\frac {26 i a^2}{3}+\frac {58}{9} a^2 \tan (c+d x)\right )}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{40 a^3} \\ & = -\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}+\frac {9 \int \frac {-\frac {58 a^2}{9}+\frac {26}{3} i a^2 \tan (c+d x)}{\sqrt [3]{a+i a \tan (c+d x)}} \, dx}{40 a^3} \\ & = -\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}+\frac {\int (a+i a \tan (c+d x))^{2/3} \, dx}{4 a^2} \\ & = -\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}-\frac {i \text {Subst}\left (\int \frac {1}{(a-x) \sqrt [3]{a+x}} \, dx,x,i a \tan (c+d x)\right )}{4 a d} \\ & = -\frac {x}{8 \sqrt [3]{2} a^{4/3}}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}-\frac {(3 i) \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 )}{8 \sqrt [3]{2} a^{4/3} d}+\frac {(3 i) \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 )}{8 a d} \\ & = -\frac {x}{8 \sqrt [3]{2} a^{4/3}}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d}-\frac {(3 i) \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 )}{4 \sqrt [3]{2} a^{4/3} d} \\ & = -\frac {x}{8 \sqrt [3]{2} a^{4/3}}+\frac {i \sqrt {3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+i a \tan (c+d x)}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{4 \sqrt [3]{2} a^{4/3} d}+\frac {i \log (\cos (c+d x))}{8 \sqrt [3]{2} a^{4/3} d}+\frac {3 i \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+i a \tan (c+d x)}\right )}{8 \sqrt [3]{2} a^{4/3} d}-\frac {39 i \tan ^2(c+d x)}{40 d (a+i a \tan (c+d x))^{4/3}}+\frac {3 \tan ^3(c+d x)}{5 d (a+i a \tan (c+d x))^{4/3}}-\frac {51 i}{10 a d \sqrt [3]{a+i a \tan (c+d x)}}-\frac {87 i (a+i a \tan (c+d x))^{2/3}}{40 a^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.40 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {15 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},\frac {1}{2} (1+i \tan (c+d x))\right ) (-i+\tan (c+d x))^2+6 \left (-97-126 i \tan (c+d x)+16 \tan ^2(c+d x)-8 i \tan ^3(c+d x)\right )}{80 a d (-i+\tan (c+d x)) \sqrt [3]{a+i a \tan (c+d x)}} \]
[In]
[Out]
Time = 0.56 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {3 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a^{2}}{4}-\frac {7 a^{2}}{4 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {a^{3}}{8 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}\right )}{d \,a^{3}}\) | \(212\) |
default | \(\frac {3 i \left (\frac {\left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{3}}}{5}-a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+\frac {\left (\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}-2^{\frac {1}{3}} a^{\frac {1}{3}}\right )}{6 a^{\frac {1}{3}}}-\frac {2^{\frac {2}{3}} \ln \left (\left (a +i a \tan \left (d x +c \right )\right )^{\frac {2}{3}}+2^{\frac {1}{3}} a^{\frac {1}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}+2^{\frac {2}{3}} a^{\frac {2}{3}}\right )}{12 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}+1\right )}{3}\right )}{6 a^{\frac {1}{3}}}\right ) a^{2}}{4}-\frac {7 a^{2}}{4 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {1}{3}}}+\frac {a^{3}}{8 \left (a +i a \tan \left (d x +c \right )\right )^{\frac {4}{3}}}\right )}{d \,a^{3}}\) | \(212\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (201) = 402\).
Time = 0.25 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.57 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=-\frac {3 \cdot 2^{\frac {2}{3}} \left (\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{\frac {2}{3}} {\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 425 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 125 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i\right )} e^{\left (\frac {4}{3} i \, d x + \frac {4}{3} i \, c\right )} - 160 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (32 \, a^{3} d^{2} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{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 ) + 80 \, {\left ({\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{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 ) + 80 \, {\left ({\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (i \, \sqrt {3} a^{2} d + a^{2} d\right )} e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {1}{3}} \log \left (-16 \, {\left (-i \, \sqrt {3} a^{3} d^{2} + a^{3} d^{2}\right )} \left (-\frac {i}{128 \, a^{4} d^{3}}\right )^{\frac {2}{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 )}{160 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]
[In]
[Out]
\[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int \frac {\tan ^{4}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {4}{3}}}\, dx \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.74 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {i \, {\left (10 \, \sqrt {3} 2^{\frac {2}{3}} a^{\frac {11}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right ) - 5 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}}\right ) + 10 \cdot 2^{\frac {2}{3}} a^{\frac {11}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {1}{3}}\right ) + 48 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{3}} a^{2} - 240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {2}{3}} a^{3} - \frac {30 \, {\left (14 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} - a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}}\right )}}{80 \, a^{5} d} \]
[In]
[Out]
\[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\int { \frac {\tan \left (d x + c\right )^{4}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {4}{3}}} \,d x } \]
[In]
[Out]
Time = 6.08 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.93 \[ \int \frac {\tan ^4(c+d x)}{(a+i a \tan (c+d x))^{4/3}} \, dx=\frac {\frac {3{}\mathrm {i}}{8\,d}-\frac {\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\,21{}\mathrm {i}}{4\,a\,d}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{4/3}}-\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{2/3}\,3{}\mathrm {i}}{a^2\,d}+\frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/3}\,3{}\mathrm {i}}{5\,a^3\,d}-\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (9\,{\left (a\,\left (1+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )\right )}^{1/3}+9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,a^{1/3}\right )}{a^{4/3}\,d}+\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (-\frac {9\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{16\,a^2\,d^2}-\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{32\,a^{5/3}\,d^2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}\,d}-\frac {{\left (\frac {1}{128}{}\mathrm {i}\right )}^{1/3}\,\ln \left (-\frac {9\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{1/3}}{16\,a^2\,d^2}+\frac {9\,{\left (-1\right )}^{1/3}\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{32\,a^{5/3}\,d^2}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{a^{4/3}\,d} \]
[In]
[Out]