Integrand size = 34, antiderivative size = 141 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-2 a^2 (A-i B) x+\frac {2 a^2 (i A+B) \log (\cos (c+d x))}{d}+\frac {2 a^2 (A-i B) \tan (c+d x)}{d}+\frac {a^2 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d} \]
[Out]
Time = 0.40 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3675, 3673, 3609, 3606, 3556} \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {a^2 (B+i A) \tan ^2(c+d x)}{d}+\frac {2 a^2 (A-i B) \tan (c+d x)}{d}+\frac {2 a^2 (B+i A) \log (\cos (c+d x))}{d}-2 a^2 x (A-i B)+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d} \]
[In]
[Out]
Rule 3556
Rule 3606
Rule 3609
Rule 3673
Rule 3675
Rubi steps \begin{align*} \text {integral}& = \frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{4} \int \tan ^2(c+d x) (a+i a \tan (c+d x)) (a (4 A-3 i B)+a (4 i A+5 B) \tan (c+d x)) \, dx \\ & = -\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{4} \int \tan ^2(c+d x) \left (8 a^2 (A-i B)+8 a^2 (i A+B) \tan (c+d x)\right ) \, dx \\ & = \frac {a^2 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}+\frac {1}{4} \int \tan (c+d x) \left (-8 a^2 (i A+B)+8 a^2 (A-i B) \tan (c+d x)\right ) \, dx \\ & = -2 a^2 (A-i B) x+\frac {2 a^2 (A-i B) \tan (c+d x)}{d}+\frac {a^2 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d}-\left (2 a^2 (i A+B)\right ) \int \tan (c+d x) \, dx \\ & = -2 a^2 (A-i B) x+\frac {2 a^2 (i A+B) \log (\cos (c+d x))}{d}+\frac {2 a^2 (A-i B) \tan (c+d x)}{d}+\frac {a^2 (i A+B) \tan ^2(c+d x)}{d}-\frac {a^2 (4 A-5 i B) \tan ^3(c+d x)}{12 d}+\frac {i B \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{4 d} \\ \end{align*}
Time = 1.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.72 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {a^2 \left (-4 i A-5 B-24 i (A-i B) \log (i+\tan (c+d x))+24 (A-i B) \tan (c+d x)+12 (i A+B) \tan ^2(c+d x)-4 (A-2 i B) \tan ^3(c+d x)-3 B \tan ^4(c+d x)\right )}{12 d} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {a^{2} \left (\frac {2 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}+i A \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i B \tan \left (d x +c \right )+B \left (\tan ^{2}\left (d x +c \right )\right )+2 A \tan \left (d x +c \right )+\frac {\left (-2 i A -2 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (2 i B -2 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(121\) |
default | \(\frac {a^{2} \left (\frac {2 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}+i A \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}-2 i B \tan \left (d x +c \right )+B \left (\tan ^{2}\left (d x +c \right )\right )+2 A \tan \left (d x +c \right )+\frac {\left (-2 i A -2 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (2 i B -2 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(121\) |
norman | \(\left (2 i B \,a^{2}-2 A \,a^{2}\right ) x +\frac {\left (i A \,a^{2}+B \,a^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{d}-\frac {\left (-2 i B \,a^{2}+A \,a^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {2 \left (-i B \,a^{2}+A \,a^{2}\right ) \tan \left (d x +c \right )}{d}-\frac {B \,a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {\left (i A \,a^{2}+B \,a^{2}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(139\) |
parts | \(\frac {\left (2 i A \,a^{2}+B \,a^{2}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (2 i B \,a^{2}-A \,a^{2}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {A \,a^{2} \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}-\frac {B \,a^{2} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(154\) |
parallelrisch | \(-\frac {-8 i B \left (\tan ^{3}\left (d x +c \right )\right ) a^{2}+3 B \,a^{2} \left (\tan ^{4}\left (d x +c \right )\right )-12 i A \left (\tan ^{2}\left (d x +c \right )\right ) a^{2}+4 A \left (\tan ^{3}\left (d x +c \right )\right ) a^{2}-24 i B x \,a^{2} d +12 i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}+24 A x \,a^{2} d +24 i B \tan \left (d x +c \right ) a^{2}-12 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{2}-24 A \tan \left (d x +c \right ) a^{2}+12 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2}}{12 d}\) | \(156\) |
risch | \(-\frac {4 i a^{2} B c}{d}+\frac {4 a^{2} A c}{d}+\frac {2 a^{2} \left (15 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+21 B \,{\mathrm e}^{6 i \left (d x +c \right )}+33 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+36 B \,{\mathrm e}^{4 i \left (d x +c \right )}+25 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+29 B \,{\mathrm e}^{2 i \left (d x +c \right )}+7 i A +8 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}+\frac {2 i a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}\) | \(170\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.67 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (3 \, {\left (-5 i \, A - 7 \, B\right )} a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, {\left (-11 i \, A - 12 \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (-25 i \, A - 29 \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-7 i \, A - 8 \, B\right )} a^{2} + 3 \, {\left ({\left (-i \, A - B\right )} a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, {\left (-i \, A - B\right )} a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (-i \, A - B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, {\left (-i \, A - B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, A - B\right )} a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
[In]
[Out]
Time = 0.46 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.67 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 i a^{2} \left (A - i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {14 i A a^{2} + 16 B a^{2} + \left (50 i A a^{2} e^{2 i c} + 58 B a^{2} e^{2 i c}\right ) e^{2 i d x} + \left (66 i A a^{2} e^{4 i c} + 72 B a^{2} e^{4 i c}\right ) e^{4 i d x} + \left (30 i A a^{2} e^{6 i c} + 42 B a^{2} e^{6 i c}\right ) e^{6 i d x}}{3 d e^{8 i c} e^{8 i d x} + 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} + 12 d e^{2 i c} e^{2 i d x} + 3 d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {3 \, B a^{2} \tan \left (d x + c\right )^{4} + 4 \, {\left (A - 2 i \, B\right )} a^{2} \tan \left (d x + c\right )^{3} + 12 \, {\left (-i \, A - B\right )} a^{2} \tan \left (d x + c\right )^{2} + 24 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{2} - 12 \, {\left (-i \, A - B\right )} a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 24 \, {\left (A - i \, B\right )} a^{2} \tan \left (d x + c\right )}{12 \, d} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (123) = 246\).
Time = 0.59 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.89 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {2 \, {\left (-3 i \, A a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 \, B a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, A a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 \, B a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, A a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 \, B a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 \, B a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 15 i \, A a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 21 \, B a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 33 i \, A a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 36 \, B a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 25 i \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 29 \, B a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 3 i \, A a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 \, B a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 7 i \, A a^{2} - 8 \, B a^{2}\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
[In]
[Out]
Time = 7.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.09 \[ \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {a^2\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {B\,a^2\,1{}\mathrm {i}}{3}\right )}{d}-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-A\,a^2+a^2\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}+B\,a^2\,1{}\mathrm {i}\right )}{d}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^2\,\left (B+A\,1{}\mathrm {i}\right )}{2}+\frac {B\,a^2}{2}+\frac {A\,a^2\,1{}\mathrm {i}}{2}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (2\,B\,a^2+A\,a^2\,2{}\mathrm {i}\right )}{d}-\frac {B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d} \]
[In]
[Out]