Integrand size = 29, antiderivative size = 226 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\left (\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x\right )-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\cos (c+d x))}{d}+\frac {b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d} \]
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Time = 0.33 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {3673, 3609, 3606, 3556} \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\left (a^2 A-2 a b B-A b^2\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {b \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right ) \tan (c+d x)}{d}-\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \log (\cos (c+d x))}{d}-x \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right )+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d} \]
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Rule 3556
Rule 3606
Rule 3609
Rule 3673
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+b \tan (c+d x))^5}{5 b d}+\int (-B+A \tan (c+d x)) (a+b \tan (c+d x))^4 \, dx \\ & = \frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d}+\int (a+b \tan (c+d x))^3 (-A b-a B+(a A-b B) \tan (c+d x)) \, dx \\ & = \frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d}+\int (a+b \tan (c+d x))^2 \left (-2 a A b-a^2 B+b^2 B+\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)\right ) \, dx \\ & = \frac {\left (a^2 A-A b^2-2 a b B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d}+\int (a+b \tan (c+d x)) \left (-3 a^2 A b+A b^3-a^3 B+3 a b^2 B+\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)\right ) \, dx \\ & = -\left (\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x\right )+\frac {b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d}+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \int \tan (c+d x) \, dx \\ & = -\left (\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) x\right )-\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \log (\cos (c+d x))}{d}+\frac {b \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)}{d}+\frac {\left (a^2 A-A b^2-2 a b B\right ) (a+b \tan (c+d x))^2}{2 d}+\frac {(a A-b B) (a+b \tan (c+d x))^3}{3 d}+\frac {A (a+b \tan (c+d x))^4}{4 d}+\frac {B (a+b \tan (c+d x))^5}{5 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.54 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 B (a+b \tan (c+d x))^5+10 (a A+b B) \left (3 i (a+i b)^4 \log (i-\tan (c+d x))-3 i (a-i b)^4 \log (i+\tan (c+d x))+6 b^2 \left (-6 a^2+b^2\right ) \tan (c+d x)-12 a b^3 \tan ^2(c+d x)-2 b^4 \tan ^3(c+d x)\right )-5 A \left (6 i (a+i b)^5 \log (i-\tan (c+d x))-6 (i a+b)^5 \log (i+\tan (c+d x))-60 a b^2 \left (2 a^2-b^2\right ) \tan (c+d x)+6 b^3 \left (-10 a^2+b^2\right ) \tan ^2(c+d x)-20 a b^4 \tan ^3(c+d x)-3 b^5 \tan ^4(c+d x)\right )}{60 b d} \]
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Time = 0.13 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.07
method | result | size |
parts | \(\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}}{2 d}+\frac {B \,b^{4} \left (\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(242\) |
norman | \(\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) x +\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}+B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5 d}+\frac {b \left (6 A \,a^{2} b -A \,b^{3}+4 B \,a^{3}-4 B a \,b^{2}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{2} \left (4 A a b +6 B \,a^{2}-B \,b^{2}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {b^{3} \left (A b +4 B a \right ) \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\) | \(250\) |
derivativedivides | \(\frac {\frac {B \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {A \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+B a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )+\frac {4 A a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 B \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {B \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 A \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 B \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )-2 B a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )-4 A a \,b^{3} \tan \left (d x +c \right )+B \tan \left (d x +c \right ) a^{4}-6 B \,a^{2} b^{2} \tan \left (d x +c \right )+B \,b^{4} \tan \left (d x +c \right )+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(309\) |
default | \(\frac {\frac {B \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {A \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4}+B a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )+\frac {4 A a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+2 B \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )-\frac {B \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{3}+3 A \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )-\frac {A \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2}+2 B \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )-2 B a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )+4 A \,a^{3} b \tan \left (d x +c \right )-4 A a \,b^{3} \tan \left (d x +c \right )+B \tan \left (d x +c \right ) a^{4}-6 B \,a^{2} b^{2} \tan \left (d x +c \right )+B \,b^{4} \tan \left (d x +c \right )+\frac {\left (A \,a^{4}-6 A \,a^{2} b^{2}+A \,b^{4}-4 B \,a^{3} b +4 B a \,b^{3}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 A \,a^{3} b +4 A a \,b^{3}-B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\) | \(309\) |
parallelrisch | \(\frac {60 B \,b^{4} \tan \left (d x +c \right )-60 B x \,a^{4} d +360 B \,a^{2} b^{2} d x -240 A \,a^{3} b d x +240 A a \,b^{3} d x +240 A \,a^{3} b \tan \left (d x +c \right )-240 A a \,b^{3} \tan \left (d x +c \right )-360 B \,a^{2} b^{2} \tan \left (d x +c \right )+30 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}-30 A \,b^{4} \left (\tan ^{2}\left (d x +c \right )\right )+15 A \,b^{4} \left (\tan ^{4}\left (d x +c \right )\right )-20 B \,b^{4} \left (\tan ^{3}\left (d x +c \right )\right )+12 B \,b^{4} \left (\tan ^{5}\left (d x +c \right )\right )+30 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{4}+60 B \tan \left (d x +c \right ) a^{4}-60 B \,b^{4} d x +60 B a \,b^{3} \left (\tan ^{4}\left (d x +c \right )\right )+80 A a \,b^{3} \left (\tan ^{3}\left (d x +c \right )\right )+120 B \,a^{2} b^{2} \left (\tan ^{3}\left (d x +c \right )\right )+180 A \,a^{2} b^{2} \left (\tan ^{2}\left (d x +c \right )\right )+120 B \,a^{3} b \left (\tan ^{2}\left (d x +c \right )\right )-120 B a \,b^{3} \left (\tan ^{2}\left (d x +c \right )\right )-180 A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{2}-120 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} b +120 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a \,b^{3}}{60 d}\) | \(357\) |
risch | \(-\frac {12 i A \,a^{2} b^{2} c}{d}+\frac {8 i B a \,b^{3} c}{d}+\frac {2 i \left (60 A \,a^{3} b -80 A a \,b^{3}-120 B \,a^{2} b^{2}+15 B \,a^{4}+23 B \,b^{4}-270 i A \,a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-180 i B \,a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+240 i B a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-270 i A \,a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-90 i A \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-60 i B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+120 i B a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-90 i A \,a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-60 i B \,a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+120 i B a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-180 i B \,a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+240 i B a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+60 B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+90 B \,a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+60 B \,a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+15 B \,a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+90 B \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+140 B \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+70 B \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+45 B \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+30 i A \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+60 i A \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+30 i A \,b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+60 i A \,b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-420 B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-120 A a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}-180 B \,a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-360 A a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-540 B \,a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+60 A \,a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+240 A \,a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+360 A \,a^{3} b \,{\mathrm e}^{4 i \left (d x +c \right )}+240 A \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-440 A a \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-660 B \,a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-280 A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+i A \,a^{4} x -\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,b^{4}}{d}-4 A \,a^{3} b x +4 A a \,b^{3} x +6 B \,a^{2} b^{2} x +4 i B a \,b^{3} x +\frac {2 i a^{4} A c}{d}-B \,a^{4} x -B \,b^{4} x +\frac {2 i A \,b^{4} c}{d}-\frac {8 i B \,a^{3} b c}{d}+\frac {6 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A \,a^{2} b^{2}}{d}+\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B \,a^{3} b}{d}-\frac {4 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a \,b^{3}}{d}+i A \,b^{4} x -6 i A \,a^{2} b^{2} x -4 i B \,a^{3} b x\) | \(908\) |
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Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.08 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 \, B b^{4} \tan \left (d x + c\right )^{5} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{3} - 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} d x + 30 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{2} - 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (214) = 428\).
Time = 0.19 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.93 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\begin {cases} \frac {A a^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - 4 A a^{3} b x + \frac {4 A a^{3} b \tan {\left (c + d x \right )}}{d} - \frac {3 A a^{2} b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {3 A a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{d} + 4 A a b^{3} x + \frac {4 A a b^{3} \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 A a b^{3} \tan {\left (c + d x \right )}}{d} + \frac {A b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {A b^{4} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {A b^{4} \tan ^{2}{\left (c + d x \right )}}{2 d} - B a^{4} x + \frac {B a^{4} \tan {\left (c + d x \right )}}{d} - \frac {2 B a^{3} b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 B a^{3} b \tan ^{2}{\left (c + d x \right )}}{d} + 6 B a^{2} b^{2} x + \frac {2 B a^{2} b^{2} \tan ^{3}{\left (c + d x \right )}}{d} - \frac {6 B a^{2} b^{2} \tan {\left (c + d x \right )}}{d} + \frac {2 B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {B a b^{3} \tan ^{4}{\left (c + d x \right )}}{d} - \frac {2 B a b^{3} \tan ^{2}{\left (c + d x \right )}}{d} - B b^{4} x + \frac {B b^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} - \frac {B b^{4} \tan ^{3}{\left (c + d x \right )}}{3 d} + \frac {B b^{4} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + B \tan {\left (c \right )}\right ) \left (a + b \tan {\left (c \right )}\right )^{4} \tan {\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.30 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {12 \, B b^{4} \tan \left (d x + c\right )^{5} + 15 \, {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} + 20 \, {\left (6 \, B a^{2} b^{2} + 4 \, A a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )^{3} + 30 \, {\left (4 \, B a^{3} b + 6 \, A a^{2} b^{2} - 4 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )^{2} - 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} {\left (d x + c\right )} + 30 \, {\left (A a^{4} - 4 \, B a^{3} b - 6 \, A a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 60 \, {\left (B a^{4} + 4 \, A a^{3} b - 6 \, B a^{2} b^{2} - 4 \, A a b^{3} + B b^{4}\right )} \tan \left (d x + c\right )}{60 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4489 vs. \(2 (218) = 436\).
Time = 4.84 (sec) , antiderivative size = 4489, normalized size of antiderivative = 19.86 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
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Time = 8.52 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.11 \[ \int \tan (c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,a^4+B\,b^4+4\,A\,a^3\,b-2\,a\,b^2\,\left (2\,A\,b+3\,B\,a\right )\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {A\,b^4}{2}+2\,B\,a\,b^3-a^2\,b\,\left (3\,A\,b+2\,B\,a\right )\right )}{d}-x\,\left (B\,a^4+4\,A\,a^3\,b-6\,B\,a^2\,b^2-4\,A\,a\,b^3+B\,b^4\right )+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (\frac {A\,b^4}{4}+B\,a\,b^3\right )}{d}+\frac {\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {A\,a^4}{2}-2\,B\,a^3\,b-3\,A\,a^2\,b^2+2\,B\,a\,b^3+\frac {A\,b^4}{2}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,b^4}{3}-\frac {2\,a\,b^2\,\left (2\,A\,b+3\,B\,a\right )}{3}\right )}{d}+\frac {B\,b^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5\,d} \]
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