Integrand size = 26, antiderivative size = 140 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=8 a^4 (A-i B) x-\frac {8 a^4 (i A+B) \log (\cos (c+d x))}{d}-\frac {4 a^4 (A-i B) \tan (c+d x)}{d}+\frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+\frac {(i A+B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3608, 3559, 3558, 3556} \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 a^4 (A-i B) \tan (c+d x)}{d}-\frac {8 a^4 (B+i A) \log (\cos (c+d x))}{d}+8 a^4 x (A-i B)+\frac {(B+i A) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac {a (B+i A) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d} \]
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Rule 3556
Rule 3558
Rule 3559
Rule 3608
Rubi steps \begin{align*} \text {integral}& = \frac {B (a+i a \tan (c+d x))^4}{4 d}-(-A+i B) \int (a+i a \tan (c+d x))^4 \, dx \\ & = \frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+(2 a (A-i B)) \int (a+i a \tan (c+d x))^3 \, dx \\ & = \frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+\frac {(i A+B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (4 a^2 (A-i B)\right ) \int (a+i a \tan (c+d x))^2 \, dx \\ & = 8 a^4 (A-i B) x-\frac {4 a^4 (A-i B) \tan (c+d x)}{d}+\frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+\frac {(i A+B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\left (8 a^4 (i A+B)\right ) \int \tan (c+d x) \, dx \\ & = 8 a^4 (A-i B) x-\frac {8 a^4 (i A+B) \log (\cos (c+d x))}{d}-\frac {4 a^4 (A-i B) \tan (c+d x)}{d}+\frac {a (i A+B) (a+i a \tan (c+d x))^3}{3 d}+\frac {B (a+i a \tan (c+d x))^4}{4 d}+\frac {(i A+B) \left (a^2+i a^2 \tan (c+d x)\right )^2}{d} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.61 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B (a+i a \tan (c+d x))^4+\frac {4}{3} a^4 (i A+B) \left (4+24 \log (i+\tan (c+d x))+21 i \tan (c+d x)-6 \tan ^2(c+d x)-i \tan ^3(c+d x)\right )}{4 d} \]
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Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {a^{4} \left (-\frac {4 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}-2 i A \left (\tan ^{2}\left (d x +c \right )\right )+\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i B \tan \left (d x +c \right )-\frac {7 B \left (\tan ^{2}\left (d x +c \right )\right )}{2}-7 A \tan \left (d x +c \right )+\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(122\) |
default | \(\frac {a^{4} \left (-\frac {4 i B \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {B \left (\tan ^{4}\left (d x +c \right )\right )}{4}-2 i A \left (\tan ^{2}\left (d x +c \right )\right )+\frac {A \left (\tan ^{3}\left (d x +c \right )\right )}{3}+8 i B \tan \left (d x +c \right )-\frac {7 B \left (\tan ^{2}\left (d x +c \right )\right )}{2}-7 A \tan \left (d x +c \right )+\frac {\left (8 i A +8 B \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-8 i B +8 A \right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(122\) |
norman | \(\left (-8 i B \,a^{4}+8 A \,a^{4}\right ) x -\frac {\left (4 i A \,a^{4}+7 B \,a^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {\left (-8 i B \,a^{4}+7 A \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {B \,a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}+\frac {4 \left (i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{d}\) | \(142\) |
parallelrisch | \(\frac {-16 i B \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}+3 B \left (\tan ^{4}\left (d x +c \right )\right ) a^{4}-24 i A \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}+4 A \left (\tan ^{3}\left (d x +c \right )\right ) a^{4}-96 i B x \,a^{4} d +48 i A \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}+96 A x \,a^{4} d +96 i B \tan \left (d x +c \right ) a^{4}-42 B \left (\tan ^{2}\left (d x +c \right )\right ) a^{4}-84 A \tan \left (d x +c \right ) a^{4}+48 B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{4}}{12 d}\) | \(156\) |
risch | \(\frac {16 i a^{4} B c}{d}-\frac {16 a^{4} A c}{d}-\frac {4 a^{4} \left (18 i A \,{\mathrm e}^{6 i \left (d x +c \right )}+30 B \,{\mathrm e}^{6 i \left (d x +c \right )}+45 i A \,{\mathrm e}^{4 i \left (d x +c \right )}+63 B \,{\mathrm e}^{4 i \left (d x +c \right )}+38 i A \,{\mathrm e}^{2 i \left (d x +c \right )}+50 B \,{\mathrm e}^{2 i \left (d x +c \right )}+11 i A +14 B \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {8 a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{d}-\frac {8 i a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) A}{d}\) | \(170\) |
parts | \(A \,a^{4} x +\frac {\left (-4 i A \,a^{4}-6 B \,a^{4}\right ) \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}+\frac {\left (-4 i B \,a^{4}+A \,a^{4}\right ) \left (\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {\left (4 i A \,a^{4}+B \,a^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {\left (4 i B \,a^{4}-6 A \,a^{4}\right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B \,a^{4} \left (\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}\right )}{d}\) | \(198\) |
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Time = 0.25 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.62 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (6 \, {\left (3 i \, A + 5 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 9 \, {\left (5 i \, A + 7 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (19 i \, A + 25 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (11 i \, A + 14 \, B\right )} a^{4} + 6 \, {\left ({\left (i \, A + B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, {\left (i \, A + B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, {\left (i \, A + B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, {\left (i \, A + B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (i \, A + B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (116) = 232\).
Time = 0.44 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=- \frac {8 i a^{4} \left (A - i B\right ) \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac {- 44 i A a^{4} - 56 B a^{4} + \left (- 152 i A a^{4} e^{2 i c} - 200 B a^{4} e^{2 i c}\right ) e^{2 i d x} + \left (- 180 i A a^{4} e^{4 i c} - 252 B a^{4} e^{4 i c}\right ) e^{4 i d x} + \left (- 72 i A a^{4} e^{6 i c} - 120 B a^{4} e^{6 i c}\right ) e^{6 i d x}}{3 d e^{8 i c} e^{8 i d x} + 12 d e^{6 i c} e^{6 i d x} + 18 d e^{4 i c} e^{4 i d x} + 12 d e^{2 i c} e^{2 i d x} + 3 d} \]
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Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {3 \, B a^{4} \tan \left (d x + c\right )^{4} + 4 \, {\left (A - 4 i \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 6 \, {\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{2} + 96 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a^{4} - 48 \, {\left (-i \, A - B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 12 \, {\left (7 \, A - 8 i \, B\right )} a^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (120) = 240\).
Time = 0.54 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.91 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=-\frac {4 \, {\left (6 i \, A a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 6 \, B a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 36 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 36 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 24 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 i \, A a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 30 \, B a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 45 i \, A a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 63 \, B a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 38 i \, A a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 50 \, B a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 i \, A a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 6 \, B a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 11 i \, A a^{4} + 14 \, B a^{4}\right )}}{3 \, {\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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Time = 7.21 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx=\frac {B\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (8\,B\,a^4+A\,a^4\,8{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (\frac {a^4\,\left (A-B\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2}+\frac {B\,a^4}{2}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )}{2}\right )}{d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-3\,a^4\,\left (A-B\,1{}\mathrm {i}\right )+a^4\,\left (B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}+B\,a^4\,1{}\mathrm {i}+a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{d}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {B\,a^4\,1{}\mathrm {i}}{3}+\frac {a^4\,\left (3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}\right )}{d} \]
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