Integrand size = 41, antiderivative size = 132 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {a^3 (i A+B) c^4 (1-i \tan (e+f x))^4}{f}-\frac {4 a^3 (i A+2 B) c^4 (1-i \tan (e+f x))^5}{5 f}+\frac {a^3 (i A+5 B) c^4 (1-i \tan (e+f x))^6}{6 f}-\frac {a^3 B c^4 (1-i \tan (e+f x))^7}{7 f} \]
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Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 78} \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {a^3 c^4 (5 B+i A) (1-i \tan (e+f x))^6}{6 f}-\frac {4 a^3 c^4 (2 B+i A) (1-i \tan (e+f x))^5}{5 f}+\frac {a^3 c^4 (B+i A) (1-i \tan (e+f x))^4}{f}-\frac {a^3 B c^4 (1-i \tan (e+f x))^7}{7 f} \]
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Rule 78
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^3 \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^3-\frac {4 a^2 (A-2 i B) (c-i c x)^4}{c}+\frac {a^2 (A-5 i B) (c-i c x)^5}{c^2}+\frac {i a^2 B (c-i c x)^6}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^3 (i A+B) c^4 (1-i \tan (e+f x))^4}{f}-\frac {4 a^3 (i A+2 B) c^4 (1-i \tan (e+f x))^5}{5 f}+\frac {a^3 (i A+5 B) c^4 (1-i \tan (e+f x))^6}{6 f}-\frac {a^3 B c^4 (1-i \tan (e+f x))^7}{7 f} \\ \end{align*}
Time = 2.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {a^3 c^4 \left (7 i A+29 B+210 A \tan (e+f x)+105 (-i A+B) \tan ^2(e+f x)+70 (2 A-i B) \tan ^3(e+f x)+105 (-i A+B) \tan ^4(e+f x)+42 (A-2 i B) \tan ^5(e+f x)+35 (-i A+B) \tan ^6(e+f x)-30 i B \tan ^7(e+f x)\right )}{210 f} \]
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Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {16 c^{4} a^{3} \left (105 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+105 B \,{\mathrm e}^{6 i \left (f x +e \right )}+147 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-21 B \,{\mathrm e}^{4 i \left (f x +e \right )}+49 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-7 B \,{\mathrm e}^{2 i \left (f x +e \right )}+7 i A -B \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{7}}\) | \(108\) |
derivativedivides | \(-\frac {i c^{4} a^{3} \left (\frac {B \tan \left (f x +e \right )^{7}}{7}+\frac {\left (i B +A \right ) \tan \left (f x +e \right )^{6}}{6}+\frac {\left (-4 B -2 i A +3 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{5}}{5}+\frac {\left (-4 A +3 i \left (-2 i A -B \right )+5 i B \right ) \tan \left (f x +e \right )^{4}}{4}+\frac {\left (3 i A +3 B -i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{3}}{3}+\frac {\left (3 A -i \left (-2 i A -B \right )\right ) \tan \left (f x +e \right )^{2}}{2}+i \tan \left (f x +e \right ) A \right )}{f}\) | \(159\) |
default | \(-\frac {i c^{4} a^{3} \left (\frac {B \tan \left (f x +e \right )^{7}}{7}+\frac {\left (i B +A \right ) \tan \left (f x +e \right )^{6}}{6}+\frac {\left (-4 B -2 i A +3 i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{5}}{5}+\frac {\left (-4 A +3 i \left (-2 i A -B \right )+5 i B \right ) \tan \left (f x +e \right )^{4}}{4}+\frac {\left (3 i A +3 B -i \left (-2 i B +A \right )\right ) \tan \left (f x +e \right )^{3}}{3}+\frac {\left (3 A -i \left (-2 i A -B \right )\right ) \tan \left (f x +e \right )^{2}}{2}+i \tan \left (f x +e \right ) A \right )}{f}\) | \(159\) |
norman | \(\frac {A \,a^{3} c^{4} \tan \left (f x +e \right )}{f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \tan \left (f x +e \right )^{4}}{2 f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \tan \left (f x +e \right )^{6}}{6 f}+\frac {\left (-2 i B \,a^{3} c^{4}+A \,a^{3} c^{4}\right ) \tan \left (f x +e \right )^{5}}{5 f}+\frac {\left (-i B \,a^{3} c^{4}+2 A \,a^{3} c^{4}\right ) \tan \left (f x +e \right )^{3}}{3 f}-\frac {i B \,a^{3} c^{4} \tan \left (f x +e \right )^{7}}{7 f}\) | \(201\) |
parallelrisch | \(-\frac {30 i B \,a^{3} c^{4} \tan \left (f x +e \right )^{7}+35 i A \tan \left (f x +e \right )^{6} a^{3} c^{4}+84 i B \tan \left (f x +e \right )^{5} a^{3} c^{4}-35 B \tan \left (f x +e \right )^{6} a^{3} c^{4}+105 i A \tan \left (f x +e \right )^{4} a^{3} c^{4}-42 A \tan \left (f x +e \right )^{5} a^{3} c^{4}+70 i B \tan \left (f x +e \right )^{3} a^{3} c^{4}-105 B \tan \left (f x +e \right )^{4} a^{3} c^{4}+105 i A \tan \left (f x +e \right )^{2} a^{3} c^{4}-140 A \tan \left (f x +e \right )^{3} a^{3} c^{4}-105 B \tan \left (f x +e \right )^{2} a^{3} c^{4}-210 A \tan \left (f x +e \right ) a^{3} c^{4}}{210 f}\) | \(215\) |
parts | \(\frac {\left (-3 i A \,a^{3} c^{4}+3 B \,a^{3} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-3 i A \,a^{3} c^{4}+3 B \,a^{3} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-3 i B \,a^{3} c^{4}+A \,a^{3} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{5}}{5}-\frac {\tan \left (f x +e \right )^{3}}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-3 i B \,a^{3} c^{4}+3 A \,a^{3} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (-i A \,a^{3} c^{4}+B \,a^{3} c^{4}\right ) \left (\frac {\tan \left (f x +e \right )^{6}}{6}-\frac {\tan \left (f x +e \right )^{4}}{4}+\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {\left (-i B \,a^{3} c^{4}+3 A \,a^{3} c^{4}\right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+A \,a^{3} c^{4} x -\frac {i B \,a^{3} c^{4} \left (\frac {\tan \left (f x +e \right )^{7}}{7}-\frac {\tan \left (f x +e \right )^{5}}{5}+\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(427\) |
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Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.29 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {16 \, {\left (105 \, {\left (-i \, A - B\right )} a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, {\left (-7 i \, A + B\right )} a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, {\left (-7 i \, A + B\right )} a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-7 i \, A + B\right )} a^{3} c^{4}\right )}}{105 \, {\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (107) = 214\).
Time = 0.64 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.17 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=\frac {112 i A a^{3} c^{4} - 16 B a^{3} c^{4} + \left (784 i A a^{3} c^{4} e^{2 i e} - 112 B a^{3} c^{4} e^{2 i e}\right ) e^{2 i f x} + \left (2352 i A a^{3} c^{4} e^{4 i e} - 336 B a^{3} c^{4} e^{4 i e}\right ) e^{4 i f x} + \left (1680 i A a^{3} c^{4} e^{6 i e} + 1680 B a^{3} c^{4} e^{6 i e}\right ) e^{6 i f x}}{105 f e^{14 i e} e^{14 i f x} + 735 f e^{12 i e} e^{12 i f x} + 2205 f e^{10 i e} e^{10 i f x} + 3675 f e^{8 i e} e^{8 i f x} + 3675 f e^{6 i e} e^{6 i f x} + 2205 f e^{4 i e} e^{4 i f x} + 735 f e^{2 i e} e^{2 i f x} + 105 f} \]
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Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {30 i \, B a^{3} c^{4} \tan \left (f x + e\right )^{7} + 35 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{6} - 42 \, {\left (A - 2 i \, B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{5} + 105 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{4} - 70 \, {\left (2 \, A - i \, B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{3} + 105 \, {\left (i \, A - B\right )} a^{3} c^{4} \tan \left (f x + e\right )^{2} - 210 \, A a^{3} c^{4} \tan \left (f x + e\right )}{210 \, f} \]
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Time = 0.96 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.63 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {16 \, {\left (-105 i \, A a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 105 \, B a^{3} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} - 147 i \, A a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 21 \, B a^{3} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} - 49 i \, A a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 7 \, B a^{3} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 7 i \, A a^{3} c^{4} + B a^{3} c^{4}\right )}}{105 \, {\left (f e^{\left (14 i \, f x + 14 i \, e\right )} + 7 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 21 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 35 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 35 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 21 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 7 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Time = 8.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx=-\frac {\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (2\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{5}-A\,a^3\,c^4\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (B+A\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6}+\frac {B\,a^3\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^7\,1{}\mathrm {i}}{7}}{f} \]
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