Integrand size = 39, antiderivative size = 64 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=\frac {a^2 A c \tan (e+f x)}{f}+\frac {a^2 (i A+B) c \tan ^2(e+f x)}{2 f}+\frac {i a^2 B c \tan ^3(e+f x)}{3 f} \]
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Time = 0.09 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3669, 45} \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=\frac {a^2 c (B+i A) \tan ^2(e+f x)}{2 f}+\frac {a^2 A c \tan (e+f x)}{f}+\frac {i a^2 B c \tan ^3(e+f x)}{3 f} \]
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Rule 45
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}(\int (a+i a x) (A+B x) \, dx,x,\tan (e+f x))}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (a A+a (i A+B) x+i a B x^2\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 A c \tan (e+f x)}{f}+\frac {a^2 (i A+B) c \tan ^2(e+f x)}{2 f}+\frac {i a^2 B c \tan ^3(e+f x)}{3 f} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.84 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=\frac {a^2 c \left (-2 B+6 A \tan (e+f x)+3 (i A+B) \tan ^2(e+f x)+2 i B \tan ^3(e+f x)\right )}{6 f} \]
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Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80
method | result | size |
derivativedivides | \(-\frac {i a^{2} c \left (-\frac {B \tan \left (f x +e \right )^{3}}{3}+\frac {\left (i B -A \right ) \tan \left (f x +e \right )^{2}}{2}+i \tan \left (f x +e \right ) A \right )}{f}\) | \(51\) |
default | \(-\frac {i a^{2} c \left (-\frac {B \tan \left (f x +e \right )^{3}}{3}+\frac {\left (i B -A \right ) \tan \left (f x +e \right )^{2}}{2}+i \tan \left (f x +e \right ) A \right )}{f}\) | \(51\) |
norman | \(\frac {a^{2} A c \tan \left (f x +e \right )}{f}+\frac {\left (i a^{2} c A +a^{2} c B \right ) \tan \left (f x +e \right )^{2}}{2 f}+\frac {i a^{2} B c \tan \left (f x +e \right )^{3}}{3 f}\) | \(64\) |
parallelrisch | \(\frac {2 i a^{2} B c \tan \left (f x +e \right )^{3}+3 i A \tan \left (f x +e \right )^{2} a^{2} c +3 B \tan \left (f x +e \right )^{2} a^{2} c +6 A \tan \left (f x +e \right ) a^{2} c}{6 f}\) | \(67\) |
risch | \(\frac {2 a^{2} c \left (6 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+6 B \,{\mathrm e}^{4 i \left (f x +e \right )}+9 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+3 B \,{\mathrm e}^{2 i \left (f x +e \right )}+3 i A +B \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{3}}\) | \(79\) |
parts | \(\frac {\left (i B \,a^{2} c +a^{2} c A \right ) \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {\left (i a^{2} c A +a^{2} c B \right ) \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {\left (i a^{2} c A +a^{2} c B \right ) \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+a^{2} c A x +\frac {i B \,a^{2} c \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(155\) |
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Time = 0.25 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.53 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=-\frac {2 \, {\left (6 \, {\left (-i \, A - B\right )} a^{2} c e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, {\left (-3 i \, A - B\right )} a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A - B\right )} a^{2} c\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (56) = 112\).
Time = 0.21 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.47 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=\frac {6 i A a^{2} c + 2 B a^{2} c + \left (18 i A a^{2} c e^{2 i e} + 6 B a^{2} c e^{2 i e}\right ) e^{2 i f x} + \left (12 i A a^{2} c e^{4 i e} + 12 B a^{2} c e^{4 i e}\right ) e^{4 i f x}}{3 f e^{6 i e} e^{6 i f x} + 9 f e^{4 i e} e^{4 i f x} + 9 f e^{2 i e} e^{2 i f x} + 3 f} \]
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Time = 0.44 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=-\frac {-2 i \, B a^{2} c \tan \left (f x + e\right )^{3} - 3 \, {\left (i \, A + B\right )} a^{2} c \tan \left (f x + e\right )^{2} - 6 \, A a^{2} c \tan \left (f x + e\right )}{6 \, f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (56) = 112\).
Time = 0.44 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.88 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=-\frac {2 \, {\left (-6 i \, A a^{2} c e^{\left (4 i \, f x + 4 i \, e\right )} - 6 \, B a^{2} c e^{\left (4 i \, f x + 4 i \, e\right )} - 9 i \, A a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, B a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, A a^{2} c - B a^{2} c\right )}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Time = 8.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.78 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x)) \, dx=\frac {a^2\,c\,\mathrm {tan}\left (e+f\,x\right )\,\left (6\,A+A\,\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+3\,B\,\mathrm {tan}\left (e+f\,x\right )+B\,{\mathrm {tan}\left (e+f\,x\right )}^2\,2{}\mathrm {i}\right )}{6\,f} \]
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