Integrand size = 36, antiderivative size = 104 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\frac {(A-i B) (c-i d) x}{4 a^2}+\frac {B (c+3 i d)+A (i c+d)}{4 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c+i d)}{4 f (a+i a \tan (e+f x))^2} \]
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Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3671, 3607, 8} \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\frac {A (d+i c)+B (c+3 i d)}{4 a^2 f (1+i \tan (e+f x))}+\frac {x (A-i B) (c-i d)}{4 a^2}+\frac {(-B+i A) (c+i d)}{4 f (a+i a \tan (e+f x))^2} \]
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Rule 8
Rule 3607
Rule 3671
Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) (c+i d)}{4 f (a+i a \tan (e+f x))^2}-\frac {i \int \frac {a (B (c+i d)+A (i c+d))+2 a B d \tan (e+f x)}{a+i a \tan (e+f x)} \, dx}{2 a^2} \\ & = \frac {B (c+3 i d)+A (i c+d)}{4 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c+i d)}{4 f (a+i a \tan (e+f x))^2}+\frac {((A-i B) (c-i d)) \int 1 \, dx}{4 a^2} \\ & = \frac {(A-i B) (c-i d) x}{4 a^2}+\frac {B (c+3 i d)+A (i c+d)}{4 a^2 f (1+i \tan (e+f x))}+\frac {(i A-B) (c+i d)}{4 f (a+i a \tan (e+f x))^2} \\ \end{align*}
Time = 2.06 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\frac {(A-i B) (c-i d) \arctan (\tan (e+f x))+\frac {(A+i B) (-i c+d)}{(-i+\tan (e+f x))^2}+\frac {A c-i B c-i A d+3 B d}{-i+\tan (e+f x)}}{4 a^2 f} \]
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Time = 0.24 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.48
method | result | size |
risch | \(-\frac {i x A d}{4 a^{2}}-\frac {i x B c}{4 a^{2}}+\frac {x c A}{4 a^{2}}-\frac {x B d}{4 a^{2}}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} c A}{4 f \,a^{2}}+\frac {i {\mathrm e}^{-2 i \left (f x +e \right )} B d}{4 f \,a^{2}}-\frac {{\mathrm e}^{-4 i \left (f x +e \right )} A d}{16 f \,a^{2}}-\frac {{\mathrm e}^{-4 i \left (f x +e \right )} B c}{16 f \,a^{2}}+\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} c A}{16 f \,a^{2}}-\frac {i {\mathrm e}^{-4 i \left (f x +e \right )} B d}{16 f \,a^{2}}\) | \(154\) |
derivativedivides | \(\frac {i B d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i A c}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i A d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i B c}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A \arctan \left (\tan \left (f x +e \right )\right ) c}{4 f \,a^{2}}-\frac {B \arctan \left (\tan \left (f x +e \right )\right ) d}{4 f \,a^{2}}-\frac {i c B \arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{2}}-\frac {i A d \arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{2}}+\frac {c A}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 B d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c B}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(244\) |
default | \(\frac {i B d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i A c}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}-\frac {i A d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}-\frac {i B c}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A \arctan \left (\tan \left (f x +e \right )\right ) c}{4 f \,a^{2}}-\frac {B \arctan \left (\tan \left (f x +e \right )\right ) d}{4 f \,a^{2}}-\frac {i c B \arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{2}}-\frac {i A d \arctan \left (\tan \left (f x +e \right )\right )}{4 f \,a^{2}}+\frac {c A}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {3 B d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )}+\frac {A d}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}+\frac {c B}{4 f \,a^{2} \left (-i+\tan \left (f x +e \right )\right )^{2}}\) | \(244\) |
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Time = 0.24 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\frac {{\left (4 \, {\left ({\left (A - i \, B\right )} c - {\left (i \, A + B\right )} d\right )} f x e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (i \, A - B\right )} c - {\left (A + i \, B\right )} d - 4 \, {\left (-i \, A c - i \, B d\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{16 \, a^{2} f} \]
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Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.85 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\begin {cases} \frac {\left (\left (16 i A a^{2} c f e^{4 i e} + 16 i B a^{2} d f e^{4 i e}\right ) e^{- 2 i f x} + \left (4 i A a^{2} c f e^{2 i e} - 4 A a^{2} d f e^{2 i e} - 4 B a^{2} c f e^{2 i e} - 4 i B a^{2} d f e^{2 i e}\right ) e^{- 4 i f x}\right ) e^{- 6 i e}}{64 a^{4} f^{2}} & \text {for}\: a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {A c - i A d - i B c - B d}{4 a^{2}} + \frac {\left (A c e^{4 i e} + 2 A c e^{2 i e} + A c - i A d e^{4 i e} + i A d - i B c e^{4 i e} + i B c - B d e^{4 i e} + 2 B d e^{2 i e} - B d\right ) e^{- 4 i e}}{4 a^{2}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (A c - i A d - i B c - B d\right )}{4 a^{2}} \]
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Exception generated. \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (80) = 160\).
Time = 0.49 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.78 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=-\frac {\frac {2 \, {\left (-i \, A c - B c - A d + i \, B d\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{2}} + \frac {2 \, {\left (i \, A c + B c + A d - i \, B d\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{2}} + \frac {-3 i \, A c \tan \left (f x + e\right )^{2} - 3 \, B c \tan \left (f x + e\right )^{2} - 3 \, A d \tan \left (f x + e\right )^{2} + 3 i \, B d \tan \left (f x + e\right )^{2} - 10 \, A c \tan \left (f x + e\right ) + 10 i \, B c \tan \left (f x + e\right ) + 10 i \, A d \tan \left (f x + e\right ) - 6 \, B d \tan \left (f x + e\right ) + 11 i \, A c + 3 \, B c + 3 \, A d + 5 i \, B d}{a^{2} {\left (\tan \left (f x + e\right ) - i\right )}^{2}}}{16 \, f} \]
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Time = 8.73 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.53 \[ \int \frac {(A+B \tan (e+f x)) (c+d \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx=-\frac {B\,d\,f\,x-A\,c\,f\,x+A\,d\,f\,x\,1{}\mathrm {i}+B\,c\,f\,x\,1{}\mathrm {i}}{4\,a^2\,f}+\frac {\left (A\,c+3\,B\,d-A\,d\,1{}\mathrm {i}-B\,c\,1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^3+\left (2\,A\,d+2\,B\,c+B\,d\,4{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\left (3\,A\,c+B\,d+A\,d\,1{}\mathrm {i}+B\,c\,1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )+A\,c\,2{}\mathrm {i}+B\,d\,2{}\mathrm {i}}{f\,\left (4\,a^2\,{\mathrm {tan}\left (e+f\,x\right )}^4+8\,a^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+4\,a^2\right )} \]
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