Integrand size = 41, antiderivative size = 62 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 A c^2 \tan ^3(e+f x)}{3 f} \]
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Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3669, 74, 655} \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {a^2 A c^2 \tan ^3(e+f x)}{3 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f} \]
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Rule 74
Rule 655
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}(\int (a+i a x) (A+B x) (c-i c x) \, dx,x,\tan (e+f x))}{f} \\ & = \frac {(a c) \text {Subst}\left (\int (A+B x) \left (a c+a c x^2\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {(a A c) \text {Subst}\left (\int \left (a c+a c x^2\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {a^2 A c^2 \tan (e+f x)}{f}+\frac {a^2 A c^2 \tan ^3(e+f x)}{3 f} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {a^2 B c^2 \sec ^4(e+f x)}{4 f}+\frac {a^2 A c^2 \left (\tan (e+f x)+\frac {1}{3} \tan ^3(e+f x)\right )}{f} \]
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Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(\frac {c^{2} a^{2} \left (\frac {B \tan \left (f x +e \right )^{4}}{4}+\frac {A \tan \left (f x +e \right )^{3}}{3}+\frac {B \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(53\) |
default | \(\frac {c^{2} a^{2} \left (\frac {B \tan \left (f x +e \right )^{4}}{4}+\frac {A \tan \left (f x +e \right )^{3}}{3}+\frac {B \tan \left (f x +e \right )^{2}}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(53\) |
risch | \(\frac {4 c^{2} a^{2} \left (3 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+3 B \,{\mathrm e}^{4 i \left (f x +e \right )}+4 i A \,{\mathrm e}^{2 i \left (f x +e \right )}+i A \right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\) | \(68\) |
parallelrisch | \(\frac {3 B \,a^{2} c^{2} \tan \left (f x +e \right )^{4}+4 A \,a^{2} c^{2} \tan \left (f x +e \right )^{3}+6 B \,a^{2} c^{2} \tan \left (f x +e \right )^{2}+12 A \,a^{2} c^{2} \tan \left (f x +e \right )}{12 f}\) | \(73\) |
norman | \(\frac {a^{2} A \,c^{2} \tan \left (f x +e \right )}{f}+\frac {B \,a^{2} c^{2} \tan \left (f x +e \right )^{2}}{2 f}+\frac {B \,a^{2} c^{2} \tan \left (f x +e \right )^{4}}{4 f}+\frac {a^{2} A \,c^{2} \tan \left (f x +e \right )^{3}}{3 f}\) | \(79\) |
parts | \(A \,a^{2} c^{2} x +\frac {A \,a^{2} c^{2} \left (\frac {\tan \left (f x +e \right )^{3}}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {B \,a^{2} c^{2} \ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2 f}+\frac {B \,a^{2} c^{2} \left (\frac {\tan \left (f x +e \right )^{4}}{4}-\frac {\tan \left (f x +e \right )^{2}}{2}+\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}+\frac {2 A \,a^{2} c^{2} \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {2 B \,a^{2} c^{2} \left (\frac {\tan \left (f x +e \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (f x +e \right )^{2}\right )}{2}\right )}{f}\) | \(180\) |
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Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.69 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=-\frac {4 \, {\left (3 \, {\left (-i \, A - B\right )} a^{2} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 4 i \, A a^{2} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, A a^{2} c^{2}\right )}}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.58 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {16 i A a^{2} c^{2} e^{2 i e} e^{2 i f x} + 4 i A a^{2} c^{2} + \left (12 i A a^{2} c^{2} e^{4 i e} + 12 B a^{2} c^{2} e^{4 i e}\right ) e^{4 i f x}}{3 f e^{8 i e} e^{8 i f x} + 12 f e^{6 i e} e^{6 i f x} + 18 f e^{4 i e} e^{4 i f x} + 12 f e^{2 i e} e^{2 i f x} + 3 f} \]
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none
Time = 0.35 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.16 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {3 \, B a^{2} c^{2} \tan \left (f x + e\right )^{4} + 4 \, A a^{2} c^{2} \tan \left (f x + e\right )^{3} + 6 \, B a^{2} c^{2} \tan \left (f x + e\right )^{2} + 12 \, A a^{2} c^{2} \tan \left (f x + e\right )}{12 \, f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (58) = 116\).
Time = 0.54 (sec) , antiderivative size = 391, normalized size of antiderivative = 6.31 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {3 \, B a^{2} c^{2} \tan \left (f x\right )^{4} \tan \left (e\right )^{4} - 12 \, A a^{2} c^{2} \tan \left (f x\right )^{4} \tan \left (e\right )^{3} - 12 \, A a^{2} c^{2} \tan \left (f x\right )^{3} \tan \left (e\right )^{4} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{4} \tan \left (e\right )^{2} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right )^{4} - 4 \, A a^{2} c^{2} \tan \left (f x\right )^{4} \tan \left (e\right ) + 24 \, A a^{2} c^{2} \tan \left (f x\right )^{3} \tan \left (e\right )^{2} + 24 \, A a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right )^{3} - 4 \, A a^{2} c^{2} \tan \left (f x\right ) \tan \left (e\right )^{4} + 3 \, B a^{2} c^{2} \tan \left (f x\right )^{4} + 12 \, B a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right )^{2} + 3 \, B a^{2} c^{2} \tan \left (e\right )^{4} + 4 \, A a^{2} c^{2} \tan \left (f x\right )^{3} - 24 \, A a^{2} c^{2} \tan \left (f x\right )^{2} \tan \left (e\right ) - 24 \, A a^{2} c^{2} \tan \left (f x\right ) \tan \left (e\right )^{2} + 4 \, A a^{2} c^{2} \tan \left (e\right )^{3} + 6 \, B a^{2} c^{2} \tan \left (f x\right )^{2} + 6 \, B a^{2} c^{2} \tan \left (e\right )^{2} + 12 \, A a^{2} c^{2} \tan \left (f x\right ) + 12 \, A a^{2} c^{2} \tan \left (e\right ) + 3 \, B a^{2} c^{2}}{12 \, {\left (f \tan \left (f x\right )^{4} \tan \left (e\right )^{4} - 4 \, f \tan \left (f x\right )^{3} \tan \left (e\right )^{3} + 6 \, f \tan \left (f x\right )^{2} \tan \left (e\right )^{2} - 4 \, f \tan \left (f x\right ) \tan \left (e\right ) + f\right )}} \]
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Time = 8.74 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.32 \[ \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^2 \, dx=\frac {a^2\,c^2\,\sin \left (e+f\,x\right )\,\left (12\,A\,{\cos \left (e+f\,x\right )}^3+6\,B\,{\cos \left (e+f\,x\right )}^2\,\sin \left (e+f\,x\right )+4\,A\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^2+3\,B\,{\sin \left (e+f\,x\right )}^3\right )}{12\,f\,{\cos \left (e+f\,x\right )}^4} \]
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