Integrand size = 41, antiderivative size = 95 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\frac {2 a^2 (A-i B)}{5 c^5 f (i+\tan (e+f x))^5}+\frac {a^2 (i A+3 B)}{4 c^5 f (i+\tan (e+f x))^4}+\frac {i a^2 B}{3 c^5 f (i+\tan (e+f x))^3} \]
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Time = 0.17 (sec) , antiderivative size = 95, 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} \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\frac {a^2 (3 B+i A)}{4 c^5 f (\tan (e+f x)+i)^4}+\frac {2 a^2 (A-i B)}{5 c^5 f (\tan (e+f x)+i)^5}+\frac {i a^2 B}{3 c^5 f (\tan (e+f x)+i)^3} \]
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Rule 78
Rule 3669
Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^6} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {(a c) \text {Subst}\left (\int \left (-\frac {2 a (A-i B)}{c^6 (i+x)^6}-\frac {i a (A-3 i B)}{c^6 (i+x)^5}-\frac {i a B}{c^6 (i+x)^4}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {2 a^2 (A-i B)}{5 c^5 f (i+\tan (e+f x))^5}+\frac {a^2 (i A+3 B)}{4 c^5 f (i+\tan (e+f x))^4}+\frac {i a^2 B}{3 c^5 f (i+\tan (e+f x))^3} \\ \end{align*}
Time = 5.36 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\frac {a^2 \left (9 A+i B+5 (3 i A+B) \tan (e+f x)+20 i B \tan ^2(e+f x)\right )}{60 c^5 f (i+\tan (e+f x))^5} \]
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Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {2 i B -2 A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {i B}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-i A -3 B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{5}}\) | \(69\) |
default | \(\frac {a^{2} \left (-\frac {2 i B -2 A}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}+\frac {i B}{3 \left (i+\tan \left (f x +e \right )\right )^{3}}-\frac {-i A -3 B}{4 \left (i+\tan \left (f x +e \right )\right )^{4}}\right )}{f \,c^{5}}\) | \(69\) |
risch | \(-\frac {a^{2} {\mathrm e}^{10 i \left (f x +e \right )} B}{80 c^{5} f}-\frac {i a^{2} {\mathrm e}^{10 i \left (f x +e \right )} A}{80 c^{5} f}-\frac {{\mathrm e}^{8 i \left (f x +e \right )} B \,a^{2}}{64 c^{5} f}-\frac {3 i {\mathrm e}^{8 i \left (f x +e \right )} A \,a^{2}}{64 c^{5} f}+\frac {{\mathrm e}^{6 i \left (f x +e \right )} B \,a^{2}}{48 c^{5} f}-\frac {i {\mathrm e}^{6 i \left (f x +e \right )} A \,a^{2}}{16 c^{5} f}+\frac {a^{2} {\mathrm e}^{4 i \left (f x +e \right )} B}{32 c^{5} f}-\frac {i a^{2} {\mathrm e}^{4 i \left (f x +e \right )} A}{32 c^{5} f}\) | \(174\) |
norman | \(\frac {\frac {A \,a^{2} \tan \left (f x +e \right )}{c f}+\frac {-9 i A \,a^{2}+B \,a^{2}}{60 c f}-\frac {\left (-7 i B \,a^{2}+12 A \,a^{2}\right ) \tan \left (f x +e \right )^{3}}{3 c f}+\frac {\left (i A \,a^{2}+7 B \,a^{2}\right ) \tan \left (f x +e \right )^{6}}{4 c f}+\frac {7 \left (-8 i B \,a^{2}+3 A \,a^{2}\right ) \tan \left (f x +e \right )^{5}}{15 c f}+\frac {\left (33 i A \,a^{2}+7 B \,a^{2}\right ) \tan \left (f x +e \right )^{2}}{12 c f}-\frac {\left (39 i A \,a^{2}+49 B \,a^{2}\right ) \tan \left (f x +e \right )^{4}}{12 c f}+\frac {i B \,a^{2} \tan \left (f x +e \right )^{7}}{3 c f}}{\left (1+\tan \left (f x +e \right )^{2}\right )^{5} c^{4}}\) | \(227\) |
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none
Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=-\frac {12 \, {\left (i \, A + B\right )} a^{2} e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, {\left (3 i \, A + B\right )} a^{2} e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, {\left (3 i \, A - B\right )} a^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 30 \, {\left (i \, A - B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )}}{960 \, c^{5} f} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (76) = 152\).
Time = 0.41 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.49 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\begin {cases} \frac {\left (- 245760 i A a^{2} c^{15} f^{3} e^{4 i e} + 245760 B a^{2} c^{15} f^{3} e^{4 i e}\right ) e^{4 i f x} + \left (- 491520 i A a^{2} c^{15} f^{3} e^{6 i e} + 163840 B a^{2} c^{15} f^{3} e^{6 i e}\right ) e^{6 i f x} + \left (- 368640 i A a^{2} c^{15} f^{3} e^{8 i e} - 122880 B a^{2} c^{15} f^{3} e^{8 i e}\right ) e^{8 i f x} + \left (- 98304 i A a^{2} c^{15} f^{3} e^{10 i e} - 98304 B a^{2} c^{15} f^{3} e^{10 i e}\right ) e^{10 i f x}}{7864320 c^{20} f^{4}} & \text {for}\: c^{20} f^{4} \neq 0 \\\frac {x \left (A a^{2} e^{10 i e} + 3 A a^{2} e^{8 i e} + 3 A a^{2} e^{6 i e} + A a^{2} e^{4 i e} - i B a^{2} e^{10 i e} - i B a^{2} e^{8 i e} + i B a^{2} e^{6 i e} + i B a^{2} e^{4 i e}\right )}{8 c^{5}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (77) = 154\).
Time = 1.13 (sec) , antiderivative size = 292, normalized size of antiderivative = 3.07 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=-\frac {2 \, {\left (15 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 45 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 15 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 150 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 10 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 225 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 55 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 306 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 24 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 225 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 55 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 150 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 10 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 45 i \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{15 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{10}} \]
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Time = 8.65 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.14 \[ \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^5} \, dx=\frac {\frac {a^2\,\left (9\,A+B\,1{}\mathrm {i}\right )}{60}+\frac {a^2\,\mathrm {tan}\left (e+f\,x\right )\,\left (5\,B+A\,15{}\mathrm {i}\right )}{60}+\frac {B\,a^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}}{3}}{c^5\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^5+{\mathrm {tan}\left (e+f\,x\right )}^4\,5{}\mathrm {i}-10\,{\mathrm {tan}\left (e+f\,x\right )}^3-{\mathrm {tan}\left (e+f\,x\right )}^2\,10{}\mathrm {i}+5\,\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )} \]
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