Integrand size = 28, antiderivative size = 234 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \]
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Time = 0.76 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3640, 3677, 3612, 3611} \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {c^2+4 i c d-7 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {x \left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right )}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^4 (d+i c)}+\frac {-3 d+i c}{8 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac {1}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
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Rule 3611
Rule 3612
Rule 3640
Rule 3677
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac {\int \frac {-3 a (i c-2 d)-3 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))} \, dx}{6 a^2 (i c-d)} \\ & = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac {\int \frac {-6 a^2 \left (c^2+3 i c d-4 d^2\right )-6 a^2 (c+3 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))} \, dx}{24 a^4 (c+i d)^2} \\ & = -\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {\int \frac {6 a^3 \left (i c^3-4 c^2 d-7 i c d^2+8 d^3\right )+6 a^3 d \left (i c^2-4 c d-7 i d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{48 a^6 (i c-d)^3} \\ & = \frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {d^4 \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (c+i d)^4 (i c+d)} \\ & = \frac {\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-7 d^4\right ) x}{8 a^3 (c-i d) (c+i d)^4}+\frac {d^4 \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c+i d)^4 (i c+d) f}-\frac {1}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac {i c-3 d}{8 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac {c^2+4 i c d-7 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )} \\ \end{align*}
Time = 1.67 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {\frac {3 i \left (\left (c^4+4 i c^3 d-6 c^2 d^2-4 i c d^3-15 d^4\right ) \log (i-\tan (e+f x))-(c+i d)^4 \log (i+\tan (e+f x))+16 d^4 \log (c+d \tan (e+f x))\right )}{(c-i d) (c+i d)^2}+\frac {8 (c+i d)}{(-i+\tan (e+f x))^3}+\frac {6 i (c+3 i d)}{(-i+\tan (e+f x))^2}-\frac {6 \left (c^2+4 i c d-7 d^2\right )}{(c+i d) (-i+\tan (e+f x))}}{48 a^3 (c+i d)^2 f} \]
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Time = 0.69 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.58
method | result | size |
risch | \(-\frac {x}{8 a^{3} \left (i d -c \right )}-\frac {5 \,{\mathrm e}^{-2 i \left (f x +e \right )} c d}{8 a^{3} \left (i d +c \right )^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {11 i {\mathrm e}^{-2 i \left (f x +e \right )} d^{2}}{16 a^{3} \left (i d +c \right )^{3} f}-\frac {5 \,{\mathrm e}^{-4 i \left (f x +e \right )} d}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c}{32 a^{3} \left (i d +c \right )^{2} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{48 a^{3} \left (i d +c \right ) f}-\frac {2 d^{4} x}{a^{3} \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {2 d^{4} e}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}-\frac {i d^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {i d +c}{i d -c}\right )}{a^{3} f \left (3 i c^{4} d +2 i c^{2} d^{3}-i d^{5}+c^{5}-2 c^{3} d^{2}-3 c \,d^{4}\right )}\) | \(370\) |
derivativedivides | \(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\) | \(702\) |
default | \(\frac {i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \,a^{3} \left (i d -c \right ) \left (i d +c \right )^{4}}+\frac {11 i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {5 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {15 \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {11 \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {7 i c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {5 i \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{16 f \,a^{3} \left (i d +c \right )^{4}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c \,d^{2}}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {15 i \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{16 f \,a^{3} \left (i d +c \right )^{4}}+\frac {i d^{3}}{6 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {c^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {11 c \,d^{2}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{2} d}{2 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {5 i c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )}+\frac {5 c^{2} d}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 d^{3}}{8 f \,a^{3} \left (i d +c \right )^{4} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{32 f \,a^{3} \left (i d +c \right )^{4}}-\frac {i \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3} \left (16 i d -16 c \right )}-\frac {\arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3} \left (16 i d -16 c \right )}\) | \(702\) |
norman | \(\frac {\frac {-16 i c d -5 c^{2}+17 d^{2}}{12 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {\left (20 i c d +7 c^{2}-17 d^{2}\right ) \tan \left (f x +e \right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{8 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (5 i c d +c^{2}-7 d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}-\frac {i d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{2 a f \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (c^{2}+5 d^{2}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{4 a f \left (i c^{3}-3 i c \,d^{2}-3 c^{2} d +d^{3}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {3 \left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}+\frac {\left (4 i c^{3} d -4 i c \,d^{3}+c^{4}-6 c^{2} d^{2}-7 d^{4}\right ) x \left (\tan ^{6}\left (f x +e \right )\right )}{8 \left (c^{2}+d^{2}\right ) a \left (3 i c^{2} d -i d^{3}+c^{3}-3 c \,d^{2}\right )}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}-\frac {d^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 a^{3} f \left (i c^{5}-2 i c^{3} d^{2}-3 i c \,d^{4}-3 c^{4} d -2 c^{2} d^{3}+d^{5}\right )}\) | \(763\) |
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Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=-\frac {{\left (96 \, d^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) - 2 \, c^{4} - 4 i \, c^{3} d - 4 i \, c d^{3} + 2 \, d^{4} - 12 \, {\left (-i \, c^{4} + 4 \, c^{3} d + 6 i \, c^{2} d^{2} - 4 \, c d^{3} + 15 i \, d^{4}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} - 6 \, {\left (3 \, c^{4} + 10 i \, c^{3} d - 8 \, c^{2} d^{2} + 10 i \, c d^{3} - 11 \, d^{4}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 3 \, {\left (3 \, c^{4} + 8 i \, c^{3} d - 2 \, c^{2} d^{2} + 8 i \, c d^{3} - 5 \, d^{4}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, {\left (-i \, a^{3} c^{5} + 3 \, a^{3} c^{4} d + 2 i \, a^{3} c^{3} d^{2} + 2 \, a^{3} c^{2} d^{3} + 3 i \, a^{3} c d^{4} - a^{3} d^{5}\right )} f} \]
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Time = 11.75 (sec) , antiderivative size = 1192, normalized size of antiderivative = 5.09 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {x \left (c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}\right )}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \begin {cases} \frac {\left (512 i a^{6} c^{5} f^{2} e^{6 i e} - 2560 a^{6} c^{4} d f^{2} e^{6 i e} - 5120 i a^{6} c^{3} d^{2} f^{2} e^{6 i e} + 5120 a^{6} c^{2} d^{3} f^{2} e^{6 i e} + 2560 i a^{6} c d^{4} f^{2} e^{6 i e} - 512 a^{6} d^{5} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{5} f^{2} e^{8 i e} - 13056 a^{6} c^{4} d f^{2} e^{8 i e} - 29184 i a^{6} c^{3} d^{2} f^{2} e^{8 i e} + 32256 a^{6} c^{2} d^{3} f^{2} e^{8 i e} + 17664 i a^{6} c d^{4} f^{2} e^{8 i e} - 3840 a^{6} d^{5} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{5} f^{2} e^{10 i e} - 29184 a^{6} c^{4} d f^{2} e^{10 i e} - 76800 i a^{6} c^{3} d^{2} f^{2} e^{10 i e} + 101376 a^{6} c^{2} d^{3} f^{2} e^{10 i e} + 66048 i a^{6} c d^{4} f^{2} e^{10 i e} - 16896 a^{6} d^{5} f^{2} e^{10 i e}\right ) e^{- 2 i f x}}{24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e}} & \text {for}\: 24576 a^{9} c^{6} f^{3} e^{12 i e} + 147456 i a^{9} c^{5} d f^{3} e^{12 i e} - 368640 a^{9} c^{4} d^{2} f^{3} e^{12 i e} - 491520 i a^{9} c^{3} d^{3} f^{3} e^{12 i e} + 368640 a^{9} c^{2} d^{4} f^{3} e^{12 i e} + 147456 i a^{9} c d^{5} f^{3} e^{12 i e} - 24576 a^{9} d^{6} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{3} + 5 i c^{2} d - 11 c d^{2} - 15 i d^{3}}{8 a^{3} c^{4} + 32 i a^{3} c^{3} d - 48 a^{3} c^{2} d^{2} - 32 i a^{3} c d^{3} + 8 a^{3} d^{4}} + \frac {c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} + 5 i c^{2} d e^{6 i e} + 13 i c^{2} d e^{4 i e} + 11 i c^{2} d e^{2 i e} + 3 i c^{2} d - 11 c d^{2} e^{6 i e} - 21 c d^{2} e^{4 i e} - 13 c d^{2} e^{2 i e} - 3 c d^{2} - 15 i d^{3} e^{6 i e} - 11 i d^{3} e^{4 i e} - 5 i d^{3} e^{2 i e} - i d^{3}}{8 a^{3} c^{4} e^{6 i e} + 32 i a^{3} c^{3} d e^{6 i e} - 48 a^{3} c^{2} d^{2} e^{6 i e} - 32 i a^{3} c d^{3} e^{6 i e} + 8 a^{3} d^{4} e^{6 i e}}\right ) & \text {otherwise} \end {cases} - \frac {i d^{4} \log {\left (\frac {c + i d}{c e^{2 i e} - i d e^{2 i e}} + e^{2 i f x} \right )}}{a^{3} f \left (c - i d\right ) \left (c + i d\right )^{4}} \]
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Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 424 vs. \(2 (198) = 396\).
Time = 0.60 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\frac {\frac {192 \, d^{5} \log \left (d \tan \left (f x + e\right ) + c\right )}{2 i \, a^{3} c^{5} d - 6 \, a^{3} c^{4} d^{2} - 4 i \, a^{3} c^{3} d^{3} - 4 \, a^{3} c^{2} d^{4} - 6 i \, a^{3} c d^{5} + 2 \, a^{3} d^{6}} + \frac {6 \, {\left (-i \, c^{3} + 5 \, c^{2} d + 11 i \, c d^{2} - 15 \, d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}} + \frac {192 \, \log \left (\tan \left (f x + e\right ) + i\right )}{-32 i \, a^{3} c - 32 \, a^{3} d} + \frac {11 i \, c^{3} \tan \left (f x + e\right )^{3} - 55 \, c^{2} d \tan \left (f x + e\right )^{3} - 121 i \, c d^{2} \tan \left (f x + e\right )^{3} + 165 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 225 i \, c^{2} d \tan \left (f x + e\right )^{2} - 495 \, c d^{2} \tan \left (f x + e\right )^{2} - 579 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 345 \, c^{2} d \tan \left (f x + e\right ) + 711 i \, c d^{2} \tan \left (f x + e\right ) - 699 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 223 i \, c^{2} d + 385 \, c d^{2} + 301 i \, d^{3}}{{\left (a^{3} c^{4} + 4 i \, a^{3} c^{3} d - 6 \, a^{3} c^{2} d^{2} - 4 i \, a^{3} c d^{3} + a^{3} d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \]
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Time = 10.89 (sec) , antiderivative size = 1952, normalized size of antiderivative = 8.34 \[ \int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \, dx=\text {Too large to display} \]
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