Integrand size = 31, antiderivative size = 64 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {3 x}{8 a^2 c^2}+\frac {3 \cos (e+f x) \sin (e+f x)}{8 a^2 c^2 f}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a^2 c^2 f} \]
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Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3603, 2715, 8} \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {\sin (e+f x) \cos ^3(e+f x)}{4 a^2 c^2 f}+\frac {3 \sin (e+f x) \cos (e+f x)}{8 a^2 c^2 f}+\frac {3 x}{8 a^2 c^2} \]
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Rule 8
Rule 2715
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \frac {\int \cos ^4(e+f x) \, dx}{a^2 c^2} \\ & = \frac {\cos ^3(e+f x) \sin (e+f x)}{4 a^2 c^2 f}+\frac {3 \int \cos ^2(e+f x) \, dx}{4 a^2 c^2} \\ & = \frac {3 \cos (e+f x) \sin (e+f x)}{8 a^2 c^2 f}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a^2 c^2 f}+\frac {3 \int 1 \, dx}{8 a^2 c^2} \\ & = \frac {3 x}{8 a^2 c^2}+\frac {3 \cos (e+f x) \sin (e+f x)}{8 a^2 c^2 f}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 a^2 c^2 f} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {12 (e+f x)+8 \sin (2 (e+f x))+\sin (4 (e+f x))}{32 a^2 c^2 f} \]
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Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {3 x}{8 a^{2} c^{2}}+\frac {\sin \left (4 f x +4 e \right )}{32 a^{2} c^{2} f}+\frac {\sin \left (2 f x +2 e \right )}{4 a^{2} c^{2} f}\) | \(51\) |
norman | \(\frac {\frac {3 x}{8 a c}+\frac {5 \tan \left (f x +e \right )}{8 a c f}+\frac {3 \left (\tan ^{3}\left (f x +e \right )\right )}{8 a c f}+\frac {3 x \left (\tan ^{2}\left (f x +e \right )\right )}{4 a c}+\frac {3 x \left (\tan ^{4}\left (f x +e \right )\right )}{8 a c}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{2} a c}\) | \(100\) |
derivativedivides | \(\frac {i}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} c^{2}}+\frac {3}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )+i\right )}-\frac {i}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {3}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )-i\right )}\) | \(110\) |
default | \(\frac {i}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{2} c^{2}}+\frac {3}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )+i\right )}-\frac {i}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {3}{16 f \,a^{2} c^{2} \left (\tan \left (f x +e \right )-i\right )}\) | \(110\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {{\left (24 \, f x e^{\left (4 i \, f x + 4 i \, e\right )} - i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 8 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 8 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{64 \, a^{2} c^{2} f} \]
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Time = 0.23 (sec) , antiderivative size = 221, normalized size of antiderivative = 3.45 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\begin {cases} \frac {\left (- 4096 i a^{6} c^{6} f^{3} e^{10 i e} e^{4 i f x} - 32768 i a^{6} c^{6} f^{3} e^{8 i e} e^{2 i f x} + 32768 i a^{6} c^{6} f^{3} e^{4 i e} e^{- 2 i f x} + 4096 i a^{6} c^{6} f^{3} e^{2 i e} e^{- 4 i f x}\right ) e^{- 6 i e}}{262144 a^{8} c^{8} f^{4}} & \text {for}\: a^{8} c^{8} f^{4} e^{6 i e} \neq 0 \\x \left (\frac {\left (e^{8 i e} + 4 e^{6 i e} + 6 e^{4 i e} + 4 e^{2 i e} + 1\right ) e^{- 4 i e}}{16 a^{2} c^{2}} - \frac {3}{8 a^{2} c^{2}}\right ) & \text {otherwise} \end {cases} + \frac {3 x}{8 a^{2} c^{2}} \]
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Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.44 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {\frac {3 \, {\left (f x + e\right )}}{a^{2} c^{2}} + \frac {3 \, \tan \left (f x + e\right )^{3} + 5 \, \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2} a^{2} c^{2}}}{8 \, f} \]
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Time = 6.21 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.59 \[ \int \frac {1}{(a+i a \tan (e+f x))^2 (c-i c \tan (e+f x))^2} \, dx=\frac {2\,\sin \left (2\,e+2\,f\,x\right )+\frac {\sin \left (4\,e+4\,f\,x\right )}{4}+3\,f\,x}{8\,a^2\,c^2\,f} \]
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