Integrand size = 31, antiderivative size = 101 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\frac {3 x}{8 a c^2}-\frac {i}{8 a f (c-i c \tan (e+f x))^2}-\frac {i}{4 a f \left (c^2-i c^2 \tan (e+f x)\right )}+\frac {i}{8 a f \left (c^2+i c^2 \tan (e+f x)\right )} \]
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Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {3603, 3568, 46, 212} \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=-\frac {i}{4 a f \left (c^2-i c^2 \tan (e+f x)\right )}+\frac {i}{8 a f \left (c^2+i c^2 \tan (e+f x)\right )}+\frac {3 x}{8 a c^2}-\frac {i}{8 a f (c-i c \tan (e+f x))^2} \]
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Rule 46
Rule 212
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\cos ^2(e+f x)}{c-i c \tan (e+f x)} \, dx}{a c} \\ & = \frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {1}{(c-x)^2 (c+x)^3} \, dx,x,-i c \tan (e+f x)\right )}{a f} \\ & = \frac {\left (i c^2\right ) \text {Subst}\left (\int \left (\frac {1}{8 c^3 (c-x)^2}+\frac {1}{4 c^2 (c+x)^3}+\frac {1}{4 c^3 (c+x)^2}+\frac {3}{8 c^3 \left (c^2-x^2\right )}\right ) \, dx,x,-i c \tan (e+f x)\right )}{a f} \\ & = -\frac {i}{8 a f (c-i c \tan (e+f x))^2}-\frac {i}{4 a f \left (c^2-i c^2 \tan (e+f x)\right )}+\frac {i}{8 a f \left (c^2+i c^2 \tan (e+f x)\right )}+\frac {(3 i) \text {Subst}\left (\int \frac {1}{c^2-x^2} \, dx,x,-i c \tan (e+f x)\right )}{8 a c f} \\ & = \frac {3 x}{8 a c^2}-\frac {i}{8 a f (c-i c \tan (e+f x))^2}-\frac {i}{4 a f \left (c^2-i c^2 \tan (e+f x)\right )}+\frac {i}{8 a f \left (c^2+i c^2 \tan (e+f x)\right )} \\ \end{align*}
Time = 0.91 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\frac {2+3 i \tan (e+f x)+3 \tan ^2(e+f x)+3 \arctan (\tan (e+f x)) (-i+\tan (e+f x)) (i+\tan (e+f x))^2}{8 a c^2 f (-i+\tan (e+f x)) (i+\tan (e+f x))^2} \]
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Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.72
method | result | size |
risch | \(\frac {3 x}{8 a \,c^{2}}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )}}{32 a \,c^{2} f}-\frac {i \cos \left (2 f x +2 e \right )}{8 a \,c^{2} f}+\frac {\sin \left (2 f x +2 e \right )}{4 a \,c^{2} f}\) | \(73\) |
derivativedivides | \(\frac {i}{8 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right )}{8 f a \,c^{2}}+\frac {1}{4 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )}+\frac {1}{8 f a \,c^{2} \left (\tan \left (f x +e \right )-i\right )}\) | \(87\) |
default | \(\frac {i}{8 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right )}{8 f a \,c^{2}}+\frac {1}{4 f a \,c^{2} \left (\tan \left (f x +e \right )+i\right )}+\frac {1}{8 f a \,c^{2} \left (\tan \left (f x +e \right )-i\right )}\) | \(87\) |
norman | \(\frac {\frac {3 x}{8 a c}+\frac {5 \tan \left (f x +e \right )}{8 a c f}+\frac {3 \left (\tan ^{3}\left (f x +e \right )\right )}{8 a c f}+\frac {3 x \left (\tan ^{2}\left (f x +e \right )\right )}{4 a c}+\frac {3 x \left (\tan ^{4}\left (f x +e \right )\right )}{8 a c}-\frac {i}{4 a c f}}{\left (1+\tan ^{2}\left (f x +e \right )\right )^{2} c}\) | \(109\) |
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Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.56 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\frac {{\left (12 \, f x e^{\left (2 i \, f x + 2 i \, e\right )} - i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 6 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{32 \, a c^{2} f} \]
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Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\begin {cases} \frac {\left (- 256 i a^{2} c^{4} f^{2} e^{6 i e} e^{4 i f x} - 1536 i a^{2} c^{4} f^{2} e^{4 i e} e^{2 i f x} + 512 i a^{2} c^{4} f^{2} e^{- 2 i f x}\right ) e^{- 2 i e}}{8192 a^{3} c^{6} f^{3}} & \text {for}\: a^{3} c^{6} f^{3} e^{2 i e} \neq 0 \\x \left (\frac {\left (e^{6 i e} + 3 e^{4 i e} + 3 e^{2 i e} + 1\right ) e^{- 2 i e}}{8 a c^{2}} - \frac {3}{8 a c^{2}}\right ) & \text {otherwise} \end {cases} + \frac {3 x}{8 a c^{2}} \]
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Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=-\frac {-\frac {6 i \, \log \left (\tan \left (f x + e\right ) + i\right )}{a c^{2}} + \frac {6 i \, \log \left (\tan \left (f x + e\right ) - i\right )}{a c^{2}} + \frac {2 \, {\left (3 \, \tan \left (f x + e\right ) - 5 i\right )}}{a c^{2} {\left (i \, \tan \left (f x + e\right ) + 1\right )}} + \frac {9 i \, \tan \left (f x + e\right )^{2} - 26 \, \tan \left (f x + e\right ) - 21 i}{a c^{2} {\left (\tan \left (f x + e\right ) + i\right )}^{2}}}{32 \, f} \]
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Time = 5.36 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(a+i a \tan (e+f x)) (c-i c \tan (e+f x))^2} \, dx=\frac {3\,x}{8\,a\,c^2}+\frac {\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,3{}\mathrm {i}}{8}-\frac {3\,\mathrm {tan}\left (e+f\,x\right )}{8}+\frac {1}{4}{}\mathrm {i}}{a\,c^2\,f\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}^2} \]
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