Integrand size = 31, antiderivative size = 55 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=-\frac {a^2 x}{c}+\frac {i a^2 \log (\cos (e+f x))}{c f}-\frac {2 i a^2}{f (c-i c \tan (e+f x))} \]
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Time = 0.13 (sec) , antiderivative size = 55, 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, 3568, 45} \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=-\frac {2 i a^2}{f (c-i c \tan (e+f x))}+\frac {i a^2 \log (\cos (e+f x))}{c f}-\frac {a^2 x}{c} \]
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Rule 45
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \left (a^2 c^2\right ) \int \frac {\sec ^4(e+f x)}{(c-i c \tan (e+f x))^3} \, dx \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int \frac {c-x}{(c+x)^2} \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = \frac {\left (i a^2\right ) \text {Subst}\left (\int \left (\frac {1}{-c-x}+\frac {2 c}{(c+x)^2}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c f} \\ & = -\frac {a^2 x}{c}+\frac {i a^2 \log (\cos (e+f x))}{c f}-\frac {2 i a^2}{f (c-i c \tan (e+f x))} \\ \end{align*}
Time = 2.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=\frac {i a^2 \left (-\log (i+\tan (e+f x))-\frac {2 c}{c-i c \tan (e+f x)}\right )}{c f} \]
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Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {i a^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{c f}+\frac {2 a^{2} e}{c f}+\frac {i a^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{c f}\) | \(59\) |
derivativedivides | \(-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f c}-\frac {a^{2} \arctan \left (\tan \left (f x +e \right )\right )}{f c}+\frac {2 a^{2}}{f c \left (\tan \left (f x +e \right )+i\right )}\) | \(65\) |
default | \(-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f c}-\frac {a^{2} \arctan \left (\tan \left (f x +e \right )\right )}{f c}+\frac {2 a^{2}}{f c \left (\tan \left (f x +e \right )+i\right )}\) | \(65\) |
norman | \(\frac {-\frac {2 i a^{2}}{c f}-\frac {a^{2} x}{c}-\frac {a^{2} x \left (\tan ^{2}\left (f x +e \right )\right )}{c}+\frac {2 a^{2} \tan \left (f x +e \right )}{c f}}{1+\tan ^{2}\left (f x +e \right )}-\frac {i a^{2} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f c}\) | \(94\) |
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none
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=\frac {-i \, a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, a^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{c f} \]
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Time = 0.18 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.24 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=\frac {i a^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c f} + \begin {cases} - \frac {i a^{2} e^{2 i e} e^{2 i f x}}{c f} & \text {for}\: c f \neq 0 \\\frac {2 a^{2} x e^{2 i e}}{c} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (49) = 98\).
Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.16 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=-\frac {-\frac {i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c} + \frac {2 i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c} - \frac {i \, a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c} + \frac {-3 i \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 10 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 i \, a^{2}}{c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{2}}}{f} \]
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Time = 5.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \frac {(a+i a \tan (e+f x))^2}{c-i c \tan (e+f x)} \, dx=\frac {2\,a^2}{c\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}-\frac {a^2\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c\,f} \]
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