Integrand size = 31, antiderivative size = 83 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=\frac {c^3 x}{a^2}+\frac {i c^3 \log (\cos (e+f x))}{a^2 f}+\frac {2 i c^3}{f (a+i a \tan (e+f x))^2}-\frac {4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )} \]
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Time = 0.14 (sec) , antiderivative size = 83, 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} \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=-\frac {4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )}+\frac {i c^3 \log (\cos (e+f x))}{a^2 f}+\frac {c^3 x}{a^2}+\frac {2 i c^3}{f (a+i a \tan (e+f x))^2} \]
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Rule 45
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(a+i a \tan (e+f x))^5} \, dx \\ & = -\frac {\left (i c^3\right ) \text {Subst}\left (\int \frac {(a-x)^2}{(a+x)^3} \, dx,x,i a \tan (e+f x)\right )}{a^2 f} \\ & = -\frac {\left (i c^3\right ) \text {Subst}\left (\int \left (\frac {4 a^2}{(a+x)^3}-\frac {4 a}{(a+x)^2}+\frac {1}{a+x}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^2 f} \\ & = \frac {c^3 x}{a^2}+\frac {i c^3 \log (\cos (e+f x))}{a^2 f}+\frac {2 i c^3}{f (a+i a \tan (e+f x))^2}-\frac {4 i c^3}{f \left (a^2+i a^2 \tan (e+f x)\right )} \\ \end{align*}
Time = 5.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.63 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=-\frac {i c^3 \left (\log (i-\tan (e+f x))+\frac {-2-4 i \tan (e+f x)}{(-i+\tan (e+f x))^2}\right )}{a^2 f} \]
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Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.05
method | result | size |
derivativedivides | \(-\frac {2 i c^{3}}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {4 c^{3}}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{2}}+\frac {c^{3} \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}\) | \(87\) |
default | \(-\frac {2 i c^{3}}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {4 c^{3}}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{2}}+\frac {c^{3} \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{2}}\) | \(87\) |
risch | \(-\frac {i c^{3} {\mathrm e}^{-2 i \left (f x +e \right )}}{a^{2} f}+\frac {i c^{3} {\mathrm e}^{-4 i \left (f x +e \right )}}{2 a^{2} f}+\frac {2 c^{3} x}{a^{2}}+\frac {2 c^{3} e}{a^{2} f}+\frac {i c^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{2} f}\) | \(89\) |
norman | \(\frac {\frac {c^{3} x}{a}-\frac {2 i c^{3}}{a f}+\frac {c^{3} x \left (\tan ^{4}\left (f x +e \right )\right )}{a}-\frac {4 c^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{a f}+\frac {2 c^{3} x \left (\tan ^{2}\left (f x +e \right )\right )}{a}-\frac {6 i c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{a f}}{a \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}-\frac {i c^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{2}}\) | \(134\) |
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.95 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=\frac {{\left (4 \, c^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 2 i \, c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 2 i \, c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{3}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{2 \, a^{2} f} \]
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Time = 0.27 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.01 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=\begin {cases} \frac {\left (- 2 i a^{2} c^{3} f e^{4 i e} e^{- 2 i f x} + i a^{2} c^{3} f e^{2 i e} e^{- 4 i f x}\right ) e^{- 6 i e}}{2 a^{4} f^{2}} & \text {for}\: a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {2 c^{3}}{a^{2}} + \frac {\left (2 c^{3} e^{4 i e} - 2 c^{3} e^{2 i e} + 2 c^{3}\right ) e^{- 4 i e}}{a^{2}}\right ) & \text {otherwise} \end {cases} + \frac {2 c^{3} x}{a^{2}} + \frac {i c^{3} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} \]
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Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (73) = 146\).
Time = 0.53 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.82 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=-\frac {-\frac {6 i \, c^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2}} + \frac {12 i \, c^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{2}} - \frac {6 i \, c^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{2}} + \frac {-25 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 100 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 198 i \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 100 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 25 i \, c^{3}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{4}}}{6 \, f} \]
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Time = 5.61 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92 \[ \int \frac {(c-i c \tan (e+f x))^3}{(a+i a \tan (e+f x))^2} \, dx=-\frac {\frac {2\,c^3}{a^2}+\frac {c^3\,\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}}{a^2}}{f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )}-\frac {c^3\,\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a^2\,f} \]
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