Integrand size = 31, antiderivative size = 71 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=-\frac {4 a^3 x}{c}+\frac {4 i a^3 \log (\cos (e+f x))}{c f}+\frac {a^3 \tan (e+f x)}{c f}-\frac {4 i a^3}{f (c-i c \tan (e+f x))} \]
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Time = 0.16 (sec) , antiderivative size = 71, 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} \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\frac {a^3 \tan (e+f x)}{c f}-\frac {4 i a^3}{f (c-i c \tan (e+f x))}+\frac {4 i a^3 \log (\cos (e+f x))}{c f}-\frac {4 a^3 x}{c} \]
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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)}{(c-i c \tan (e+f x))^4} \, dx \\ & = \frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {(c-x)^2}{(c+x)^2} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f} \\ & = \frac {\left (i a^3\right ) \text {Subst}\left (\int \left (1+\frac {4 c^2}{(c+x)^2}-\frac {4 c}{c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f} \\ & = -\frac {4 a^3 x}{c}+\frac {4 i a^3 \log (\cos (e+f x))}{c f}+\frac {a^3 \tan (e+f x)}{c f}-\frac {4 i a^3}{f (c-i c \tan (e+f x))} \\ \end{align*}
Time = 3.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.75 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\frac {i a^3 \left (-4 \log (i+\tan (e+f x))-i \tan (e+f x)-\frac {4 i}{i+\tan (e+f x)}\right )}{c f} \]
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Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {a^{3} \tan \left (f x +e \right )}{c f}-\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right )}{f c}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f c}+\frac {4 a^{3}}{f c \left (\tan \left (f x +e \right )+i\right )}\) | \(81\) |
default | \(\frac {a^{3} \tan \left (f x +e \right )}{c f}-\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right )}{f c}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f c}+\frac {4 a^{3}}{f c \left (\tan \left (f x +e \right )+i\right )}\) | \(81\) |
risch | \(-\frac {2 i a^{3} {\mathrm e}^{2 i \left (f x +e \right )}}{c f}+\frac {8 a^{3} e}{c f}+\frac {2 i a^{3}}{f c \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f c}\) | \(84\) |
norman | \(\frac {-\frac {4 i a^{3}}{c f}+\frac {a^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{c f}-\frac {4 a^{3} x}{c}-\frac {4 a^{3} x \left (\tan ^{2}\left (f x +e \right )\right )}{c}+\frac {5 a^{3} \tan \left (f x +e \right )}{c f}}{1+\tan ^{2}\left (f x +e \right )}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f c}\) | \(112\) |
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none
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.24 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=-\frac {2 \, {\left (i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{3} + 2 \, {\left (-i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{3}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\right )}}{c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f} \]
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Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\frac {2 i a^{3}}{c f e^{2 i e} e^{2 i f x} + c f} + \frac {4 i a^{3} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c f} + \begin {cases} - \frac {2 i a^{3} e^{2 i e} e^{2 i f x}}{c f} & \text {for}\: c f \neq 0 \\\frac {4 a^{3} x e^{2 i e}}{c} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (65) = 130\).
Time = 0.47 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.46 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\frac {2 \, {\left (\frac {2 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c} - \frac {4 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c} + \frac {2 i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c} + \frac {-2 i \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 i \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c} - \frac {2 \, {\left (-3 i \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 i \, a^{3}\right )}}{c {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{2}}\right )}}{f} \]
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Time = 5.35 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int \frac {(a+i a \tan (e+f x))^3}{c-i c \tan (e+f x)} \, dx=\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )}{c\,f}+\frac {4\,a^3}{c\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}-\frac {a^3\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{c\,f} \]
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