Integrand size = 31, antiderivative size = 114 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=-\frac {c^4 x}{a^3}-\frac {i c^4 \log (\cos (e+f x))}{a^3 f}+\frac {8 i c^4}{3 f (a+i a \tan (e+f x))^3}-\frac {6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac {6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )} \]
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Time = 0.16 (sec) , antiderivative size = 114, 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} \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=\frac {6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac {i c^4 \log (\cos (e+f x))}{a^3 f}-\frac {c^4 x}{a^3}-\frac {6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac {8 i c^4}{3 f (a+i a \tan (e+f x))^3} \]
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Rule 45
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \left (a^4 c^4\right ) \int \frac {\sec ^8(e+f x)}{(a+i a \tan (e+f x))^7} \, dx \\ & = -\frac {\left (i c^4\right ) \text {Subst}\left (\int \frac {(a-x)^3}{(a+x)^4} \, dx,x,i a \tan (e+f x)\right )}{a^3 f} \\ & = -\frac {\left (i c^4\right ) \text {Subst}\left (\int \left (\frac {1}{-a-x}+\frac {8 a^3}{(a+x)^4}-\frac {12 a^2}{(a+x)^3}+\frac {6 a}{(a+x)^2}\right ) \, dx,x,i a \tan (e+f x)\right )}{a^3 f} \\ & = -\frac {c^4 x}{a^3}-\frac {i c^4 \log (\cos (e+f x))}{a^3 f}+\frac {8 i c^4}{3 f (a+i a \tan (e+f x))^3}-\frac {6 i c^4}{a f (a+i a \tan (e+f x))^2}+\frac {6 i c^4}{f \left (a^3+i a^3 \tan (e+f x)\right )} \\ \end{align*}
Time = 5.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.61 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=-\frac {i c^4 \left (-\log (i-\tan (e+f x))+\frac {2 \left (-4 i+9 \tan (e+f x)+9 i \tan ^2(e+f x)\right )}{3 (-i+\tan (e+f x))^3}\right )}{a^3 f} \]
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.96
method | result | size |
derivativedivides | \(\frac {6 c^{4}}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {6 i c^{4}}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {8 c^{4}}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i c^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3}}-\frac {c^{4} \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3}}\) | \(110\) |
default | \(\frac {6 c^{4}}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {6 i c^{4}}{f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {8 c^{4}}{3 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i c^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3}}-\frac {c^{4} \arctan \left (\tan \left (f x +e \right )\right )}{f \,a^{3}}\) | \(110\) |
risch | \(\frac {i c^{4} {\mathrm e}^{-2 i \left (f x +e \right )}}{a^{3} f}-\frac {i c^{4} {\mathrm e}^{-4 i \left (f x +e \right )}}{2 a^{3} f}+\frac {i c^{4} {\mathrm e}^{-6 i \left (f x +e \right )}}{3 a^{3} f}-\frac {2 c^{4} x}{a^{3}}-\frac {2 c^{4} e}{a^{3} f}-\frac {i c^{4} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{a^{3} f}\) | \(110\) |
norman | \(\frac {\frac {4 i c^{4} \left (\tan ^{2}\left (f x +e \right )\right )}{a f}-\frac {c^{4} x}{a}+\frac {8 i c^{4}}{3 a f}-\frac {8 c^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{3 a f}+\frac {6 c^{4} \left (\tan ^{5}\left (f x +e \right )\right )}{a f}-\frac {3 c^{4} x \left (\tan ^{2}\left (f x +e \right )\right )}{a}-\frac {3 c^{4} x \left (\tan ^{4}\left (f x +e \right )\right )}{a}-\frac {c^{4} x \left (\tan ^{6}\left (f x +e \right )\right )}{a}+\frac {2 c^{4} \tan \left (f x +e \right )}{a f}+\frac {12 i c^{4} \left (\tan ^{4}\left (f x +e \right )\right )}{a f}}{a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {i c^{4} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \,a^{3}}\) | \(209\) |
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Time = 0.24 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.82 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=-\frac {{\left (12 \, c^{4} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 6 i \, c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 6 i \, c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 3 i \, c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 i \, c^{4}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{6 \, a^{3} f} \]
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Time = 0.31 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.86 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=\begin {cases} \frac {\left (6 i a^{6} c^{4} f^{2} e^{10 i e} e^{- 2 i f x} - 3 i a^{6} c^{4} f^{2} e^{8 i e} e^{- 4 i f x} + 2 i a^{6} c^{4} f^{2} e^{6 i e} e^{- 6 i f x}\right ) e^{- 12 i e}}{6 a^{9} f^{3}} & \text {for}\: a^{9} f^{3} e^{12 i e} \neq 0 \\x \left (\frac {2 c^{4}}{a^{3}} + \frac {\left (- 2 c^{4} e^{6 i e} + 2 c^{4} e^{4 i e} - 2 c^{4} e^{2 i e} + 2 c^{4}\right ) e^{- 6 i e}}{a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {2 c^{4} x}{a^{3}} - \frac {i c^{4} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{3} f} \]
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Exception generated. \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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Time = 0.78 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.61 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=-\frac {\frac {30 i \, c^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3}} - \frac {60 i \, c^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{3}} + \frac {30 i \, c^{4} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{3}} + \frac {147 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 1002 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2445 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 3820 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2445 i \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1002 \, c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 147 i \, c^{4}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{6}}}{30 \, f} \]
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Time = 5.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {(c-i c \tan (e+f x))^4}{(a+i a \tan (e+f x))^3} \, dx=-\frac {\frac {6\,c^4\,\mathrm {tan}\left (e+f\,x\right )}{a^3}-\frac {c^4\,8{}\mathrm {i}}{3\,a^3}+\frac {c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2\,6{}\mathrm {i}}{a^3}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )}+\frac {c^4\,\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,1{}\mathrm {i}}{a^3\,f} \]
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