Integrand size = 31, antiderivative size = 82 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {i a^4 c^3 \sec ^6(e+f x)}{6 f}+\frac {a^4 c^3 \tan (e+f x)}{f}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^4 c^3 \tan ^5(e+f x)}{5 f} \]
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Time = 0.12 (sec) , antiderivative size = 82, 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, 3567, 3852} \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {a^4 c^3 \tan ^5(e+f x)}{5 f}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^4 c^3 \tan (e+f x)}{f}+\frac {i a^4 c^3 \sec ^6(e+f x)}{6 f} \]
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Rule 3567
Rule 3603
Rule 3852
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \sec ^6(e+f x) (a+i a \tan (e+f x)) \, dx \\ & = \frac {i a^4 c^3 \sec ^6(e+f x)}{6 f}+\left (a^4 c^3\right ) \int \sec ^6(e+f x) \, dx \\ & = \frac {i a^4 c^3 \sec ^6(e+f x)}{6 f}-\frac {\left (a^4 c^3\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{f} \\ & = \frac {i a^4 c^3 \sec ^6(e+f x)}{6 f}+\frac {a^4 c^3 \tan (e+f x)}{f}+\frac {2 a^4 c^3 \tan ^3(e+f x)}{3 f}+\frac {a^4 c^3 \tan ^5(e+f x)}{5 f} \\ \end{align*}
Time = 2.74 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.80 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {a^4 c^3 \sec ^6(e+f x) (3+8 \cos (2 (e+f x))-7 i \sin (2 (e+f x))) (-i \cos (4 (e+f x))+\sin (4 (e+f x)))}{30 f} \]
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Time = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74
method | result | size |
risch | \(\frac {16 i a^{4} c^{3} \left (20 \,{\mathrm e}^{6 i \left (f x +e \right )}+15 \,{\mathrm e}^{4 i \left (f x +e \right )}+6 \,{\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}\) | \(61\) |
derivativedivides | \(-\frac {i a^{4} c^{3} \left (i \tan \left (f x +e \right )-\frac {\left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {i \left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{2}+\frac {2 i \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}\) | \(75\) |
default | \(-\frac {i a^{4} c^{3} \left (i \tan \left (f x +e \right )-\frac {\left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {i \left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{2}+\frac {2 i \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}\) | \(75\) |
parallelrisch | \(\frac {5 i a^{4} c^{3} \left (\tan ^{6}\left (f x +e \right )\right )+15 i a^{4} c^{3} \left (\tan ^{4}\left (f x +e \right )\right )+6 \left (\tan ^{5}\left (f x +e \right )\right ) a^{4} c^{3}+15 i a^{4} c^{3} \left (\tan ^{2}\left (f x +e \right )\right )+20 \left (\tan ^{3}\left (f x +e \right )\right ) a^{4} c^{3}+30 \tan \left (f x +e \right ) a^{4} c^{3}}{30 f}\) | \(104\) |
norman | \(\frac {a^{4} c^{3} \tan \left (f x +e \right )}{f}+\frac {2 a^{4} c^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {a^{4} c^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}+\frac {i a^{4} c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {i a^{4} c^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{2 f}+\frac {i a^{4} c^{3} \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}\) | \(116\) |
parts | \(a^{4} c^{3} x +\frac {a^{4} c^{3} \left (\frac {\left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}+\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {i a^{4} c^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {i a^{4} c^{3} \left (\frac {\left (\tan ^{6}\left (f x +e \right )\right )}{6}-\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {3 i a^{4} c^{3} \left (\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {3 i a^{4} c^{3} \left (\frac {\left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f}+\frac {3 a^{4} c^{3} \left (\tan \left (f x +e \right )-\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}+\frac {3 a^{4} c^{3} \left (\frac {\left (\tan ^{3}\left (f x +e \right )\right )}{3}-\tan \left (f x +e \right )+\arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(281\) |
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Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.67 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=-\frac {16 \, {\left (-20 i \, a^{4} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - 15 i \, a^{4} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 i \, a^{4} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{4} c^{3}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (73) = 146\).
Time = 0.35 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.45 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {320 i a^{4} c^{3} e^{6 i e} e^{6 i f x} + 240 i a^{4} c^{3} e^{4 i e} e^{4 i f x} + 96 i a^{4} c^{3} e^{2 i e} e^{2 i f x} + 16 i a^{4} c^{3}}{15 f e^{12 i e} e^{12 i f x} + 90 f e^{10 i e} e^{10 i f x} + 225 f e^{8 i e} e^{8 i f x} + 300 f e^{6 i e} e^{6 i f x} + 225 f e^{4 i e} e^{4 i f x} + 90 f e^{2 i e} e^{2 i f x} + 15 f} \]
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Time = 0.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {5 i \, a^{4} c^{3} \tan \left (f x + e\right )^{6} + 6 \, a^{4} c^{3} \tan \left (f x + e\right )^{5} + 15 i \, a^{4} c^{3} \tan \left (f x + e\right )^{4} + 20 \, a^{4} c^{3} \tan \left (f x + e\right )^{3} + 15 i \, a^{4} c^{3} \tan \left (f x + e\right )^{2} + 30 \, a^{4} c^{3} \tan \left (f x + e\right )}{30 \, f} \]
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Time = 0.67 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.67 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=-\frac {16 \, {\left (-20 i \, a^{4} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} - 15 i \, a^{4} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 6 i \, a^{4} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - i \, a^{4} c^{3}\right )}}{15 \, {\left (f e^{\left (12 i \, f x + 12 i \, e\right )} + 6 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 15 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 20 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 15 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 6 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
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Time = 6.00 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.43 \[ \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^3 \, dx=\frac {a^4\,c^3\,\sin \left (e+f\,x\right )\,\left (30\,{\cos \left (e+f\,x\right )}^5+{\cos \left (e+f\,x\right )}^4\,\sin \left (e+f\,x\right )\,15{}\mathrm {i}+20\,{\cos \left (e+f\,x\right )}^3\,{\sin \left (e+f\,x\right )}^2+{\cos \left (e+f\,x\right )}^2\,{\sin \left (e+f\,x\right )}^3\,15{}\mathrm {i}+6\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^4+{\sin \left (e+f\,x\right )}^5\,5{}\mathrm {i}\right )}{30\,f\,{\cos \left (e+f\,x\right )}^6} \]
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