Optimal. Leaf size=100 \[ \frac {a^5 c^4 \tan ^7(e+f x)}{7 f}+\frac {3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^5 c^4 \tan ^3(e+f x)}{f}+\frac {a^5 c^4 \tan (e+f x)}{f}+\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f} \]
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Rubi [A] time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3522, 3486, 3767} \[ \frac {a^5 c^4 \tan ^7(e+f x)}{7 f}+\frac {3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^5 c^4 \tan ^3(e+f x)}{f}+\frac {a^5 c^4 \tan (e+f x)}{f}+\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f} \]
Antiderivative was successfully verified.
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Rule 3486
Rule 3522
Rule 3767
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^5 (c-i c \tan (e+f x))^4 \, dx &=\left (a^4 c^4\right ) \int \sec ^8(e+f x) (a+i a \tan (e+f x)) \, dx\\ &=\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f}+\left (a^5 c^4\right ) \int \sec ^8(e+f x) \, dx\\ &=\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f}-\frac {\left (a^5 c^4\right ) \operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (e+f x)\right )}{f}\\ &=\frac {i a^5 c^4 \sec ^8(e+f x)}{8 f}+\frac {a^5 c^4 \tan (e+f x)}{f}+\frac {a^5 c^4 \tan ^3(e+f x)}{f}+\frac {3 a^5 c^4 \tan ^5(e+f x)}{5 f}+\frac {a^5 c^4 \tan ^7(e+f x)}{7 f}\\ \end {align*}
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Mathematica [A] time = 5.42, size = 74, normalized size = 0.74 \[ \frac {a^5 c^4 \sec (e) \sec ^8(e+f x) (56 \sin (e+2 f x)+28 \sin (3 e+4 f x)+8 \sin (5 e+6 f x)+\sin (7 e+8 f x)-35 \sin (e)+35 i \cos (e))}{280 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 178, normalized size = 1.78 \[ \frac {2240 i \, a^{5} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 1792 i \, a^{5} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 896 i \, a^{5} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 256 i \, a^{5} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{5} c^{4}}{35 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.09, size = 190, normalized size = 1.90 \[ \frac {2240 i \, a^{5} c^{4} e^{\left (8 i \, f x + 8 i \, e\right )} + 1792 i \, a^{5} c^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 896 i \, a^{5} c^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 256 i \, a^{5} c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, a^{5} c^{4}}{35 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 90, normalized size = 0.90 \[ \frac {a^{5} c^{4} \left (\tan \left (f x +e \right )+\frac {i \left (\tan ^{8}\left (f x +e \right )\right )}{8}+\frac {\left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {i \left (\tan ^{6}\left (f x +e \right )\right )}{2}+\frac {3 \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {3 i \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\tan ^{3}\left (f x +e \right )+\frac {i \left (\tan ^{2}\left (f x +e \right )\right )}{2}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 132, normalized size = 1.32 \[ \frac {105 i \, a^{5} c^{4} \tan \left (f x + e\right )^{8} + 120 \, a^{5} c^{4} \tan \left (f x + e\right )^{7} + 420 i \, a^{5} c^{4} \tan \left (f x + e\right )^{6} + 504 \, a^{5} c^{4} \tan \left (f x + e\right )^{5} + 630 i \, a^{5} c^{4} \tan \left (f x + e\right )^{4} + 840 \, a^{5} c^{4} \tan \left (f x + e\right )^{3} + 420 i \, a^{5} c^{4} \tan \left (f x + e\right )^{2} + 840 \, a^{5} c^{4} \tan \left (f x + e\right )}{840 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.74, size = 95, normalized size = 0.95 \[ \frac {a^5\,c^4\,\left (-{\cos \left (e+f\,x\right )}^8\,35{}\mathrm {i}+128\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^7+64\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^5+48\,\sin \left (e+f\,x\right )\,{\cos \left (e+f\,x\right )}^3+40\,\sin \left (e+f\,x\right )\,\cos \left (e+f\,x\right )+35{}\mathrm {i}\right )}{280\,f\,{\cos \left (e+f\,x\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.05, size = 272, normalized size = 2.72 \[ \frac {- 2240 a^{5} c^{4} e^{8 i e} e^{8 i f x} - 1792 a^{5} c^{4} e^{6 i e} e^{6 i f x} - 896 a^{5} c^{4} e^{4 i e} e^{4 i f x} - 256 a^{5} c^{4} e^{2 i e} e^{2 i f x} - 32 a^{5} c^{4}}{35 i f e^{16 i e} e^{16 i f x} + 280 i f e^{14 i e} e^{14 i f x} + 980 i f e^{12 i e} e^{12 i f x} + 1960 i f e^{10 i e} e^{10 i f x} + 2450 i f e^{8 i e} e^{8 i f x} + 1960 i f e^{6 i e} e^{6 i f x} + 980 i f e^{4 i e} e^{4 i f x} + 280 i f e^{2 i e} e^{2 i f x} + 35 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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