Optimal. Leaf size=58 \[ \frac {i c^2 (a+i a \tan (e+f x))^5}{5 a f}-\frac {i c^2 (a+i a \tan (e+f x))^4}{2 f} \]
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Rubi [A] time = 0.10, antiderivative size = 58, 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, 3487, 43} \[ \frac {i c^2 (a+i a \tan (e+f x))^5}{5 a f}-\frac {i c^2 (a+i a \tan (e+f x))^4}{2 f} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3487
Rule 3522
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^4 (c-i c \tan (e+f x))^2 \, dx &=\left (a^2 c^2\right ) \int \sec ^4(e+f x) (a+i a \tan (e+f x))^2 \, dx\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int (a-x) (a+x)^3 \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {\left (i c^2\right ) \operatorname {Subst}\left (\int \left (2 a (a+x)^3-(a+x)^4\right ) \, dx,x,i a \tan (e+f x)\right )}{a f}\\ &=-\frac {i c^2 (a+i a \tan (e+f x))^4}{2 f}+\frac {i c^2 (a+i a \tan (e+f x))^5}{5 a f}\\ \end {align*}
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Mathematica [A] time = 3.33, size = 80, normalized size = 1.38 \[ \frac {a^4 c^2 \sec (e) \sec ^5(e+f x) (-5 \sin (2 e+f x)+5 \sin (2 e+3 f x)+\sin (4 e+5 f x)+5 i \cos (2 e+f x)+5 \sin (f x)+5 i \cos (f x))}{20 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 125, normalized size = 2.16 \[ \frac {80 i \, a^{4} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 80 i \, a^{4} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 40 i \, a^{4} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{4} c^{2}}{5 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.29, size = 133, normalized size = 2.29 \[ \frac {80 i \, a^{4} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 80 i \, a^{4} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 40 i \, a^{4} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{4} c^{2}}{5 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 50, normalized size = 0.86 \[ \frac {a^{4} c^{2} \left (\tan \left (f x +e \right )-\frac {\left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {i \left (\tan ^{4}\left (f x +e \right )\right )}{2}+i \left (\tan ^{2}\left (f x +e \right )\right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 68, normalized size = 1.17 \[ -\frac {6 \, a^{4} c^{2} \tan \left (f x + e\right )^{5} - 15 i \, a^{4} c^{2} \tan \left (f x + e\right )^{4} - 30 i \, a^{4} c^{2} \tan \left (f x + e\right )^{2} - 30 \, a^{4} c^{2} \tan \left (f x + e\right )}{30 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.80, size = 80, normalized size = 1.38 \[ \frac {a^4\,c^2\,\sin \left (e+f\,x\right )\,\left (10\,{\cos \left (e+f\,x\right )}^4+{\cos \left (e+f\,x\right )}^3\,\sin \left (e+f\,x\right )\,10{}\mathrm {i}+\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^3\,5{}\mathrm {i}-2\,{\sin \left (e+f\,x\right )}^4\right )}{10\,f\,{\cos \left (e+f\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.58, size = 185, normalized size = 3.19 \[ \frac {- 80 i a^{4} c^{2} e^{6 i e} e^{6 i f x} - 80 i a^{4} c^{2} e^{4 i e} e^{4 i f x} - 40 i a^{4} c^{2} e^{2 i e} e^{2 i f x} - 8 i a^{4} c^{2}}{- 5 f e^{10 i e} e^{10 i f x} - 25 f e^{8 i e} e^{8 i f x} - 50 f e^{6 i e} e^{6 i f x} - 50 f e^{4 i e} e^{4 i f x} - 25 f e^{2 i e} e^{2 i f x} - 5 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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