Optimal. Leaf size=99 \[ -\frac {a^2 c^3 (3 B+i A) (1-i \tan (e+f x))^4}{4 f}+\frac {2 a^2 c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}+\frac {a^2 B c^3 (1-i \tan (e+f x))^5}{5 f} \]
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Rubi [A] time = 0.15, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac {a^2 c^3 (3 B+i A) (1-i \tan (e+f x))^4}{4 f}+\frac {2 a^2 c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}+\frac {a^2 B c^3 (1-i \tan (e+f x))^5}{5 f} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^3 \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x) (A+B x) (c-i c x)^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (2 a (A-i B) (c-i c x)^2-\frac {a (A-3 i B) (c-i c x)^3}{c}-\frac {i a B (c-i c x)^4}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a^2 (i A+B) c^3 (1-i \tan (e+f x))^3}{3 f}-\frac {a^2 (i A+3 B) c^3 (1-i \tan (e+f x))^4}{4 f}+\frac {a^2 B c^3 (1-i \tan (e+f x))^5}{5 f}\\ \end {align*}
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Mathematica [A] time = 6.34, size = 146, normalized size = 1.47 \[ \frac {a^2 c^3 \sec (e) \sec ^5(e+f x) (15 (B-i A) \cos (2 e+f x)+15 (B-i A) \cos (f x)-15 A \sin (2 e+f x)+25 A \sin (2 e+3 f x)+5 A \sin (4 e+5 f x)+35 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)+i B \sin (4 e+5 f x)-5 i B \sin (f x))}{120 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.30, size = 126, normalized size = 1.27 \[ \frac {{\left (80 i \, A + 80 \, B\right )} a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (100 i \, A - 20 \, B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (20 i \, A - 4 \, B\right )} a^{2} c^{3}}{15 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.25, size = 165, normalized size = 1.67 \[ \frac {80 i \, A a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 80 \, B a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 100 i \, A a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 20 \, B a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 i \, A a^{2} c^{3} - 4 \, B a^{2} c^{3}}{15 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 101, normalized size = 1.02 \[ \frac {c^{3} a^{2} \left (-\frac {i B \left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {i A \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {i B \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {i A \left (\tan ^{2}\left (f x +e \right )\right )}{2}+\frac {A \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 103, normalized size = 1.04 \[ -\frac {12 i \, B a^{2} c^{3} \tan \left (f x + e\right )^{5} + {\left (15 i \, A - 15 \, B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{4} - 20 \, {\left (A - i \, B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{3} + {\left (30 i \, A - 30 \, B\right )} a^{2} c^{3} \tan \left (f x + e\right )^{2} - 60 \, A a^{2} c^{3} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.02, size = 108, normalized size = 1.09 \[ -\frac {-A\,a^2\,c^3\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^2\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^2\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+\frac {a^2\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {B\,a^2\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^5\,1{}\mathrm {i}}{5}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.87, size = 218, normalized size = 2.20 \[ \frac {20 A a^{2} c^{3} + 4 i B a^{2} c^{3} + \left (100 A a^{2} c^{3} e^{2 i e} + 20 i B a^{2} c^{3} e^{2 i e}\right ) e^{2 i f x} + \left (80 A a^{2} c^{3} e^{4 i e} - 80 i B a^{2} c^{3} e^{4 i e}\right ) e^{4 i f x}}{- 15 i f e^{10 i e} e^{10 i f x} - 75 i f e^{8 i e} e^{8 i f x} - 150 i f e^{6 i e} e^{6 i f x} - 150 i f e^{4 i e} e^{4 i f x} - 75 i f e^{2 i e} e^{2 i f x} - 15 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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