Optimal. Leaf size=59 \[ \frac {a c^4 (B+i A) (1-i \tan (e+f x))^4}{4 f}-\frac {a B c^4 (1-i \tan (e+f x))^5}{5 f} \]
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Rubi [A] time = 0.08, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {3588, 43} \[ \frac {a c^4 (B+i A) (1-i \tan (e+f x))^4}{4 f}-\frac {a B c^4 (1-i \tan (e+f x))^5}{5 f} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (A+B \tan (e+f x)) (c-i c \tan (e+f x))^4 \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (A+B x) (c-i c x)^3 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left ((A-i B) (c-i c x)^3+\frac {i B (c-i c x)^4}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a (i A+B) c^4 (1-i \tan (e+f x))^4}{4 f}-\frac {a B c^4 (1-i \tan (e+f x))^5}{5 f}\\ \end {align*}
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Mathematica [B] time = 3.55, size = 226, normalized size = 3.83 \[ \frac {a c^4 \sec (e) \sec ^5(e+f x) (5 (3 B-5 i A) \cos (2 e+f x)+5 (3 B-5 i A) \cos (f x)-25 A \sin (2 e+f x)+15 A \sin (2 e+3 f x)-10 A \sin (4 e+3 f x)+5 A \sin (4 e+5 f x)-10 i A \cos (2 e+3 f x)-10 i A \cos (4 e+3 f x)+25 A \sin (f x)-15 i B \sin (2 e+f x)+5 i B \sin (2 e+3 f x)-10 i B \sin (4 e+3 f x)+3 i B \sin (4 e+5 f x)+10 B \cos (2 e+3 f x)+10 B \cos (4 e+3 f x)+15 i B \sin (f x))}{40 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 99, normalized size = 1.68 \[ \frac {{\left (20 i \, A + 20 \, B\right )} a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (20 i \, A - 12 \, B\right )} a c^{4}}{5 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.26, size = 119, normalized size = 2.02 \[ \frac {20 i \, A a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 \, B a c^{4} e^{\left (2 i \, f x + 2 i \, e\right )} + 20 i \, A a c^{4} - 12 \, B a c^{4}}{5 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 1.68 \[ \frac {a \,c^{4} \left (\frac {i B \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {i A \left (\tan ^{4}\left (f x +e \right )\right )}{4}-i B \left (\tan ^{3}\left (f x +e \right )\right )-\frac {3 B \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {3 i A \left (\tan ^{2}\left (f x +e \right )\right )}{2}-A \left (\tan ^{3}\left (f x +e \right )\right )+\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 93, normalized size = 1.58 \[ \frac {12 i \, B a c^{4} \tan \left (f x + e\right )^{5} + {\left (15 i \, A - 45 \, B\right )} a c^{4} \tan \left (f x + e\right )^{4} - 60 \, {\left (A + i \, B\right )} a c^{4} \tan \left (f x + e\right )^{3} + {\left (-90 i \, A + 30 \, B\right )} a c^{4} \tan \left (f x + e\right )^{2} + 60 \, A a c^{4} \tan \left (f x + e\right )}{60 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.50, size = 100, normalized size = 1.69 \[ \frac {\frac {1{}\mathrm {i}\,B\,a\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^5}{5}+\frac {1{}\mathrm {i}\,a\,\left (A+B\,3{}\mathrm {i}\right )\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4}+1{}\mathrm {i}\,a\,\left (-B+A\,1{}\mathrm {i}\right )\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^3-\frac {1{}\mathrm {i}\,a\,\left (3\,A+B\,1{}\mathrm {i}\right )\,c^4\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2}+A\,a\,c^4\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.68, size = 158, normalized size = 2.68 \[ \frac {- 20 i A a c^{4} + 12 B a c^{4} + \left (- 20 i A a c^{4} e^{2 i e} - 20 B a c^{4} e^{2 i e}\right ) e^{2 i f x}}{- 5 f e^{10 i e} e^{10 i f x} - 25 f e^{8 i e} e^{8 i f x} - 50 f e^{6 i e} e^{6 i f x} - 50 f e^{4 i e} e^{4 i f x} - 25 f e^{2 i e} e^{2 i f x} - 5 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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