Optimal. Leaf size=59 \[ \frac {a c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}-\frac {a B c^3 (1-i \tan (e+f x))^4}{4 f} \]
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Rubi [A] time = 0.09, 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} \[ \frac {a c^3 (B+i A) (1-i \tan (e+f x))^3}{3 f}-\frac {a B c^3 (1-i \tan (e+f x))^4}{4 f} \]
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))^3 \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (A+B x) (c-i c x)^2 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left ((A-i B) (c-i c x)^2+\frac {i B (c-i c x)^3}{c}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a (i A+B) c^3 (1-i \tan (e+f x))^3}{3 f}-\frac {a B c^3 (1-i \tan (e+f x))^4}{4 f}\\ \end {align*}
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Mathematica [B] time = 3.67, size = 161, normalized size = 2.73 \[ \frac {a c^3 \sec (e) \sec ^4(e+f x) (3 (B-i A) \cos (e+2 f x)+3 (B-2 i A) \cos (e)+5 A \sin (e+2 f x)-3 A \sin (3 e+2 f x)+2 A \sin (3 e+4 f x)-3 i A \cos (3 e+2 f x)-6 A \sin (e)+i B \sin (e+2 f x)-3 i B \sin (3 e+2 f x)+i B \sin (3 e+4 f x)+3 B \cos (3 e+2 f x)-3 i B \sin (e))}{12 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 87, normalized size = 1.47 \[ \frac {{\left (8 i \, A + 8 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (8 i \, A - 4 \, B\right )} a c^{3}}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.29, size = 106, normalized size = 1.80 \[ \frac {8 i \, A a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, B a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, A a c^{3} - 4 \, B a c^{3}}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 1.27 \[ \frac {a \,c^{3} \left (-\frac {2 i B \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {B \left (\tan ^{4}\left (f x +e \right )\right )}{4}-i A \left (\tan ^{2}\left (f x +e \right )\right )-\frac {A \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {B \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 73, normalized size = 1.24 \[ -\frac {3 \, B a c^{3} \tan \left (f x + e\right )^{4} + 4 \, {\left (A + 2 i \, B\right )} a c^{3} \tan \left (f x + e\right )^{3} + {\left (12 i \, A - 6 \, B\right )} a c^{3} \tan \left (f x + e\right )^{2} - 12 \, A a c^{3} \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.47, size = 76, normalized size = 1.29 \[ -\frac {\frac {B\,a\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^4}{4}+\frac {a\,\left (A+B\,2{}\mathrm {i}\right )\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3}+\frac {a\,\left (-B+A\,2{}\mathrm {i}\right )\,c^3\,{\mathrm {tan}\left (e+f\,x\right )}^2}{2}-A\,a\,c^3\,\mathrm {tan}\left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.57, size = 144, normalized size = 2.44 \[ \frac {- 8 A a c^{3} - 4 i B a c^{3} + \left (- 8 A a c^{3} e^{2 i e} + 8 i B a c^{3} e^{2 i e}\right ) e^{2 i f x}}{3 i f e^{8 i e} e^{8 i f x} + 12 i f e^{6 i e} e^{6 i f x} + 18 i f e^{4 i e} e^{4 i f x} + 12 i f e^{2 i e} e^{2 i f x} + 3 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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