Optimal. Leaf size=99 \[ \frac {2 a^2 (i A+B) c^3 (1-i \tan (e+f x))^3}{3 f}-\frac {a^2 (i A+3 B) c^3 (1-i \tan (e+f x))^4}{4 f}+\frac {a^2 B c^3 (1-i \tan (e+f x))^5}{5 f} \]
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Rubi [A]
time = 0.10, 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 = {3669, 78}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
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) \text {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) \text {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 = 1.66, size = 146, normalized size = 1.47 \begin {gather*} \frac {a^2 c^3 \sec (e) \sec ^5(e+f x) (15 (-i A+B) \cos (f x)+15 (-i A+B) \cos (2 e+f x)+35 A \sin (f x)-5 i B \sin (f x)-15 A \sin (2 e+f x)-15 i B \sin (2 e+f x)+25 A \sin (2 e+3 f x)+5 i B \sin (2 e+3 f x)+5 A \sin (4 e+5 f x)+i B \sin (4 e+5 f x))}{120 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 98, normalized size = 0.99
method | result | size |
risch | \(\frac {4 c^{3} a^{2} \left (20 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+20 B \,{\mathrm e}^{4 i \left (f x +e \right )}+25 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-5 B \,{\mathrm e}^{2 i \left (f x +e \right )}+5 i A -B \right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{5}}\) | \(83\) |
derivativedivides | \(\frac {i c^{3} a^{2} \left (-\frac {B \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-i B -A \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (i A +2 i \left (i B -A \right )+B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (-i B -A \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}-i A \tan \left (f x +e \right )\right )}{f}\) | \(98\) |
default | \(\frac {i c^{3} a^{2} \left (-\frac {B \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (-i B -A \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (i A +2 i \left (i B -A \right )+B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (-i B -A \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}-i A \tan \left (f x +e \right )\right )}{f}\) | \(98\) |
norman | \(\frac {A \,a^{2} c^{3} \tan \left (f x +e \right )}{f}+\frac {\left (-i A \,a^{2} c^{3}+B \,a^{2} c^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (-i A \,a^{2} c^{3}+B \,a^{2} c^{3}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {\left (-i B \,a^{2} c^{3}+A \,a^{2} c^{3}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}-\frac {i B \,a^{2} c^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 106, normalized size = 1.07 \begin {gather*} -\frac {12 i \, B a^{2} c^{3} \tan \left (f x + e\right )^{5} - 15 \, {\left (-i \, A + 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} - 30 \, {\left (-i \, A + 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.19, size = 131, normalized size = 1.32 \begin {gather*} -\frac {4 \, {\left (20 \, {\left (-i \, A - B\right )} a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, {\left (-5 i \, A + B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + B\right )} a^{2} c^{3}\right )}}{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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 206 vs. \(2 (80) = 160\).
time = 0.48, size = 206, normalized size = 2.08 \begin {gather*} \frac {20 i A a^{2} c^{3} - 4 B a^{2} c^{3} + \left (100 i A a^{2} c^{3} e^{2 i e} - 20 B a^{2} c^{3} e^{2 i e}\right ) e^{2 i f x} + \left (80 i A a^{2} c^{3} e^{4 i e} + 80 B a^{2} c^{3} e^{4 i e}\right ) e^{4 i f x}}{15 f e^{10 i e} e^{10 i f x} + 75 f e^{8 i e} e^{8 i f x} + 150 f e^{6 i e} e^{6 i f x} + 150 f e^{4 i e} e^{4 i f x} + 75 f e^{2 i e} e^{2 i f x} + 15 f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 164, normalized size = 1.66 \begin {gather*} -\frac {4 \, {\left (-20 i \, A a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 20 \, B a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 25 i \, A a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 5 \, B a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 5 i \, A a^{2} c^{3} + B a^{2} c^{3}\right )}}{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 )}} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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