Optimal. Leaf size=135 \[ \frac {4 a^3 (i A+B) c^5 (1-i \tan (e+f x))^5}{5 f}-\frac {2 a^3 (i A+2 B) c^5 (1-i \tan (e+f x))^6}{3 f}+\frac {a^3 (i A+5 B) c^5 (1-i \tan (e+f x))^7}{7 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 f} \]
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Rubi [A]
time = 0.13, antiderivative size = 135, 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^3 c^5 (5 B+i A) (1-i \tan (e+f x))^7}{7 f}-\frac {2 a^3 c^5 (2 B+i A) (1-i \tan (e+f x))^6}{3 f}+\frac {4 a^3 c^5 (B+i A) (1-i \tan (e+f x))^5}{5 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 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))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^5 \, dx &=\frac {(a c) \text {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^4 \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (4 a^2 (A-i B) (c-i c x)^4-\frac {4 a^2 (A-2 i B) (c-i c x)^5}{c}+\frac {a^2 (A-5 i B) (c-i c x)^6}{c^2}+\frac {i a^2 B (c-i c x)^7}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {4 a^3 (i A+B) c^5 (1-i \tan (e+f x))^5}{5 f}-\frac {2 a^3 (i A+2 B) c^5 (1-i \tan (e+f x))^6}{3 f}+\frac {a^3 (i A+5 B) c^5 (1-i \tan (e+f x))^7}{7 f}-\frac {a^3 B c^5 (1-i \tan (e+f x))^8}{8 f}\\ \end {align*}
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Mathematica [A]
time = 2.03, size = 215, normalized size = 1.59 \begin {gather*} \frac {a^3 c^5 \sec (e) \sec ^8(e+f x) (35 (-4 i A+B) \cos (e)+70 (-i A+B) \cos (e+2 f x)-70 i A \cos (3 e+2 f x)+70 B \cos (3 e+2 f x)-140 A \sin (e)-35 i B \sin (e)+154 A \sin (e+2 f x)-14 i B \sin (e+2 f x)-70 A \sin (3 e+2 f x)-70 i B \sin (3 e+2 f x)+112 A \sin (3 e+4 f x)+28 i B \sin (3 e+4 f x)+32 A \sin (5 e+6 f x)+8 i B \sin (5 e+6 f x)+4 A \sin (7 e+8 f x)+i B \sin (7 e+8 f x))}{840 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 177, normalized size = 1.31
method | result | size |
risch | \(\frac {32 c^{5} a^{3} \left (84 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+84 B \,{\mathrm e}^{6 i \left (f x +e \right )}+112 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-28 B \,{\mathrm e}^{4 i \left (f x +e \right )}+32 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-8 B \,{\mathrm e}^{2 i \left (f x +e \right )}+4 i A -B \right )}{105 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{8}}\) | \(108\) |
derivativedivides | \(\frac {c^{5} a^{3} \left (-\frac {B \left (\tan ^{8}\left (f x +e \right )\right )}{8}-\frac {\left (2 i B +A \right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7}-\frac {\left (-7 B -2 i A +4 i \left (-2 i B +A \right )\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}-\frac {\left (-7 A +4 i \left (-2 i A -B \right )+8 i B \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {\left (8 i A +7 B -4 i \left (-2 i B +A \right )\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (7 A -4 i \left (-2 i A -B \right )-2 i B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {\left (2 i A -B \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(177\) |
default | \(\frac {c^{5} a^{3} \left (-\frac {B \left (\tan ^{8}\left (f x +e \right )\right )}{8}-\frac {\left (2 i B +A \right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7}-\frac {\left (-7 B -2 i A +4 i \left (-2 i B +A \right )\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}-\frac {\left (-7 A +4 i \left (-2 i A -B \right )+8 i B \right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}-\frac {\left (8 i A +7 B -4 i \left (-2 i B +A \right )\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {\left (7 A -4 i \left (-2 i A -B \right )-2 i B \right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {\left (2 i A -B \right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}+A \tan \left (f x +e \right )\right )}{f}\) | \(177\) |
norman | \(\frac {A \,a^{3} c^{5} \tan \left (f x +e \right )}{f}-\frac {\left (2 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}-\frac {\left (2 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7 f}-\frac {\left (4 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}+\frac {\left (-4 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}+\frac {\left (-2 i A \,a^{3} c^{5}+B \,a^{3} c^{5}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {\left (-2 i B \,a^{3} c^{5}+A \,a^{3} c^{5}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}-\frac {B \,a^{3} c^{5} \left (\tan ^{8}\left (f x +e \right )\right )}{8 f}\) | \(231\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 174, normalized size = 1.29 \begin {gather*} -\frac {105 \, B a^{3} c^{5} \tan \left (f x + e\right )^{8} + 120 \, {\left (A + 2 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{7} - 140 \, {\left (-2 i \, A - B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{6} + 168 \, {\left (A + 4 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{5} - 210 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{4} - 280 \, {\left (A - 2 i \, B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{3} - 420 \, {\left (-2 i \, A + B\right )} a^{3} c^{5} \tan \left (f x + e\right )^{2} - 840 \, A a^{3} c^{5} \tan \left (f x + e\right )}{840 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.33, size = 193, normalized size = 1.43 \begin {gather*} -\frac {32 \, {\left (84 \, {\left (-i \, A - B\right )} a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, {\left (-4 i \, A + B\right )} a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-4 i \, A + B\right )} a^{3} c^{5}\right )}}{105 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, 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. 306 vs. \(2 (110) = 220\).
time = 1.06, size = 306, normalized size = 2.27 \begin {gather*} \frac {128 i A a^{3} c^{5} - 32 B a^{3} c^{5} + \left (1024 i A a^{3} c^{5} e^{2 i e} - 256 B a^{3} c^{5} e^{2 i e}\right ) e^{2 i f x} + \left (3584 i A a^{3} c^{5} e^{4 i e} - 896 B a^{3} c^{5} e^{4 i e}\right ) e^{4 i f x} + \left (2688 i A a^{3} c^{5} e^{6 i e} + 2688 B a^{3} c^{5} e^{6 i e}\right ) e^{6 i f x}}{105 f e^{16 i e} e^{16 i f x} + 840 f e^{14 i e} e^{14 i f x} + 2940 f e^{12 i e} e^{12 i f x} + 5880 f e^{10 i e} e^{10 i f x} + 7350 f e^{8 i e} e^{8 i f x} + 5880 f e^{6 i e} e^{6 i f x} + 2940 f e^{4 i e} e^{4 i f x} + 840 f e^{2 i e} e^{2 i f x} + 105 f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 241 vs. \(2 (117) = 234\).
time = 1.30, size = 241, normalized size = 1.79 \begin {gather*} -\frac {32 \, {\left (-84 i \, A a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} - 84 \, B a^{3} c^{5} e^{\left (6 i \, f x + 6 i \, e\right )} - 112 i \, A a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} + 28 \, B a^{3} c^{5} e^{\left (4 i \, f x + 4 i \, e\right )} - 32 i \, A a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, B a^{3} c^{5} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 i \, A a^{3} c^{5} + B a^{3} c^{5}\right )}}{105 \, {\left (f e^{\left (16 i \, f x + 16 i \, e\right )} + 8 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 28 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 56 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 70 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 56 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 28 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, 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 = 8.65, size = 174, normalized size = 1.29 \begin {gather*} -\frac {\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (-B+A\,2{}\mathrm {i}\right )}{2}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (-B+A\,4{}\mathrm {i}\right )}{4}-A\,a^3\,c^5\,\mathrm {tan}\left (e+f\,x\right )-\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (A-B\,2{}\mathrm {i}\right )}{3}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (B+A\,2{}\mathrm {i}\right )}{6}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (A+B\,4{}\mathrm {i}\right )}{5}+\frac {a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (A+B\,2{}\mathrm {i}\right )}{7}+\frac {B\,a^3\,c^5\,{\mathrm {tan}\left (e+f\,x\right )}^8}{8}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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