Optimal. Leaf size=135 \[ \frac {2 a^3 (i A+B) c^6 (1-i \tan (e+f x))^6}{3 f}-\frac {4 a^3 (i A+2 B) c^6 (1-i \tan (e+f x))^7}{7 f}+\frac {a^3 (i A+5 B) c^6 (1-i \tan (e+f x))^8}{8 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 f} \]
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Rubi [A]
time = 0.14, 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^6 (5 B+i A) (1-i \tan (e+f x))^8}{8 f}-\frac {4 a^3 c^6 (2 B+i A) (1-i \tan (e+f x))^7}{7 f}+\frac {2 a^3 c^6 (B+i A) (1-i \tan (e+f x))^6}{3 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 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))^6 \, dx &=\frac {(a c) \text {Subst}\left (\int (a+i a x)^2 (A+B x) (c-i c x)^5 \, 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)^5-\frac {4 a^2 (A-2 i B) (c-i c x)^6}{c}+\frac {a^2 (A-5 i B) (c-i c x)^7}{c^2}+\frac {i a^2 B (c-i c x)^8}{c^3}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {2 a^3 (i A+B) c^6 (1-i \tan (e+f x))^6}{3 f}-\frac {4 a^3 (i A+2 B) c^6 (1-i \tan (e+f x))^7}{7 f}+\frac {a^3 (i A+5 B) c^6 (1-i \tan (e+f x))^8}{8 f}-\frac {a^3 B c^6 (1-i \tan (e+f x))^9}{9 f}\\ \end {align*}
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Mathematica [A]
time = 3.07, size = 262, normalized size = 1.94 \begin {gather*} \frac {a^3 c^6 \sec (e) \sec ^9(e+f x) (63 (-3 i A+B) \cos (f x)+63 (-3 i A+B) \cos (2 e+f x)-84 i A \cos (2 e+3 f x)+84 B \cos (2 e+3 f x)-84 i A \cos (4 e+3 f x)+84 B \cos (4 e+3 f x)+189 A \sin (f x)+63 i B \sin (f x)-189 A \sin (2 e+f x)-63 i B \sin (2 e+f x)+168 A \sin (2 e+3 f x)-84 A \sin (4 e+3 f x)-84 i B \sin (4 e+3 f x)+108 A \sin (4 e+5 f x)+36 i B \sin (4 e+5 f x)+27 A \sin (6 e+7 f x)+9 i B \sin (6 e+7 f x)+3 A \sin (8 e+9 f x)+i B \sin (8 e+9 f x))}{1008 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 211, normalized size = 1.56
method | result | size |
risch | \(\frac {32 c^{6} a^{3} \left (84 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+84 B \,{\mathrm e}^{6 i \left (f x +e \right )}+108 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-36 B \,{\mathrm e}^{4 i \left (f x +e \right )}+27 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-9 B \,{\mathrm e}^{2 i \left (f x +e \right )}+3 i A -B \right )}{63 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{9}}\) | \(108\) |
derivativedivides | \(\frac {i c^{6} a^{3} \left (\frac {B \left (\tan ^{9}\left (f x +e \right )\right )}{9}+\frac {\left (3 i B +A \right ) \left (\tan ^{8}\left (f x +e \right )\right )}{8}+\frac {\left (-11 B -2 i A +5 i \left (-2 i B +A \right )\right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (-11 A +5 i \left (-2 i A -B \right )+10 i B \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (15 i A +15 B -10 i \left (-2 i B +A \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (15 A -10 i \left (-2 i A -B \right )-9 i B \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (-5 B +i \left (-2 i B +A \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (-5 A +i \left (-2 i A -B \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}-i A \tan \left (f x +e \right )\right )}{f}\) | \(211\) |
default | \(\frac {i c^{6} a^{3} \left (\frac {B \left (\tan ^{9}\left (f x +e \right )\right )}{9}+\frac {\left (3 i B +A \right ) \left (\tan ^{8}\left (f x +e \right )\right )}{8}+\frac {\left (-11 B -2 i A +5 i \left (-2 i B +A \right )\right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7}+\frac {\left (-11 A +5 i \left (-2 i A -B \right )+10 i B \right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6}+\frac {\left (15 i A +15 B -10 i \left (-2 i B +A \right )\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (15 A -10 i \left (-2 i A -B \right )-9 i B \right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4}+\frac {\left (-5 B +i \left (-2 i B +A \right )\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3}+\frac {\left (-5 A +i \left (-2 i A -B \right )\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2}-i A \tan \left (f x +e \right )\right )}{f}\) | \(211\) |
norman | \(\frac {A \,a^{3} c^{6} \tan \left (f x +e \right )}{f}-\frac {\left (3 i B \,a^{3} c^{6}+A \,a^{3} c^{6}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}-\frac {\left (5 i A \,a^{3} c^{6}+B \,a^{3} c^{6}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}-\frac {\left (-i A \,a^{3} c^{6}+3 B \,a^{3} c^{6}\right ) \left (\tan ^{8}\left (f x +e \right )\right )}{8 f}-\frac {\left (i B \,a^{3} c^{6}+A \,a^{3} c^{6}\right ) \left (\tan ^{5}\left (f x +e \right )\right )}{f}-\frac {\left (i B \,a^{3} c^{6}+3 A \,a^{3} c^{6}\right ) \left (\tan ^{7}\left (f x +e \right )\right )}{7 f}-\frac {\left (i A \,a^{3} c^{6}+5 B \,a^{3} c^{6}\right ) \left (\tan ^{6}\left (f x +e \right )\right )}{6 f}+\frac {\left (-3 i A \,a^{3} c^{6}+B \,a^{3} c^{6}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}+\frac {i B \,a^{3} c^{6} \left (\tan ^{9}\left (f x +e \right )\right )}{9 f}\) | \(267\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 202, normalized size = 1.50 \begin {gather*} -\frac {-56 i \, B a^{3} c^{6} \tan \left (f x + e\right )^{9} + 63 \, {\left (-i \, A + 3 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{8} + 72 \, {\left (3 \, A + i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{7} + 84 \, {\left (i \, A + 5 \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{6} + 504 \, {\left (A + i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{5} + 126 \, {\left (5 i \, A + B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{4} + 168 \, {\left (A + 3 i \, B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{3} + 252 \, {\left (3 i \, A - B\right )} a^{3} c^{6} \tan \left (f x + e\right )^{2} - 504 \, A a^{3} c^{6} \tan \left (f x + e\right )}{504 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.32, size = 206, normalized size = 1.53 \begin {gather*} -\frac {32 \, {\left (84 \, {\left (-i \, A - B\right )} a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} + 36 \, {\left (-3 i \, A + B\right )} a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, {\left (-3 i \, A + B\right )} a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-3 i \, A + B\right )} a^{3} c^{6}\right )}}{63 \, {\left (f e^{\left (18 i \, f x + 18 i \, e\right )} + 9 \, f e^{\left (16 i \, f x + 16 i \, e\right )} + 36 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 84 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 126 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 126 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 84 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 36 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, 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. 325 vs. \(2 (110) = 220\).
time = 1.23, size = 325, normalized size = 2.41 \begin {gather*} \frac {96 i A a^{3} c^{6} - 32 B a^{3} c^{6} + \left (864 i A a^{3} c^{6} e^{2 i e} - 288 B a^{3} c^{6} e^{2 i e}\right ) e^{2 i f x} + \left (3456 i A a^{3} c^{6} e^{4 i e} - 1152 B a^{3} c^{6} e^{4 i e}\right ) e^{4 i f x} + \left (2688 i A a^{3} c^{6} e^{6 i e} + 2688 B a^{3} c^{6} e^{6 i e}\right ) e^{6 i f x}}{63 f e^{18 i e} e^{18 i f x} + 567 f e^{16 i e} e^{16 i f x} + 2268 f e^{14 i e} e^{14 i f x} + 5292 f e^{12 i e} e^{12 i f x} + 7938 f e^{10 i e} e^{10 i f x} + 7938 f e^{8 i e} e^{8 i f x} + 5292 f e^{6 i e} e^{6 i f x} + 2268 f e^{4 i e} e^{4 i f x} + 567 f e^{2 i e} e^{2 i f x} + 63 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. 254 vs. \(2 (117) = 234\).
time = 1.44, size = 254, normalized size = 1.88 \begin {gather*} -\frac {32 \, {\left (-84 i \, A a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} - 84 \, B a^{3} c^{6} e^{\left (6 i \, f x + 6 i \, e\right )} - 108 i \, A a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} + 36 \, B a^{3} c^{6} e^{\left (4 i \, f x + 4 i \, e\right )} - 27 i \, A a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} + 9 \, B a^{3} c^{6} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, A a^{3} c^{6} + B a^{3} c^{6}\right )}}{63 \, {\left (f e^{\left (18 i \, f x + 18 i \, e\right )} + 9 \, f e^{\left (16 i \, f x + 16 i \, e\right )} + 36 \, f e^{\left (14 i \, f x + 14 i \, e\right )} + 84 \, f e^{\left (12 i \, f x + 12 i \, e\right )} + 126 \, f e^{\left (10 i \, f x + 10 i \, e\right )} + 126 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 84 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 36 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 9 \, 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 = 8.70, size = 208, normalized size = 1.54 \begin {gather*} \frac {A\,a^3\,c^6\,\mathrm {tan}\left (e+f\,x\right )+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (-3\,B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}+a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^5\,\left (-B+A\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (5\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^7\,\left (-B+A\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{7}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (3\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}-\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^6\,\left (A-B\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{6}+\frac {a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^8\,\left (A+B\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{8}+\frac {B\,a^3\,c^6\,{\mathrm {tan}\left (e+f\,x\right )}^9\,1{}\mathrm {i}}{9}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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