Optimal. Leaf size=190 \[ 4 a^3 (c-i d)^3 x+\frac {4 a^3 (i c+d)^3 \log (\cos (e+f x))}{f}+\frac {4 i a^3 (c-i d)^2 d \tan (e+f x)}{f}+\frac {2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f} \]
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Rubi [A]
time = 0.24, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3637, 3673,
3609, 3606, 3556} \begin {gather*} \frac {a^3 (-11 d+i c) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {2 a^3 (d+i c) (c+d \tan (e+f x))^2}{f}+\frac {4 i a^3 d (c-i d)^2 \tan (e+f x)}{f}+\frac {4 a^3 (d+i c)^3 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 3606
Rule 3609
Rule 3637
Rule 3673
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3 \, dx &=-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac {a \int (a+i a \tan (e+f x)) (a (i c+9 d)+a (c+11 i d) \tan (e+f x)) (c+d \tan (e+f x))^3 \, dx}{5 d}\\ &=\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac {a \int (c+d \tan (e+f x))^3 \left (20 a^2 d+20 i a^2 d \tan (e+f x)\right ) \, dx}{5 d}\\ &=\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac {a \int (c+d \tan (e+f x))^2 \left (20 a^2 (c-i d) d+20 a^2 d (i c+d) \tan (e+f x)\right ) \, dx}{5 d}\\ &=\frac {2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}+\frac {a \int (c+d \tan (e+f x)) \left (20 a^2 (c-i d)^2 d+20 i a^2 (c-i d)^2 d \tan (e+f x)\right ) \, dx}{5 d}\\ &=4 a^3 (c-i d)^3 x+\frac {4 i a^3 (c-i d)^2 d \tan (e+f x)}{f}+\frac {2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}-\left (4 a^3 (i c+d)^3\right ) \int \tan (e+f x) \, dx\\ &=4 a^3 (c-i d)^3 x+\frac {4 a^3 (i c+d)^3 \log (\cos (e+f x))}{f}+\frac {4 i a^3 (c-i d)^2 d \tan (e+f x)}{f}+\frac {2 a^3 (i c+d) (c+d \tan (e+f x))^2}{f}+\frac {4 i a^3 (c+d \tan (e+f x))^3}{3 f}+\frac {a^3 (i c-11 d) (c+d \tan (e+f x))^4}{20 d^2 f}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^4}{5 d f}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1564\) vs. \(2(190)=380\).
time = 8.15, size = 1564, normalized size = 8.23 \begin {gather*} \frac {\cos ^3(e+f x) \left (-i c^3 \cos \left (\frac {3 e}{2}\right )-3 c^2 d \cos \left (\frac {3 e}{2}\right )+3 i c d^2 \cos \left (\frac {3 e}{2}\right )+d^3 \cos \left (\frac {3 e}{2}\right )-c^3 \sin \left (\frac {3 e}{2}\right )+3 i c^2 d \sin \left (\frac {3 e}{2}\right )+3 c d^2 \sin \left (\frac {3 e}{2}\right )-i d^3 \sin \left (\frac {3 e}{2}\right )\right ) \left (2 \cos \left (\frac {3 e}{2}\right ) \log \left (\cos ^2(e+f x)\right )-2 i \log \left (\cos ^2(e+f x)\right ) \sin \left (\frac {3 e}{2}\right )\right ) (a+i a \tan (e+f x))^3}{f (\cos (f x)+i \sin (f x))^3}+\frac {\sec (e) \sec ^2(e+f x) \left (\frac {1}{240} \cos (3 e)-\frac {1}{240} i \sin (3 e)\right ) \left (-45 i c^3 \cos (f x)-405 c^2 d \cos (f x)+585 i c d^2 \cos (f x)+225 d^3 \cos (f x)+300 c^3 f x \cos (f x)-900 i c^2 d f x \cos (f x)-900 c d^2 f x \cos (f x)+300 i d^3 f x \cos (f x)-45 i c^3 \cos (2 e+f x)-405 c^2 d \cos (2 e+f x)+585 i c d^2 \cos (2 e+f x)+225 d^3 \cos (2 e+f x)+300 c^3 f x \cos (2 e+f x)-900 i c^2 d f x \cos (2 e+f x)-900 c d^2 f x \cos (2 e+f x)+300 i d^3 f x \cos (2 e+f x)-15 i c^3 \cos (2 e+3 f x)-135 c^2 d \cos (2 e+3 f x)+225 i c d^2 \cos (2 e+3 f x)+105 d^3 \cos (2 e+3 f x)+150 c^3 f x \cos (2 e+3 f x)-450 i c^2 d f x \cos (2 e+3 f x)-450 c d^2 f x \cos (2 e+3 f x)+150 i d^3 f x \cos (2 e+3 f x)-15 i c^3 \cos (4 e+3 f x)-135 c^2 d \cos (4 e+3 f x)+225 i c d^2 \cos (4 e+3 f x)+105 d^3 \cos (4 e+3 f x)+150 c^3 f x \cos (4 e+3 f x)-450 i c^2 d f x \cos (4 e+3 f x)-450 c d^2 f x \cos (4 e+3 f x)+150 i d^3 f x \cos (4 e+3 f x)+30 c^3 f x \cos (4 e+5 f x)-90 i c^2 d f x \cos (4 e+5 f x)-90 c d^2 f x \cos (4 e+5 f x)+30 i d^3 f x \cos (4 e+5 f x)+30 c^3 f x \cos (6 e+5 f x)-90 i c^2 d f x \cos (6 e+5 f x)-90 c d^2 f x \cos (6 e+5 f x)+30 i d^3 f x \cos (6 e+5 f x)-270 c^3 \sin (f x)+1140 i c^2 d \sin (f x)+1260 c d^2 \sin (f x)-470 i d^3 \sin (f x)+180 c^3 \sin (2 e+f x)-810 i c^2 d \sin (2 e+f x)-990 c d^2 \sin (2 e+f x)+360 i d^3 \sin (2 e+f x)-180 c^3 \sin (2 e+3 f x)+750 i c^2 d \sin (2 e+3 f x)+810 c d^2 \sin (2 e+3 f x)-280 i d^3 \sin (2 e+3 f x)+45 c^3 \sin (4 e+3 f x)-225 i c^2 d \sin (4 e+3 f x)-315 c d^2 \sin (4 e+3 f x)+135 i d^3 \sin (4 e+3 f x)-45 c^3 \sin (4 e+5 f x)+195 i c^2 d \sin (4 e+5 f x)+225 c d^2 \sin (4 e+5 f x)-83 i d^3 \sin (4 e+5 f x)\right ) (a+i a \tan (e+f x))^3}{f (\cos (f x)+i \sin (f x))^3}+\frac {x \cos ^3(e+f x) \left (-2 c^3 \cos (e)+6 i c^2 d \cos (e)+6 c d^2 \cos (e)-2 i d^3 \cos (e)+2 c^3 \cos ^3(e)-6 i c^2 d \cos ^3(e)-6 c d^2 \cos ^3(e)+2 i d^3 \cos ^3(e)+4 i c^3 \sin (e)+12 c^2 d \sin (e)-12 i c d^2 \sin (e)-4 d^3 \sin (e)-8 i c^3 \cos ^2(e) \sin (e)-24 c^2 d \cos ^2(e) \sin (e)+24 i c d^2 \cos ^2(e) \sin (e)+8 d^3 \cos ^2(e) \sin (e)-12 c^3 \cos (e) \sin ^2(e)+36 i c^2 d \cos (e) \sin ^2(e)+36 c d^2 \cos (e) \sin ^2(e)-12 i d^3 \cos (e) \sin ^2(e)+8 i c^3 \sin ^3(e)+24 c^2 d \sin ^3(e)-24 i c d^2 \sin ^3(e)-8 d^3 \sin ^3(e)+2 c^3 \sin (e) \tan (e)-6 i c^2 d \sin (e) \tan (e)-6 c d^2 \sin (e) \tan (e)+2 i d^3 \sin (e) \tan (e)+2 c^3 \sin ^3(e) \tan (e)-6 i c^2 d \sin ^3(e) \tan (e)-6 c d^2 \sin ^3(e) \tan (e)+2 i d^3 \sin ^3(e) \tan (e)+(-i c-d)^3 (4 \cos (3 e)-4 i \sin (3 e)) \tan (e)\right ) (a+i a \tan (e+f x))^3}{(\cos (f x)+i \sin (f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 269, normalized size = 1.42
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {i c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-4 i d^{3} \tan \left (f x +e \right )-\frac {i d^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5}+12 i c^{2} d \tan \left (f x +e \right )-\frac {3 d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+6 i c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )-i c^{2} d \left (\tan ^{3}\left (f x +e \right )\right )-3 c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {4 i d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {3 i c \,d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {9 c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+2 d^{3} \left (\tan ^{2}\left (f x +e \right )\right )-3 c^{3} \tan \left (f x +e \right )+12 c \,d^{2} \tan \left (f x +e \right )+\frac {\left (4 i c^{3}-12 i c \,d^{2}+12 c^{2} d -4 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-12 i c^{2} d +4 i d^{3}+4 c^{3}-12 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(269\) |
default | \(\frac {a^{3} \left (-\frac {i c^{3} \left (\tan ^{2}\left (f x +e \right )\right )}{2}-4 i d^{3} \tan \left (f x +e \right )-\frac {i d^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5}+12 i c^{2} d \tan \left (f x +e \right )-\frac {3 d^{3} \left (\tan ^{4}\left (f x +e \right )\right )}{4}+6 i c \,d^{2} \left (\tan ^{2}\left (f x +e \right )\right )-i c^{2} d \left (\tan ^{3}\left (f x +e \right )\right )-3 c \,d^{2} \left (\tan ^{3}\left (f x +e \right )\right )+\frac {4 i d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3}-\frac {3 i c \,d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4}-\frac {9 c^{2} d \left (\tan ^{2}\left (f x +e \right )\right )}{2}+2 d^{3} \left (\tan ^{2}\left (f x +e \right )\right )-3 c^{3} \tan \left (f x +e \right )+12 c \,d^{2} \tan \left (f x +e \right )+\frac {\left (4 i c^{3}-12 i c \,d^{2}+12 c^{2} d -4 d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2}+\left (-12 i c^{2} d +4 i d^{3}+4 c^{3}-12 c \,d^{2}\right ) \arctan \left (\tan \left (f x +e \right )\right )\right )}{f}\) | \(269\) |
norman | \(\left (-12 i a^{3} c^{2} d +4 i a^{3} d^{3}+4 a^{3} c^{3}-12 a^{3} c \,d^{2}\right ) x -\frac {3 \left (i a^{3} c \,d^{2}+a^{3} d^{3}\right ) \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}-\frac {\left (-12 i a^{3} c^{2} d +4 i a^{3} d^{3}+3 a^{3} c^{3}-12 a^{3} c \,d^{2}\right ) \tan \left (f x +e \right )}{f}+\frac {\left (-i a^{3} c^{3}+12 i a^{3} c \,d^{2}-9 a^{3} c^{2} d +4 a^{3} d^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{2 f}-\frac {i a^{3} d^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{5 f}-\frac {i a^{3} d \left (-9 i c d +3 c^{2}-4 d^{2}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}-\frac {2 \left (-i a^{3} c^{3}+3 i a^{3} c \,d^{2}-3 a^{3} c^{2} d +a^{3} d^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{f}\) | \(276\) |
risch | \(\frac {12 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c \,d^{2}}{f}+\frac {24 i a^{3} c^{2} d e}{f}-\frac {8 i a^{3} d^{3} e}{f}-\frac {4 i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c^{3}}{f}-\frac {8 a^{3} c^{3} e}{f}+\frac {24 a^{3} c \,d^{2} e}{f}+\frac {2 a^{3} \left (-60 i c^{3} {\mathrm e}^{8 i \left (f x +e \right )}-315 i c^{3} {\mathrm e}^{4 i \left (f x +e \right )}-360 c^{2} d \,{\mathrm e}^{8 i \left (f x +e \right )}+240 d^{3} {\mathrm e}^{8 i \left (f x +e \right )}+1575 i c \,d^{2} {\mathrm e}^{6 i \left (f x +e \right )}-45 i c^{3}-1215 c^{2} d \,{\mathrm e}^{6 i \left (f x +e \right )}+585 d^{3} {\mathrm e}^{6 i \left (f x +e \right )}-195 i c^{3} {\mathrm e}^{2 i \left (f x +e \right )}+1035 i c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-1545 c^{2} d \,{\mathrm e}^{4 i \left (f x +e \right )}+695 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+1845 i c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-225 i c^{3} {\mathrm e}^{6 i \left (f x +e \right )}-885 c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+385 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+540 i c \,d^{2} {\mathrm e}^{8 i \left (f x +e \right )}+225 i c \,d^{2}-195 c^{2} d +83 d^{3}\right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{5}}-\frac {12 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) c^{2} d}{f}+\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) d^{3}}{f}\) | \(436\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 269, normalized size = 1.42 \begin {gather*} -\frac {12 i \, a^{3} d^{3} \tan \left (f x + e\right )^{5} + 45 \, {\left (i \, a^{3} c d^{2} + a^{3} d^{3}\right )} \tan \left (f x + e\right )^{4} + 20 \, {\left (3 i \, a^{3} c^{2} d + 9 \, a^{3} c d^{2} - 4 i \, a^{3} d^{3}\right )} \tan \left (f x + e\right )^{3} + 30 \, {\left (i \, a^{3} c^{3} + 9 \, a^{3} c^{2} d - 12 i \, a^{3} c d^{2} - 4 \, a^{3} d^{3}\right )} \tan \left (f x + e\right )^{2} - 240 \, {\left (a^{3} c^{3} - 3 i \, a^{3} c^{2} d - 3 \, a^{3} c d^{2} + i \, a^{3} d^{3}\right )} {\left (f x + e\right )} + 120 \, {\left (-i \, a^{3} c^{3} - 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} + a^{3} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + 60 \, {\left (3 \, a^{3} c^{3} - 12 i \, a^{3} c^{2} d - 12 \, a^{3} c d^{2} + 4 i \, a^{3} d^{3}\right )} \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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 577 vs. \(2 (177) = 354\).
time = 0.93, size = 577, normalized size = 3.04 \begin {gather*} -\frac {2 \, {\left (45 i \, a^{3} c^{3} + 195 \, a^{3} c^{2} d - 225 i \, a^{3} c d^{2} - 83 \, a^{3} d^{3} + 60 \, {\left (i \, a^{3} c^{3} + 6 \, a^{3} c^{2} d - 9 i \, a^{3} c d^{2} - 4 \, a^{3} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 45 \, {\left (5 i \, a^{3} c^{3} + 27 \, a^{3} c^{2} d - 35 i \, a^{3} c d^{2} - 13 \, a^{3} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 5 \, {\left (63 i \, a^{3} c^{3} + 309 \, a^{3} c^{2} d - 369 i \, a^{3} c d^{2} - 139 \, a^{3} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, {\left (39 i \, a^{3} c^{3} + 177 \, a^{3} c^{2} d - 207 i \, a^{3} c d^{2} - 77 \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + 30 \, {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3} + {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, {\left (i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d - 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )\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. 476 vs. \(2 (167) = 334\).
time = 0.81, size = 476, normalized size = 2.51 \begin {gather*} - \frac {4 i a^{3} \left (c - i d\right )^{3} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {- 90 i a^{3} c^{3} - 390 a^{3} c^{2} d + 450 i a^{3} c d^{2} + 166 a^{3} d^{3} + \left (- 390 i a^{3} c^{3} e^{2 i e} - 1770 a^{3} c^{2} d e^{2 i e} + 2070 i a^{3} c d^{2} e^{2 i e} + 770 a^{3} d^{3} e^{2 i e}\right ) e^{2 i f x} + \left (- 630 i a^{3} c^{3} e^{4 i e} - 3090 a^{3} c^{2} d e^{4 i e} + 3690 i a^{3} c d^{2} e^{4 i e} + 1390 a^{3} d^{3} e^{4 i e}\right ) e^{4 i f x} + \left (- 450 i a^{3} c^{3} e^{6 i e} - 2430 a^{3} c^{2} d e^{6 i e} + 3150 i a^{3} c d^{2} e^{6 i e} + 1170 a^{3} d^{3} e^{6 i e}\right ) e^{6 i f x} + \left (- 120 i a^{3} c^{3} e^{8 i e} - 720 a^{3} c^{2} d e^{8 i e} + 1080 i a^{3} c d^{2} e^{8 i e} + 480 a^{3} d^{3} e^{8 i e}\right ) e^{8 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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1117 vs. \(2 (177) = 354\).
time = 1.03, size = 1117, normalized size = 5.88 \begin {gather*} -\frac {2 \, {\left (30 i \, a^{3} c^{3} e^{\left (10 i \, f x + 10 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 90 \, a^{3} c^{2} d e^{\left (10 i \, f x + 10 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 90 i \, a^{3} c d^{2} e^{\left (10 i \, f x + 10 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 30 \, a^{3} d^{3} e^{\left (10 i \, f x + 10 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 150 i \, a^{3} c^{3} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 450 \, a^{3} c^{2} d e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 450 i \, a^{3} c d^{2} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 150 \, a^{3} d^{3} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 300 i \, a^{3} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 900 \, a^{3} c^{2} d e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 900 i \, a^{3} c d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 300 \, a^{3} d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 300 i \, a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 900 \, a^{3} c^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 900 i \, a^{3} c d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 300 \, a^{3} d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 150 i \, a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 450 \, a^{3} c^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 450 i \, a^{3} c d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 150 \, a^{3} d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 60 i \, a^{3} c^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 360 \, a^{3} c^{2} d e^{\left (8 i \, f x + 8 i \, e\right )} - 540 i \, a^{3} c d^{2} e^{\left (8 i \, f x + 8 i \, e\right )} - 240 \, a^{3} d^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 225 i \, a^{3} c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 1215 \, a^{3} c^{2} d e^{\left (6 i \, f x + 6 i \, e\right )} - 1575 i \, a^{3} c d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 585 \, a^{3} d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 315 i \, a^{3} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 1545 \, a^{3} c^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} - 1845 i \, a^{3} c d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 695 \, a^{3} d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 195 i \, a^{3} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 885 \, a^{3} c^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} - 1035 i \, a^{3} c d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 385 \, a^{3} d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 30 i \, a^{3} c^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 90 \, a^{3} c^{2} d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 90 i \, a^{3} c d^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 30 \, a^{3} d^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 45 i \, a^{3} c^{3} + 195 \, a^{3} c^{2} d - 225 i \, a^{3} c d^{2} - 83 \, a^{3} d^{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 = 5.19, size = 356, normalized size = 1.87 \begin {gather*} -\frac {{\mathrm {tan}\left (e+f\,x\right )}^4\,\left (\frac {a^3\,d^3}{4}+\frac {a^3\,d^2\,\left (2\,d+c\,3{}\mathrm {i}\right )}{4}\right )}{f}-\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^3\,d^3\,1{}\mathrm {i}-a^3\,c\,\left (c^2\,1{}\mathrm {i}+6\,c\,d-d^2\,3{}\mathrm {i}\right )\,1{}\mathrm {i}-a^3\,d\,\left (c^2\,3{}\mathrm {i}+6\,c\,d-d^2\,1{}\mathrm {i}\right )+a^3\,c^2\,\left (2\,c-d\,3{}\mathrm {i}\right )+a^3\,d^2\,\left (2\,d+c\,3{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (a^3\,c^3\,4{}\mathrm {i}+12\,a^3\,c^2\,d-a^3\,c\,d^2\,12{}\mathrm {i}-4\,a^3\,d^3\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a^3\,d^3\,1{}\mathrm {i}}{3}-\frac {a^3\,d\,\left (c^2\,3{}\mathrm {i}+6\,c\,d-d^2\,1{}\mathrm {i}\right )}{3}+\frac {a^3\,d^2\,\left (2\,d+c\,3{}\mathrm {i}\right )\,1{}\mathrm {i}}{3}\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^3\,d^3}{2}-\frac {a^3\,c\,\left (c^2\,1{}\mathrm {i}+6\,c\,d-d^2\,3{}\mathrm {i}\right )}{2}+\frac {a^3\,d\,\left (c^2\,3{}\mathrm {i}+6\,c\,d-d^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {a^3\,d^2\,\left (2\,d+c\,3{}\mathrm {i}\right )}{2}\right )}{f}-\frac {a^3\,d^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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