Optimal. Leaf size=225 \[ -\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{4 a^4 (B+i A) \tan ^2(c+d x)}{d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{8 a^4 (A-i B) \tan (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
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Rubi [A] time = 0.642305, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.147, Rules used = {3594, 3592, 3528, 3525, 3475} \[ -\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{4 a^4 (B+i A) \tan ^2(c+d x)}{d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{8 a^4 (A-i B) \tan (c+d x)}{d}+\frac{8 a^4 (B+i A) \log (\cos (c+d x))}{d}-8 a^4 x (A-i B)+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d} \]
Antiderivative was successfully verified.
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Rule 3594
Rule 3592
Rule 3528
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}+\frac{1}{6} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^3 (3 a (2 A-i B)+3 a (2 i A+3 B) \tan (c+d x)) \, dx\\ &=\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}+\frac{1}{30} \int \tan ^2(c+d x) (a+i a \tan (c+d x))^2 \left (6 a^2 (8 A-7 i B)+6 a^2 (12 i A+13 B) \tan (c+d x)\right ) \, dx\\ &=\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \tan ^2(c+d x) (a+i a \tan (c+d x)) \left (6 a^3 (68 A-67 i B)+6 a^3 (92 i A+93 B) \tan (c+d x)\right ) \, dx\\ &=-\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \tan ^2(c+d x) \left (960 a^4 (A-i B)+960 a^4 (i A+B) \tan (c+d x)\right ) \, dx\\ &=\frac{4 a^4 (i A+B) \tan ^2(c+d x)}{d}-\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}+\frac{1}{120} \int \tan (c+d x) \left (-960 a^4 (i A+B)+960 a^4 (A-i B) \tan (c+d x)\right ) \, dx\\ &=-8 a^4 (A-i B) x+\frac{8 a^4 (A-i B) \tan (c+d x)}{d}+\frac{4 a^4 (i A+B) \tan ^2(c+d x)}{d}-\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}-\left (8 a^4 (i A+B)\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 (A-i B) x+\frac{8 a^4 (i A+B) \log (\cos (c+d x))}{d}+\frac{8 a^4 (A-i B) \tan (c+d x)}{d}+\frac{4 a^4 (i A+B) \tan ^2(c+d x)}{d}-\frac{a^4 (92 A-93 i B) \tan ^3(c+d x)}{60 d}+\frac{i a B \tan ^3(c+d x) (a+i a \tan (c+d x))^3}{6 d}-\frac{(2 A-3 i B) \tan ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^2}{10 d}-\frac{(12 A-13 i B) \tan ^3(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{20 d}\\ \end{align*}
Mathematica [B] time = 8.66036, size = 951, normalized size = 4.23 \[ \frac{x \left (-4 A \cos ^4(c)+4 i B \cos ^4(c)+20 i A \sin (c) \cos ^3(c)+20 B \sin (c) \cos ^3(c)+40 A \sin ^2(c) \cos ^2(c)-40 i B \sin ^2(c) \cos ^2(c)+4 A \cos ^2(c)-4 i B \cos ^2(c)-40 i A \sin ^3(c) \cos (c)-40 B \sin ^3(c) \cos (c)-12 i A \sin (c) \cos (c)-12 B \sin (c) \cos (c)-20 A \sin ^4(c)+20 i B \sin ^4(c)-12 A \sin ^2(c)+12 i B \sin ^2(c)+4 i A \sin ^4(c) \tan (c)+4 B \sin ^4(c) \tan (c)+4 i A \sin ^2(c) \tan (c)+4 B \sin ^2(c) \tan (c)-i (A-i B) (8 \cos (4 c)-8 i \sin (4 c)) \tan (c)\right ) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{(\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(i A \cos (2 c)+B \cos (2 c)+A \sin (2 c)-i B \sin (2 c)) \left (4 \cos (2 c) \log \left (\cos ^2(c+d x)\right )-4 i \log \left (\cos ^2(c+d x)\right ) \sin (2 c)\right ) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x)) \cos ^5(c+d x)}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\sec (c) \sec (c+d x) \left (\frac{1}{240} \cos (4 c)-\frac{1}{240} i \sin (4 c)\right ) (420 i A \cos (c)+490 B \cos (c)-600 A d x \cos (c)+600 i B d x \cos (c)+300 i A \cos (c+2 d x)+345 B \cos (c+2 d x)-450 A d x \cos (c+2 d x)+450 i B d x \cos (c+2 d x)+300 i A \cos (3 c+2 d x)+345 B \cos (3 c+2 d x)-450 A d x \cos (3 c+2 d x)+450 i B d x \cos (3 c+2 d x)+90 i A \cos (3 c+4 d x)+120 B \cos (3 c+4 d x)-180 A d x \cos (3 c+4 d x)+180 i B d x \cos (3 c+4 d x)+90 i A \cos (5 c+4 d x)+120 B \cos (5 c+4 d x)-180 A d x \cos (5 c+4 d x)+180 i B d x \cos (5 c+4 d x)-30 A d x \cos (5 c+6 d x)+30 i B d x \cos (5 c+6 d x)-30 A d x \cos (7 c+6 d x)+30 i B d x \cos (7 c+6 d x)-790 A \sin (c)+860 i B \sin (c)+720 A \sin (c+2 d x)-780 i B \sin (c+2 d x)-465 A \sin (3 c+2 d x)+510 i B \sin (3 c+2 d x)+354 A \sin (3 c+4 d x)-366 i B \sin (3 c+4 d x)-120 A \sin (5 c+4 d x)+150 i B \sin (5 c+4 d x)+79 A \sin (5 c+6 d x)-86 i B \sin (5 c+6 d x)) (i \tan (c+d x) a+a)^4 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^4 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 264, normalized size = 1.2 \begin{align*}{\frac{-{\frac{4\,i}{5}}{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{d}}+{\frac{{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{6}}{6\,d}}-{\frac{i{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}+{\frac{{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}+{\frac{{\frac{8\,i}{3}}{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{7\,{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}+{\frac{4\,i{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{7\,{a}^{4}A \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{8\,i{a}^{4}B\tan \left ( dx+c \right ) }{d}}+4\,{\frac{{a}^{4}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}+8\,{\frac{{a}^{4}A\tan \left ( dx+c \right ) }{d}}-{\frac{4\,i{a}^{4}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-4\,{\frac{{a}^{4}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+{\frac{8\,i{a}^{4}B\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-8\,{\frac{{a}^{4}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.17629, size = 208, normalized size = 0.92 \begin{align*} \frac{10 \, B a^{4} \tan \left (d x + c\right )^{6} +{\left (12 \, A - 48 i \, B\right )} a^{4} \tan \left (d x + c\right )^{5} - 15 \,{\left (4 i \, A + 7 \, B\right )} a^{4} \tan \left (d x + c\right )^{4} -{\left (140 \, A - 160 i \, B\right )} a^{4} \tan \left (d x + c\right )^{3} - 240 \,{\left (-i \, A - B\right )} a^{4} \tan \left (d x + c\right )^{2} - 60 \,{\left (d x + c\right )}{\left (8 \, A - 8 i \, B\right )} a^{4} - 240 \,{\left (i \, A + B\right )} a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) +{\left (480 \, A - 480 i \, B\right )} a^{4} \tan \left (d x + c\right )}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7098, size = 1019, normalized size = 4.53 \begin{align*} \frac{{\left (840 i \, A + 1080 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (3060 i \, A + 3420 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (4840 i \, A + 5400 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (4080 i \, A + 4500 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (1776 i \, A + 1944 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (316 i \, A + 344 \, B\right )} a^{4} +{\left ({\left (120 i \, A + 120 \, B\right )} a^{4} e^{\left (12 i \, d x + 12 i \, c\right )} +{\left (720 i \, A + 720 \, B\right )} a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (1800 i \, A + 1800 \, B\right )} a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (2400 i \, A + 2400 \, B\right )} a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (1800 i \, A + 1800 \, B\right )} a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (720 i \, A + 720 \, B\right )} a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (120 i \, A + 120 \, B\right )} a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{15 \,{\left (d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.98238, size = 810, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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