Optimal. Leaf size=138 \[ \frac{2 a^3 (B+i A) \tan (c+d x)}{d}-\frac{4 a^3 (A-i B) \log (\cos (c+d x))}{d}-4 a^3 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d} \]
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Rubi [A] time = 0.134645, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3592, 3527, 3478, 3477, 3475} \[ \frac{2 a^3 (B+i A) \tan (c+d x)}{d}-\frac{4 a^3 (A-i B) \log (\cos (c+d x))}{d}-4 a^3 x (B+i A)+\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d} \]
Antiderivative was successfully verified.
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Rule 3592
Rule 3527
Rule 3478
Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int \tan (c+d x) (a+i a \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}+\int (a+i a \tan (c+d x))^3 (-B+A \tan (c+d x)) \, dx\\ &=\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}-(i A+B) \int (a+i a \tan (c+d x))^3 \, dx\\ &=\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}-(2 a (i A+B)) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-4 a^3 (i A+B) x+\frac{2 a^3 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}+\left (4 a^3 (A-i B)\right ) \int \tan (c+d x) \, dx\\ &=-4 a^3 (i A+B) x-\frac{4 a^3 (A-i B) \log (\cos (c+d x))}{d}+\frac{2 a^3 (i A+B) \tan (c+d x)}{d}+\frac{a (A-i B) (a+i a \tan (c+d x))^2}{2 d}+\frac{A (a+i a \tan (c+d x))^3}{3 d}-\frac{i B (a+i a \tan (c+d x))^4}{4 a d}\\ \end{align*}
Mathematica [B] time = 7.61, size = 980, normalized size = 7.1 \[ \frac{x \left (-2 i A \cos ^3(c)-2 B \cos ^3(c)-8 A \sin (c) \cos ^2(c)+8 i B \sin (c) \cos ^2(c)+12 i A \sin ^2(c) \cos (c)+12 B \sin ^2(c) \cos (c)+2 i A \cos (c)+2 B \cos (c)+8 A \sin ^3(c)-8 i B \sin ^3(c)+4 A \sin (c)-4 i B \sin (c)-2 i A \sin ^3(c) \tan (c)-2 B \sin ^3(c) \tan (c)-2 i A \sin (c) \tan (c)-2 B \sin (c) \tan (c)+(A-i B) (4 \cos (3 c)-4 i \sin (3 c)) \tan (c)\right ) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{(\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left (A \cos \left (\frac{3 c}{2}\right )-i B \cos \left (\frac{3 c}{2}\right )-i A \sin \left (\frac{3 c}{2}\right )-B \sin \left (\frac{3 c}{2}\right )\right ) \left (2 i \log \left (\cos ^2(c+d x)\right ) \sin \left (\frac{3 c}{2}\right )-2 \cos \left (\frac{3 c}{2}\right ) \log \left (\cos ^2(c+d x)\right )\right ) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(A-i B) (-4 i d x \cos (3 c)-4 d x \sin (3 c)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^4(c+d x)}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left (\frac{1}{3} \cos (3 c)-\frac{1}{3} i \sin (3 c)\right ) (13 i A \sin (d x)+15 B \sin (d x)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^3(c+d x)}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{(-9 A \cos (c)+15 i B \cos (c)-2 i A \sin (c)-6 B \sin (c)) \left (\frac{1}{6} \cos (3 c)-\frac{1}{6} i \sin (3 c)\right ) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos ^2(c+d x)}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left (\frac{1}{3} \cos (3 c)-\frac{1}{3} i \sin (3 c)\right ) (-i A \sin (d x)-3 B \sin (d x)) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x)) \cos (c+d x)}{d \left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{c}{2}\right )+\sin \left (\frac{c}{2}\right )\right ) (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))}+\frac{\left (-\frac{1}{4} i B \cos (3 c)-\frac{1}{4} B \sin (3 c)\right ) (i \tan (c+d x) a+a)^3 (A+B \tan (c+d x))}{d (\cos (d x)+i \sin (d x))^3 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 195, normalized size = 1.4 \begin{align*}{\frac{-{\frac{i}{4}}{a}^{3}B \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{{\frac{i}{3}}{a}^{3}A \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{2\,i{a}^{3}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{{a}^{3}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d}}+{\frac{4\,i{a}^{3}A\tan \left ( dx+c \right ) }{d}}-{\frac{3\,{a}^{3}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+4\,{\frac{{a}^{3}B\tan \left ( dx+c \right ) }{d}}-{\frac{2\,i{a}^{3}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+2\,{\frac{{a}^{3}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{4\,i{a}^{3}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-4\,{\frac{{a}^{3}B\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 = 1.69836, size = 155, normalized size = 1.12 \begin{align*} -\frac{3 i \, B a^{3} \tan \left (d x + c\right )^{4} + 4 \,{\left (i \, A + 3 \, B\right )} a^{3} \tan \left (d x + c\right )^{3} +{\left (18 \, A - 24 i \, B\right )} a^{3} \tan \left (d x + c\right )^{2} + 48 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{3} - 12 \,{\left (2 \, A - 2 i \, B\right )} a^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 48 \,{\left (-i \, A - B\right )} a^{3} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4785, size = 626, normalized size = 4.54 \begin{align*} -\frac{2 \,{\left (12 \,{\left (2 \, A - 3 i \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \,{\left (19 \, A - 23 i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \,{\left (23 \, A - 27 i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (13 \, A - 15 i \, B\right )} a^{3} + 6 \,{\left ({\left (A - i \, B\right )} a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \,{\left (A - i \, B\right )} a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \,{\left (A - i \, B\right )} a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \,{\left (A - i \, B\right )} a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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 [A] time = 25.5057, size = 223, normalized size = 1.62 \begin{align*} \frac{4 a^{3} \left (- A + i B\right ) \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (16 A a^{3} - 24 i B a^{3}\right ) e^{- 2 i c} e^{6 i d x}}{d} - \frac{\left (26 A a^{3} - 30 i B a^{3}\right ) e^{- 8 i c}}{3 d} - \frac{\left (38 A a^{3} - 46 i B a^{3}\right ) e^{- 4 i c} e^{4 i d x}}{d} - \frac{\left (92 A a^{3} - 108 i B a^{3}\right ) e^{- 6 i c} e^{2 i d x}}{3 d}}{e^{8 i d x} + 4 e^{- 2 i c} e^{6 i d x} + 6 e^{- 4 i c} e^{4 i d x} + 4 e^{- 6 i c} e^{2 i d x} + e^{- 8 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.39743, size = 551, normalized size = 3.99 \begin{align*} -\frac{12 \, A a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, B a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 48 \, A a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 48 i \, B a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 72 \, A a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 72 i \, B a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 48 \, A a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 48 i \, B a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 48 \, A a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 72 i \, B a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} + 114 \, A a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 138 i \, B a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 92 \, A a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - 108 i \, B a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 12 \, A a^{3} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, B a^{3} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 26 \, A a^{3} - 30 i \, B a^{3}}{3 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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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