Optimal. Leaf size=107 \[ \frac{a^2 (B+i A) \tan (c+d x)}{d}-\frac{2 a^2 (A-i B) \log (\cos (c+d x))}{d}-2 a^2 x (B+i A)+\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d} \]
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Rubi [A] time = 0.115465, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3592, 3527, 3477, 3475} \[ \frac{a^2 (B+i A) \tan (c+d x)}{d}-\frac{2 a^2 (A-i B) \log (\cos (c+d x))}{d}-2 a^2 x (B+i A)+\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d} \]
Antiderivative was successfully verified.
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Rule 3592
Rule 3527
Rule 3477
Rule 3475
Rubi steps
\begin{align*} \int \tan (c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac{i B (a+i a \tan (c+d x))^3}{3 a d}+\int (a+i a \tan (c+d x))^2 (-B+A \tan (c+d x)) \, dx\\ &=\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d}-(i A+B) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-2 a^2 (i A+B) x+\frac{a^2 (i A+B) \tan (c+d x)}{d}+\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d}+\left (2 a^2 (A-i B)\right ) \int \tan (c+d x) \, dx\\ &=-2 a^2 (i A+B) x-\frac{2 a^2 (A-i B) \log (\cos (c+d x))}{d}+\frac{a^2 (i A+B) \tan (c+d x)}{d}+\frac{A (a+i a \tan (c+d x))^2}{2 d}-\frac{i B (a+i a \tan (c+d x))^3}{3 a d}\\ \end{align*}
Mathematica [B] time = 3.91559, size = 273, normalized size = 2.55 \[ \frac{(a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \left ((A-i B) \cos ^3(c+d x) (-4 d x \sin (2 c)-4 i d x \cos (2 c))-(A-i B) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \log \left (\cos ^2(c+d x)\right )+2 (B+i A) (\cos (2 c)-i \sin (2 c)) \cos ^3(c+d x) \tan ^{-1}(\tan (3 c+d x))+\frac{1}{3} (6 A-7 i B) \sec (c) (\sin (2 c)+i \cos (2 c)) \sin (d x) \cos ^2(c+d x)-\frac{1}{6} (\cos (2 c)-i \sin (2 c)) (3 A+2 B \tan (c)-6 i B) \cos (c+d x)+\frac{1}{3} B \cos (c) (\tan (c)+i)^2 \sin (d x)\right )}{d (\cos (d x)+i \sin (d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 158, normalized size = 1.5 \begin{align*}{\frac{i{a}^{2}B \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{{a}^{2}B \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{2\,i{a}^{2}A\tan \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}A \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}+2\,{\frac{{a}^{2}B\tan \left ( dx+c \right ) }{d}}-{\frac{i{a}^{2}B\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}+{\frac{{a}^{2}A\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-{\frac{2\,i{a}^{2}A\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}}-2\,{\frac{{a}^{2}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 = 1.70474, size = 126, normalized size = 1.18 \begin{align*} -\frac{2 \, B a^{2} \tan \left (d x + c\right )^{3} +{\left (3 \, A - 6 i \, B\right )} a^{2} \tan \left (d x + c\right )^{2} + 12 \,{\left (d x + c\right )}{\left (i \, A + B\right )} a^{2} - 6 \,{\left (A - i \, B\right )} a^{2} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 12 \,{\left (-i \, A - B\right )} a^{2} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32, size = 474, normalized size = 4.43 \begin{align*} -\frac{2 \,{\left (3 \,{\left (3 \, A - 5 i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \,{\left (5 \, A - 6 i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (6 \, A - 7 i \, B\right )} a^{2} + 3 \,{\left ({\left (A - i \, B\right )} a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \,{\left (A - i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \,{\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} +{\left (A - i \, B\right )} a^{2}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )\right )}}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 = 15.407, size = 172, normalized size = 1.61 \begin{align*} \frac{2 a^{2} \left (- A + i B\right ) \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{- \frac{\left (6 A a^{2} - 10 i B a^{2}\right ) e^{- 2 i c} e^{4 i d x}}{d} - \frac{\left (10 A a^{2} - 12 i B a^{2}\right ) e^{- 4 i c} e^{2 i d x}}{d} - \frac{\left (12 A a^{2} - 14 i B a^{2}\right ) e^{- 6 i c}}{3 d}}{e^{6 i d x} + 3 e^{- 2 i c} e^{4 i d x} + 3 e^{- 4 i c} e^{2 i d x} + e^{- 6 i c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.47892, size = 421, normalized size = 3.93 \begin{align*} -\frac{6 \, A a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 6 i \, B a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, A a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, B a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, B a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 18 \, A a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 30 i \, B a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 30 \, A a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 36 i \, B a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 6 \, A a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 6 i \, B a^{2} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 12 \, A a^{2} - 14 i \, B a^{2}}{3 \,{\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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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