Optimal. Leaf size=67 \[ -\frac{A+i B}{2 a d (1+i \tan (c+d x))}-\frac{x (-B+i A)}{2 a}+\frac{i B \log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.093083, antiderivative size = 67, 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 = {3589, 3475, 12, 3526, 8} \[ -\frac{A+i B}{2 a d (1+i \tan (c+d x))}-\frac{x (-B+i A)}{2 a}+\frac{i B \log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3589
Rule 3475
Rule 12
Rule 3526
Rule 8
Rubi steps
\begin{align*} \int \frac{\tan (c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx &=-\frac{i \int \frac{a (i A-B) \tan (c+d x)}{a+i a \tan (c+d x)} \, dx}{a}-\frac{(i B) \int \tan (c+d x) \, dx}{a}\\ &=\frac{i B \log (\cos (c+d x))}{a d}-(-A-i B) \int \frac{\tan (c+d x)}{a+i a \tan (c+d x)} \, dx\\ &=\frac{i B \log (\cos (c+d x))}{a d}-\frac{A+i B}{2 d (a+i a \tan (c+d x))}-\frac{(i A-B) \int 1 \, dx}{2 a}\\ &=-\frac{(i A-B) x}{2 a}+\frac{i B \log (\cos (c+d x))}{a d}-\frac{A+i B}{2 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.918988, size = 148, normalized size = 2.21 \[ \frac{\cos (c+d x) (A+B \tan (c+d x)) \left (\tan (c+d x) \left (-2 i A d x+A+2 i B \log \left (\cos ^2(c+d x)\right )-2 B d x+i B\right )-2 A d x+i A+4 B \tan ^{-1}(\tan (d x)) (\tan (c+d x)-i)+2 B \log \left (\cos ^2(c+d x)\right )+2 i B d x-B\right )}{4 a d (\tan (c+d x)-i) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 121, normalized size = 1.8 \begin{align*} -{\frac{\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{4\,ad}}-{\frac{{\frac{3\,i}{4}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{ad}}+{\frac{{\frac{i}{2}}A}{ad \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{B}{2\,ad \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{4\,ad}}-{\frac{{\frac{i}{4}}B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55483, size = 189, normalized size = 2.82 \begin{align*} \frac{{\left ({\left (-2 i \, A + 6 \, B\right )} d x e^{\left (2 i \, d x + 2 i \, c\right )} + 4 i \, B e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - A - i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.19816, size = 114, normalized size = 1.7 \begin{align*} \frac{i B \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{a d} - \frac{\left (\begin{cases} i A x e^{2 i c} + \frac{A e^{- 2 i d x}}{2 d} - 3 B x e^{2 i c} + \frac{i B e^{- 2 i d x}}{2 d} & \text{for}\: d \neq 0 \\x \left (i A e^{2 i c} - i A - 3 B e^{2 i c} + B\right ) & \text{otherwise} \end{cases}\right ) e^{- 2 i c}}{2 a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44311, size = 111, normalized size = 1.66 \begin{align*} -\frac{\frac{{\left (A + 3 i \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a} - \frac{{\left (A - i \, B\right )} \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a} - \frac{A \tan \left (d x + c\right ) + 3 i \, B \tan \left (d x + c\right ) + i \, A + B}{a{\left (\tan \left (d x + c\right ) - i\right )}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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