Optimal. Leaf size=102 \[ -\frac{(3 A+i B) \cot (c+d x)}{2 a d}-\frac{(-B+i A) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{x (3 A+i B)}{2 a} \]
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Rubi [A] time = 0.174708, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3596, 3529, 3531, 3475} \[ -\frac{(3 A+i B) \cot (c+d x)}{2 a d}-\frac{(-B+i A) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{x (3 A+i B)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3596
Rule 3529
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{a+i a \tan (c+d x)} \, dx &=\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{\int \cot ^2(c+d x) (a (3 A+i B)-2 a (i A-B) \tan (c+d x)) \, dx}{2 a^2}\\ &=-\frac{(3 A+i B) \cot (c+d x)}{2 a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}+\frac{\int \cot (c+d x) (-2 a (i A-B)-a (3 A+i B) \tan (c+d x)) \, dx}{2 a^2}\\ &=-\frac{(3 A+i B) x}{2 a}-\frac{(3 A+i B) \cot (c+d x)}{2 a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}-\frac{(i A-B) \int \cot (c+d x) \, dx}{a}\\ &=-\frac{(3 A+i B) x}{2 a}-\frac{(3 A+i B) \cot (c+d x)}{2 a d}-\frac{(i A-B) \log (\sin (c+d x))}{a d}+\frac{(A+i B) \cot (c+d x)}{2 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [B] time = 2.70133, size = 225, normalized size = 2.21 \[ \frac{(\cos (d x)+i \sin (d x)) (A+B \tan (c+d x)) \left (\frac{1}{2} (B-i A) (\cos (c)-i \sin (c)) \cos (2 d x)-\frac{1}{2} (A+i B) (\cos (c)-i \sin (c)) \sin (2 d x)+2 d x (A+i B) (\cos (c)+i \sin (c))-d x (3 A+i B) (\cos (c)+i \sin (c))+(B-i A) (\cos (c)+i \sin (c)) \log \left (\sin ^2(c+d x)\right )-2 (A+i B) (\cos (c)+i \sin (c)) \tan ^{-1}(\tan (d x))+2 A (\cot (c)+i) \sin (d x) \csc (c+d x)\right )}{2 d (a+i a \tan (c+d x)) (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.096, size = 170, normalized size = 1.7 \begin{align*} -{\frac{A}{2\,ad \left ( \tan \left ( dx+c \right ) -i \right ) }}-{\frac{{\frac{i}{2}}B}{ad \left ( \tan \left ( dx+c \right ) -i \right ) }}+{\frac{{\frac{5\,i}{4}}\ln \left ( \tan \left ( dx+c \right ) -i \right ) A}{ad}}-{\frac{3\,\ln \left ( \tan \left ( dx+c \right ) -i \right ) B}{4\,ad}}-{\frac{B\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{4\,ad}}-{\frac{{\frac{i}{4}}A\ln \left ( \tan \left ( dx+c \right ) +i \right ) }{ad}}-{\frac{A}{ad\tan \left ( dx+c \right ) }}-{\frac{iA\ln \left ( \tan \left ( dx+c \right ) \right ) }{ad}}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \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 = 1.54263, size = 365, normalized size = 3.58 \begin{align*} -\frac{2 \,{\left (5 \, A + 3 i \, B\right )} d x e^{\left (4 i \, d x + 4 i \, c\right )} -{\left (2 \,{\left (5 \, A + 3 i \, B\right )} d x - 9 i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left ({\left (-4 i \, A + 4 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (4 i \, A - 4 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - i \, A + B}{4 \,{\left (a d e^{\left (4 i \, d x + 4 i \, c\right )} - a d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.99142, size = 151, normalized size = 1.48 \begin{align*} - \frac{2 i A e^{- 2 i c}}{a d \left (e^{2 i d x} - e^{- 2 i c}\right )} - \frac{\left (\begin{cases} 5 A x e^{2 i c} + \frac{i A e^{- 2 i d x}}{2 d} + 3 i B x e^{2 i c} - \frac{B e^{- 2 i d x}}{2 d} & \text{for}\: d \neq 0 \\x \left (5 A e^{2 i c} + A + 3 i B e^{2 i c} + i B\right ) & \text{otherwise} \end{cases}\right ) e^{- 2 i c}}{2 a} + \frac{\left (- i A + B\right ) \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40134, size = 184, normalized size = 1.8 \begin{align*} -\frac{\frac{2 \,{\left (-5 i \, A + 3 \, B\right )} \log \left (\tan \left (d x + c\right ) - i\right )}{a} + \frac{2 \,{\left (i \, A + B\right )} \log \left (-i \, \tan \left (d x + c\right ) + 1\right )}{a} + \frac{8 \,{\left (i \, A - B\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a} + \frac{A \tan \left (d x + c\right )^{2} - i \, B \tan \left (d x + c\right )^{2} - 13 i \, A \tan \left (d x + c\right ) + 3 \, B \tan \left (d x + c\right ) - 8 \, A}{{\left (-i \, \tan \left (d x + c\right )^{2} - \tan \left (d x + c\right )\right )} a}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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