Optimal. Leaf size=48 \[ -\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 i \sec (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A] time = 0.0471635, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3500, 3770} \[ -\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 i \sec (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=\frac{2 i \sec (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{\int \sec (c+d x) \, dx}{a^2}\\ &=-\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 i \sec (c+d x)}{d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 0.177561, size = 184, normalized size = 3.83 \[ -\frac{\sec ^2(c+d x) \left (\cos \left (\frac{3}{2} (c+d x)\right )+i \sin \left (\frac{3}{2} (c+d x)\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 i\right )+\sin \left (\frac{1}{2} (c+d x)\right ) \left (i \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-i \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+2\right )\right )}{a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.073, size = 63, normalized size = 1.3 \begin{align*} -{\frac{1}{{a}^{2}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{1}{{a}^{2}d}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+4\,{\frac{1}{{a}^{2}d \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49095, size = 158, normalized size = 3.29 \begin{align*} -\frac{-2 i \, \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - 2 i \, \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) - 4 i \, \cos \left (d x + c\right ) + \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) - 4 \, \sin \left (d x + c\right )}{2 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.30481, size = 161, normalized size = 3.35 \begin{align*} -\frac{{\left (e^{\left (i \, d x + i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - e^{\left (i \, d x + i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 2 i\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26314, size = 80, normalized size = 1.67 \begin{align*} -\frac{\frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} - \frac{\log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} - \frac{4}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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