Optimal. Leaf size=65 \[ \frac{i \sec (c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2} \]
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Rubi [A] time = 0.0539423, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3502, 3488} \[ \frac{i \sec (c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=\frac{i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}+\frac{\int \frac{\sec (c+d x)}{a+i a \tan (c+d x)} \, dx}{3 a}\\ &=\frac{i \sec (c+d x)}{3 d (a+i a \tan (c+d x))^2}+\frac{i \sec (c+d x)}{3 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0738067, size = 38, normalized size = 0.58 \[ \frac{(\tan (c+d x)-2 i) \sec (c+d x)}{3 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 57, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{{a}^{2}d} \left ({\frac{i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}-2/3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981063, size = 61, normalized size = 0.94 \begin{align*} \frac{i \, \cos \left (3 \, d x + 3 \, c\right ) + 3 i \, \cos \left (d x + c\right ) + \sin \left (3 \, d x + 3 \, c\right ) + 3 \, \sin \left (d x + c\right )}{6 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21306, size = 86, normalized size = 1.32 \begin{align*} \frac{{\left (3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{6 \, 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 = 1.15592, size = 63, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2\right )}}{3 \, a^{2} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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