Optimal. Leaf size=28 \[ \frac{i \sec (c+d x)}{d (a+i a \tan (c+d x))} \]
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Rubi [A] time = 0.0228633, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3488} \[ \frac{i \sec (c+d x)}{d (a+i a \tan (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{a+i a \tan (c+d x)} \, dx &=\frac{i \sec (c+d x)}{d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0267033, size = 25, normalized size = 0.89 \[ \frac{\sec (c+d x)}{a d (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 23, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{ad \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 [A] time = 0.985451, size = 39, normalized size = 1.39 \begin{align*} \frac{2}{{\left (-i \, a + \frac{a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16242, size = 35, normalized size = 1.25 \begin{align*} \frac{i \, e^{\left (-i \, d x - i \, c\right )}}{a 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 = 1.1525, size = 28, normalized size = 1. \begin{align*} \frac{2}{a d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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