Optimal. Leaf size=23 \[ \frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.0408845, antiderivative size = 23, 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 = {3487, 31} \[ \frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 31
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{x}{a}+\frac{i \log (\cos (c+d x))}{a d}\\ \end{align*}
Mathematica [A] time = 0.0893654, size = 31, normalized size = 1.35 \[ \frac{2 \tan ^{-1}(\tan (d x))+i \log \left (\cos ^2(c+d x)\right )}{2 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 23, normalized size = 1. \begin{align*}{\frac{-i\ln \left ( a+ia\tan \left ( dx+c \right ) \right ) }{ad}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16437, size = 27, normalized size = 1.17 \begin{align*} -\frac{i \, \log \left (i \, a \tan \left (d x + c\right ) + a\right )}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21461, size = 65, normalized size = 2.83 \begin{align*} \frac{2 \, d x + i \, \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\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 [B] time = 1.18413, size = 80, normalized size = 3.48 \begin{align*} -\frac{\frac{2 i \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}{a} - \frac{i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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