Optimal. Leaf size=26 \[ \frac{i}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A] time = 0.0432143, antiderivative size = 26, 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, 32} \[ \frac{i}{d \left (a^2+i a^2 \tan (c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \frac{\sec ^2(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{i}{d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.0401591, size = 32, normalized size = 1.23 \[ \frac{i \sec ^2(c+d x)}{2 d (a+i a \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 24, normalized size = 0.9 \begin{align*}{\frac{i}{ad \left ( a+ia\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955199, size = 28, normalized size = 1.08 \begin{align*} \frac{i}{{\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 = 2.26377, size = 49, normalized size = 1.88 \begin{align*} \frac{i \, e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, 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 = 1.19236, size = 41, normalized size = 1.58 \begin{align*} -\frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{2} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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