Optimal. Leaf size=27 \[ \frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.0646933, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 32} \[ \frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}} \]
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))^{3/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.143462, size = 27, normalized size = 1. \[ \frac{2 i}{a d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 24, normalized size = 0.9 \begin{align*}{\frac{2\,i}{ad}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09316, size = 28, normalized size = 1.04 \begin{align*} \frac{2 i}{\sqrt{i \, a \tan \left (d x + c\right ) + a} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.12964, size = 130, normalized size = 4.81 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{2}{\left (c + d x \right )}}{\left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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