Optimal. Leaf size=29 \[ \frac{2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.0628563, antiderivative size = 29, 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} \[ \frac{2 i}{3 a d (a+i a \tan (c+d x))^{3/2}} \]
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))^{5/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{1}{(a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=\frac{2 i}{3 a d (a+i a \tan (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.213731, size = 39, normalized size = 1.34 \[ \frac{2}{3 a^2 d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 24, normalized size = 0.8 \begin{align*}{\frac{{\frac{2\,i}{3}}}{ad} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08588, size = 28, normalized size = 0.97 \begin{align*} \frac{2 i}{3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.16009, size = 176, normalized size = 6.07 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2 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^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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