Optimal. Leaf size=55 \[ \frac{2 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}+\frac{4 i}{a^2 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.0701862, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3487, 43} \[ \frac{2 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}+\frac{4 i}{a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{(a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (\frac{2 a}{(a+x)^{3/2}}-\frac{1}{\sqrt{a+x}}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=\frac{4 i}{a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{2 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.254093, size = 36, normalized size = 0.65 \[ \frac{-2 \tan (c+d x)+6 i}{a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.266, size = 65, normalized size = 1.2 \begin{align*} 2\,{\frac{2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +i+2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d{a}^{3}}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11321, size = 59, normalized size = 1.07 \begin{align*} \frac{2 i \,{\left (\frac{\sqrt{i \, a \tan \left (d x + c\right ) + a}}{a^{2}} + \frac{2}{\sqrt{i \, a \tan \left (d x + c\right ) + a} a}\right )}}{a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14453, size = 135, normalized size = 2.45 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )} e^{\left (-i \, d x - i \, c\right )}}{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 )^{4}}{{\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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