Optimal. Leaf size=57 \[ \frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}-\frac{4 i \sqrt{a+i a \tan (c+d x)}}{a^2 d} \]
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Rubi [A] time = 0.0716966, antiderivative size = 57, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}-\frac{4 i \sqrt{a+i a \tan (c+d x)}}{a^2 d} \]
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))^{3/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{a-x}{\sqrt{a+x}} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (\frac{2 a}{\sqrt{a+x}}-\sqrt{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{4 i \sqrt{a+i a \tan (c+d x)}}{a^2 d}+\frac{2 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.191338, size = 80, normalized size = 1.4 \[ \frac{2 i (\tan (c+d x)+5 i) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{3 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.262, size = 61, normalized size = 1.1 \begin{align*} -{\frac{10\,i\cos \left ( dx+c \right ) +2\,\sin \left ( dx+c \right ) }{3\,{a}^{2}d\cos \left ( dx+c \right ) }\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 = 1.05252, size = 51, normalized size = 0.89 \begin{align*} \frac{2 i \,{\left ({\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a\right )}}{3 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07209, size = 180, normalized size = 3.16 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 12 i\right )} e^{\left (i \, d x + i \, c\right )}}{3 \,{\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )}} \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 ^{4}{\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 )^{4}}{{\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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