Optimal. Leaf size=59 \[ \frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{7/2}}{7 a^2 d} \]
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Rubi [A] time = 0.0687435, antiderivative size = 59, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}-\frac{4 i (a+i a \tan (c+d x))^{7/2}}{7 a^2 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^{3/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x) (a+x)^{5/2} \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (2 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac{4 i (a+i a \tan (c+d x))^{7/2}}{7 a^2 d}+\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.473612, size = 81, normalized size = 1.37 \[ -\frac{2 a (7 \tan (c+d x)+11 i) \sec ^3(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (\cos (3 c+4 d x)+i \sin (3 c+4 d x))}{63 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.289, size = 98, normalized size = 1.7 \begin{align*} -{\frac{2\,a \left ( 16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-16\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +2\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-10\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -7\,i \right ) }{63\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\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 = 1.02943, size = 54, normalized size = 0.92 \begin{align*} \frac{2 i \,{\left (7 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 18 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a\right )}}{63 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35138, size = 311, normalized size = 5.27 \begin{align*} \frac{\sqrt{2}{\left (-64 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 288 i \, a e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{63 \,{\left (d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \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{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \sec \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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