Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d} \]
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Rubi [A] time = 0.078767, antiderivative size = 88, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 \sqrt{a+x} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 \sqrt{a+x}-4 a (a+x)^{3/2}+(a+x)^{5/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{5/2}}{5 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{7/2}}{7 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.284994, size = 92, normalized size = 1.05 \[ -\frac{2 \sec ^5(c+d x) (-27 i \sin (2 (c+d x))+43 \cos (2 (c+d x))+28) (\cos (3 (c+d x))+i \sin (3 (c+d x)))}{105 a d (\tan (c+d x)-i) \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.279, size = 90, normalized size = 1. \begin{align*}{\frac{-64\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -78\,i\cos \left ( dx+c \right ) -30\,\sin \left ( dx+c \right ) }{105\,{a}^{2}d \left ( \cos \left ( dx+c \right ) \right ) ^{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 = 1.08481, size = 78, normalized size = 0.89 \begin{align*} -\frac{2 i \,{\left (15 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} - 84 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a + 140 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{2}\right )}}{105 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23479, size = 332, normalized size = 3.77 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-128 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 448 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 560 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{105 \,{\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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(-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 )^{6}}{{\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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