Optimal. Leaf size=86 \[ -\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^3 d} \]
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Rubi [A] time = 0.0793853, antiderivative size = 86, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^3 d} \]
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))^{5/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(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 (\frac{4 a^2}{\sqrt{a+x}}-4 a \sqrt{a+x}+(a+x)^{3/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i \sqrt{a+i a \tan (c+d x)}}{a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.263347, size = 94, normalized size = 1.09 \[ \frac{2 \sec ^5(c+d x) (7 \sin (2 (c+d x))+23 i \cos (2 (c+d x))+20 i) (\cos (3 (c+d x))+i \sin (3 (c+d x)))}{15 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.273, size = 73, normalized size = 0.9 \begin{align*} -{\frac{92\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+28\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -6\,i}{15\,d{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}\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 = 1.06895, size = 78, normalized size = 0.91 \begin{align*} -\frac{2 i \,{\left (3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 20 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 60 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{2}\right )}}{15 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11693, size = 262, normalized size = 3.05 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-64 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 160 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 120 i\right )} e^{\left (i \, d x + i \, c\right )}}{15 \,{\left (a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} 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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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