Optimal. Leaf size=117 \[ \frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d} \]
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Rubi [A] time = 0.0856808, antiderivative size = 117, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 \sqrt{a+x} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 \sqrt{a+x}-12 a^2 (a+x)^{3/2}+6 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{16 i (a+i a \tan (c+d x))^{3/2}}{3 a^4 d}+\frac{24 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}-\frac{12 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}+\frac{2 i (a+i a \tan (c+d x))^{9/2}}{9 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.494226, size = 108, normalized size = 0.92 \[ \frac{2 \sec ^6(c+d x) (\cos (4 (c+d x))+i \sin (4 (c+d x))) (242 i \cos (2 (c+d x))+54 \tan (c+d x)+89 \sin (3 (c+d x)) \sec (c+d x)+77 i)}{315 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.3, size = 100, normalized size = 0.9 \begin{align*} -{\frac{256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-256\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +452\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+260\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -70\,i}{315\,d{a}^{3} \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 = 1.18941, size = 103, normalized size = 0.88 \begin{align*} \frac{2 i \,{\left (35 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 270 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 756 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 840 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3}\right )}}{315 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21845, size = 414, normalized size = 3.54 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-512 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 2304 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 4032 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 3360 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{315 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, 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 )^{8}}{{\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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