Optimal. Leaf size=117 \[ \frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{11/2}}{11 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{3 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d} \]
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Rubi [A] time = 0.0797237, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{11/2}}{11 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{3 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^8(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^{5/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{5/2}-12 a^2 (a+x)^{7/2}+6 a (a+x)^{9/2}-(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{16 i (a+i a \tan (c+d x))^{7/2}}{7 a^4 d}+\frac{8 i (a+i a \tan (c+d x))^{9/2}}{3 a^5 d}-\frac{12 i (a+i a \tan (c+d x))^{11/2}}{11 a^6 d}+\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.466889, size = 95, normalized size = 0.81 \[ \frac{2 \sec ^7(c+d x) (-7 i (26 \sin (c+d x)+59 \sin (3 (c+d x)))+390 \cos (c+d x)+445 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{3003 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.583, size = 127, normalized size = 1.1 \begin{align*} -{\frac{1024\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1024\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-640\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +56\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-504\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +462\,i}{3003\,ad \left ( \cos \left ( dx+c \right ) \right ) ^{6}}\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 [B] time = 1.12363, size = 401, normalized size = 3.43 \begin{align*} -\frac{2 i \,{\left (15015 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} - \frac{3003 \,{\left (3 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{2}\right )}}{a^{2}} + \frac{143 \,{\left (35 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} - 180 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a + 378 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{2} - 420 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{3} + 315 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{4}\right )}}{a^{4}} - \frac{5 \,{\left (231 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 1638 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 5005 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 8580 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4} - 6006 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a^{5} + 3003 \, \sqrt{i \, a \tan \left (d x + c\right ) + a} a^{6}\right )}}{a^{6}}\right )}}{15015 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09395, size = 498, normalized size = 4.26 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-2048 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 13312 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 36608 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 54912 i \, e^{\left (6 i \, d x + 6 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{3003 \,{\left (a d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a 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}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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