Optimal. Leaf size=117 \[ \frac{2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d} \]
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Rubi [A] time = 0.0860853, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^{11/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{11/2}-12 a^2 (a+x)^{13/2}+6 a (a+x)^{15/2}-(a+x)^{17/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{16 i (a+i a \tan (c+d x))^{13/2}}{13 a^4 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{5 a^5 d}-\frac{12 i (a+i a \tan (c+d x))^{17/2}}{17 a^6 d}+\frac{2 i (a+i a \tan (c+d x))^{19/2}}{19 a^7 d}\\ \end{align*}
Mathematica [A] time = 1.41642, size = 113, normalized size = 0.97 \[ -\frac{2 a^2 \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} (\cos (6 c+8 d x)+i \sin (6 c+8 d x)) (3262 i \cos (2 (c+d x))+494 \tan (c+d x)+1599 \sin (3 (c+d x)) \sec (c+d x)-833 i)}{20995 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 15.969, size = 171, normalized size = 1.5 \begin{align*} -{\frac{2\,{a}^{2} \left ( 4096\,i \left ( \cos \left ( dx+c \right ) \right ) ^{9}-4096\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+512\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}-2560\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +224\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}-2016\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+132\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}-1716\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -2535\,i\cos \left ( dx+c \right ) +1105\,\sin \left ( dx+c \right ) \right ) }{20995\,d \left ( \cos \left ( dx+c \right ) \right ) ^{9}}\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 = 1.11067, size = 103, normalized size = 0.88 \begin{align*} \frac{2 i \,{\left (1105 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{19}{2}} - 7410 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{17}{2}} a + 16796 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a^{2} - 12920 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{3}\right )}}{20995 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.52358, size = 635, normalized size = 5.43 \begin{align*} \frac{\sqrt{2}{\left (-16384 i \, a^{2} e^{\left (18 i \, d x + 18 i \, c\right )} - 155648 i \, a^{2} e^{\left (16 i \, d x + 16 i \, c\right )} - 661504 i \, a^{2} e^{\left (14 i \, d x + 14 i \, c\right )} - 1653760 i \, a^{2} e^{\left (12 i \, d x + 12 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 )}}{20995 \,{\left (d e^{\left (18 i \, d x + 18 i \, c\right )} + 9 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 36 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 84 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 126 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 126 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 84 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 36 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 9 \, 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{5}{2}} \sec \left (d x + c\right )^{8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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