Optimal. Leaf size=117 \[ \frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^7 d}-\frac{4 i (a+i a \tan (c+d x))^{15/2}}{5 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d} \]
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Rubi [A] time = 0.0893523, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^7 d}-\frac{4 i (a+i a \tan (c+d x))^{15/2}}{5 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d} \]
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))^{3/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^{9/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{9/2}-12 a^2 (a+x)^{11/2}+6 a (a+x)^{13/2}-(a+x)^{15/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^4 d}+\frac{24 i (a+i a \tan (c+d x))^{13/2}}{13 a^5 d}-\frac{4 i (a+i a \tan (c+d x))^{15/2}}{5 a^6 d}+\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^7 d}\\ \end{align*}
Mathematica [A] time = 1.12643, size = 111, normalized size = 0.95 \[ \frac{2 a \sec ^8(c+d x) (\cos (d x)-i \sin (d x)) \sqrt{a+i a \tan (c+d x)} (-11 i (34 \sin (c+d x)+99 \sin (3 (c+d x)))+646 \cos (c+d x)+1121 \cos (3 (c+d x))) (\sin (5 c+6 d x)-i \cos (5 c+6 d x))}{12155 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 5.216, size = 152, normalized size = 1.3 \begin{align*} -{\frac{2\,a \left ( 2048\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}-2048\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-1280\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +112\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-1008\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +66\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-858\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -715\,i \right ) }{12155\,d \left ( \cos \left ( dx+c \right ) \right ) ^{8}}\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 = 1.04075, size = 103, normalized size = 0.88 \begin{align*} \frac{2 i \,{\left (715 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{17}{2}} - 4862 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a + 11220 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{2} - 8840 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a^{3}\right )}}{12155 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.59432, size = 578, normalized size = 4.94 \begin{align*} \frac{\sqrt{2}{\left (-8192 i \, a e^{\left (16 i \, d x + 16 i \, c\right )} - 69632 i \, a e^{\left (14 i \, d x + 14 i \, c\right )} - 261120 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 565760 i \, a e^{\left (10 i \, d x + 10 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 )}}{12155 \,{\left (d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, 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{3}{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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