Optimal. Leaf size=88 \[ -\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]
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Rubi [A] time = 0.0766982, antiderivative size = 88, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d}\\ &=-\frac{8 i (a+i a \tan (c+d x))^{13/2}}{13 a^3 d}+\frac{8 i (a+i a \tan (c+d x))^{15/2}}{15 a^4 d}-\frac{2 i (a+i a \tan (c+d x))^{17/2}}{17 a^5 d}\\ \end{align*}
Mathematica [A] time = 0.858398, size = 97, normalized size = 1.1 \[ \frac{2 a^3 \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} (-247 i \sin (2 (c+d x))+263 \cos (2 (c+d x))+68) (\sin (6 c+9 d x)-i \cos (6 c+9 d x))}{3315 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.628, size = 154, normalized size = 1.8 \begin{align*} -{\frac{2\,{a}^{3} \left ( 1024\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}-1024\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}-640\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +56\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}-504\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -1072\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+676\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +195\,i \right ) }{3315\,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 = 1.11964, size = 78, normalized size = 0.89 \begin{align*} -\frac{2 i \,{\left (195 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{17}{2}} - 884 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{15}{2}} a + 1020 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05958, size = 537, normalized size = 6.1 \begin{align*} \frac{\sqrt{2}{\left (-4096 i \, a^{3} e^{\left (16 i \, d x + 16 i \, c\right )} - 34816 i \, a^{3} e^{\left (14 i \, d x + 14 i \, c\right )} - 130560 i \, a^{3} 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 )}}{3315 \,{\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{7}{2}} \sec \left (d x + c\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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