Optimal. Leaf size=117 \[ \frac{2 i (a+i a \tan (c+d x))^{15/2}}{15 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{13/2}}{13 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d} \]
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Rubi [A] time = 0.0754535, 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} \[ \frac{2 i (a+i a \tan (c+d x))^{15/2}}{15 a^7 d}-\frac{12 i (a+i a \tan (c+d x))^{13/2}}{13 a^6 d}+\frac{24 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \sec ^8(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^3 (a+x)^{7/2} \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (8 a^3 (a+x)^{7/2}-12 a^2 (a+x)^{9/2}+6 a (a+x)^{11/2}-(a+x)^{13/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^7 d}\\ &=-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{9 a^4 d}+\frac{24 i (a+i a \tan (c+d x))^{11/2}}{11 a^5 d}-\frac{12 i (a+i a \tan (c+d x))^{13/2}}{13 a^6 d}+\frac{2 i (a+i a \tan (c+d x))^{15/2}}{15 a^7 d}\\ \end{align*}
Mathematica [A] time = 0.818673, size = 95, normalized size = 0.81 \[ \frac{2 \sec ^7(c+d x) \sqrt{a+i a \tan (c+d x)} (-3 i (90 \sin (c+d x)+233 \sin (3 (c+d x)))+510 \cos (c+d x)+731 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{6435 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.026, size = 141, normalized size = 1.2 \begin{align*}{\frac{-2048\,i \left ( \cos \left ( dx+c \right ) \right ) ^{7}+2048\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) -256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+1280\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-112\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+1008\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -66\,i\cos \left ( dx+c \right ) +858\,\sin \left ( dx+c \right ) }{6435\,d \left ( \cos \left ( dx+c \right ) \right ) ^{7}}\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 = 0.979973, size = 103, normalized size = 0.88 \begin{align*} \frac{2 i \,{\left (429 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{15}{2}} - 2970 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} a + 7020 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a^{2} - 5720 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{3}\right )}}{6435 \, a^{7} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37633, size = 522, normalized size = 4.46 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-4096 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 30720 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 99840 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 183040 i \, e^{\left (8 i \, d x + 8 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{6435 \,{\left (d e^{\left (14 i \, d x + 14 i \, c\right )} + 7 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 21 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 35 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 35 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 21 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 7 \, 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 \sqrt{i \, a \tan \left (d x + c\right ) + a} \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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