Optimal. Leaf size=146 \[ -\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d}+\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac{64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac{32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d} \]
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Rubi [A] time = 0.0931937, antiderivative size = 146, 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} \[ -\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d}+\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac{64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac{32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 43
Rubi steps
\begin{align*} \int \frac{\sec ^{10}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=-\frac{i \operatorname{Subst}\left (\int (a-x)^4 (a+x)^{3/2} \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (16 a^4 (a+x)^{3/2}-32 a^3 (a+x)^{5/2}+24 a^2 (a+x)^{7/2}-8 a (a+x)^{9/2}+(a+x)^{11/2}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d}\\ &=-\frac{32 i (a+i a \tan (c+d x))^{5/2}}{5 a^5 d}+\frac{64 i (a+i a \tan (c+d x))^{7/2}}{7 a^6 d}-\frac{16 i (a+i a \tan (c+d x))^{9/2}}{3 a^7 d}+\frac{16 i (a+i a \tan (c+d x))^{11/2}}{11 a^8 d}-\frac{2 i (a+i a \tan (c+d x))^{13/2}}{13 a^9 d}\\ \end{align*}
Mathematica [A] time = 0.692974, size = 116, normalized size = 0.79 \[ \frac{2 \sec ^9(c+d x) (2600 \sin (2 (c+d x))+2875 \sin (4 (c+d x))+4264 i \cos (2 (c+d x))+3131 i \cos (4 (c+d x))+2288 i) (\cos (5 (c+d x))+i \sin (5 (c+d x)))}{15015 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.349, size = 127, normalized size = 0.9 \begin{align*}{\frac{-8192\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+8192\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -1024\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+5120\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -12460\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-7980\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2310\,i}{15015\,d{a}^{3} \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 [A] time = 0.97875, size = 127, normalized size = 0.87 \begin{align*} -\frac{2 i \,{\left (1155 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{13}{2}} - 10920 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{11}{2}} a + 40040 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a^{2} - 68640 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}} a^{3} + 48048 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}} a^{4}\right )}}{15015 \, a^{9} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27935, size = 566, normalized size = 3.88 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-16384 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 106496 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 292864 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 439296 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 384384 i \, e^{\left (4 i \, d x + 4 i \, c\right )}\right )} e^{\left (i \, d x + i \, c\right )}}{15015 \,{\left (a^{3} d e^{\left (12 i \, d x + 12 i \, c\right )} + 6 \, a^{3} d e^{\left (10 i \, d x + 10 i \, c\right )} + 15 \, a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} + 20 \, a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )} + 15 \, a^{3} d e^{\left (4 i \, d x + 4 i \, c\right )} + 6 \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{3} 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 )^{10}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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