Optimal. Leaf size=73 \[ \frac{8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}} \]
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Rubi [A] time = 0.125539, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3494, 3493} \[ \frac{8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}}+\frac{1}{11} (4 a) \int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\\ &=\frac{8 i a^2 \sec ^9(c+d x)}{99 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^9(c+d x)}{11 d (a+i a \tan (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.434218, size = 80, normalized size = 1.1 \[ \frac{2 (9 \tan (c+d x)-13 i) \sec ^7(c+d x) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{99 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.416, size = 127, normalized size = 1.7 \begin{align*}{\frac{128\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+128\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+48\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -104\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-64\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +18\,i}{99\,d{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}\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 [B] time = 2.17587, size = 845, normalized size = 11.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06984, size = 373, normalized size = 5.11 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (704 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i\right )} e^{\left (i \, d x + i \, c\right )}}{99 \,{\left (a^{3} d e^{\left (11 i \, d x + 11 i \, c\right )} + 5 \, a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} + 10 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + 10 \, a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )} + 5 \, a^{3} d e^{\left (3 i \, d x + 3 i \, c\right )} + a^{3} d e^{\left (i \, d x + i \, c\right )}\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 )^{9}}{{\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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