Optimal. Leaf size=110 \[ \frac{16 i a^2 \sec ^{11}(c+d x)}{195 d (a+i a \tan (c+d x))^{9/2}}+\frac{64 i a^3 \sec ^{11}(c+d x)}{2145 d (a+i a \tan (c+d x))^{11/2}}+\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}} \]
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Rubi [A] time = 0.192844, antiderivative size = 110, 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 = {3494, 3493} \[ \frac{16 i a^2 \sec ^{11}(c+d x)}{195 d (a+i a \tan (c+d x))^{9/2}}+\frac{64 i a^3 \sec ^{11}(c+d x)}{2145 d (a+i a \tan (c+d x))^{11/2}}+\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}}+\frac{1}{15} (8 a) \int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\\ &=\frac{16 i a^2 \sec ^{11}(c+d x)}{195 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}}+\frac{1}{195} \left (32 a^2\right ) \int \frac{\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^{9/2}} \, dx\\ &=\frac{64 i a^3 \sec ^{11}(c+d x)}{2145 d (a+i a \tan (c+d x))^{11/2}}+\frac{16 i a^2 \sec ^{11}(c+d x)}{195 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{11}(c+d x)}{15 d (a+i a \tan (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.591508, size = 94, normalized size = 0.85 \[ \frac{\sec ^{10}(c+d x) (187 i \sin (2 (c+d x))+203 \cos (2 (c+d x))+60) (-2 \sin (3 (c+d x))-2 i \cos (3 (c+d x)))}{2145 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.819, size = 154, normalized size = 1.4 \begin{align*}{\frac{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}+768\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -80\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+560\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -1472\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-968\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +286\,i}{2145\,d{a}^{3} \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 [B] time = 2.50036, size = 1031, normalized size = 9.37 \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 = 2.21804, size = 509, normalized size = 4.63 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (49920 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15360 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2048 i\right )} e^{\left (i \, d x + i \, c\right )}}{2145 \,{\left (a^{3} d e^{\left (15 i \, d x + 15 i \, c\right )} + 7 \, a^{3} d e^{\left (13 i \, d x + 13 i \, c\right )} + 21 \, a^{3} d e^{\left (11 i \, d x + 11 i \, c\right )} + 35 \, a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} + 35 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + 21 \, a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )} + 7 \, 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 )^{11}}{{\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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