Optimal. Leaf size=147 \[ \frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{64 i a^3 \sec ^{13}(c+d x)}{1615 d (a+i a \tan (c+d x))^{11/2}}+\frac{256 i a^4 \sec ^{13}(c+d x)}{20995 d (a+i a \tan (c+d x))^{13/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}} \]
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Rubi [A] time = 0.265408, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3494, 3493} \[ \frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{64 i a^3 \sec ^{13}(c+d x)}{1615 d (a+i a \tan (c+d x))^{11/2}}+\frac{256 i a^4 \sec ^{13}(c+d x)}{20995 d (a+i a \tan (c+d x))^{13/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}}+\frac{1}{19} (12 a) \int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\\ &=\frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}}+\frac{1}{323} \left (96 a^2\right ) \int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{9/2}} \, dx\\ &=\frac{64 i a^3 \sec ^{13}(c+d x)}{1615 d (a+i a \tan (c+d x))^{11/2}}+\frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}}+\frac{\left (128 a^3\right ) \int \frac{\sec ^{13}(c+d x)}{(a+i a \tan (c+d x))^{11/2}} \, dx}{1615}\\ &=\frac{256 i a^4 \sec ^{13}(c+d x)}{20995 d (a+i a \tan (c+d x))^{13/2}}+\frac{64 i a^3 \sec ^{13}(c+d x)}{1615 d (a+i a \tan (c+d x))^{11/2}}+\frac{24 i a^2 \sec ^{13}(c+d x)}{323 d (a+i a \tan (c+d x))^{9/2}}+\frac{2 i a \sec ^{13}(c+d x)}{19 d (a+i a \tan (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.952615, size = 112, normalized size = 0.76 \[ \frac{\sec ^{12}(c+d x) (13 i (38 \sin (c+d x)+123 \sin (3 (c+d x)))+798 \cos (c+d x)+1631 \cos (3 (c+d x))) (-2 \sin (4 (c+d x))-2 i \cos (4 (c+d x)))}{20995 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 15.287, size = 181, normalized size = 1.2 \begin{align*}{\frac{16384\,i \left ( \cos \left ( dx+c \right ) \right ) ^{10}+16384\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{9}-2048\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}+6144\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}-640\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+4480\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -336\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3696\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -10712\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-7280\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2210\,i}{20995\,d{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{9}}\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 = 2.87335, size = 1218, normalized size = 8.29 \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 [A] time = 2.24708, size = 649, normalized size = 4.41 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (1653760 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 661504 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 155648 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 16384 i\right )} e^{\left (i \, d x + i \, c\right )}}{20995 \,{\left (a^{3} d e^{\left (19 i \, d x + 19 i \, c\right )} + 9 \, a^{3} d e^{\left (17 i \, d x + 17 i \, c\right )} + 36 \, a^{3} d e^{\left (15 i \, d x + 15 i \, c\right )} + 84 \, a^{3} d e^{\left (13 i \, d x + 13 i \, c\right )} + 126 \, a^{3} d e^{\left (11 i \, d x + 11 i \, c\right )} + 126 \, a^{3} d e^{\left (9 i \, d x + 9 i \, c\right )} + 84 \, a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + 36 \, a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )} + 9 \, 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 )^{13}}{{\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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