Optimal. Leaf size=110 \[ \frac{16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac{64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.177871, 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} \[ \frac{16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac{64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \sec ^5(c+d x) \sqrt{a+i a \tan (c+d x)} \, dx &=\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{9} (8 a) \int \frac{\sec ^5(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx\\ &=\frac{16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}}+\frac{1}{63} \left (32 a^2\right ) \int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac{64 i a^3 \sec ^5(c+d x)}{315 d (a+i a \tan (c+d x))^{5/2}}+\frac{16 i a^2 \sec ^5(c+d x)}{63 d (a+i a \tan (c+d x))^{3/2}}+\frac{2 i a \sec ^5(c+d x)}{9 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.369822, size = 77, normalized size = 0.7 \[ \frac{2 \sec ^4(c+d x) \sqrt{a+i a \tan (c+d x)} (55 i \sin (2 (c+d x))+71 \cos (2 (c+d x))+36) (\sin (3 (c+d x))+i \cos (3 (c+d x)))}{315 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.385, size = 114, normalized size = 1. \begin{align*}{\frac{256\,i \left ( \cos \left ( dx+c \right ) \right ) ^{5}+256\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-32\,i \left ( \cos \left ( dx+c \right ) \right ) ^{3}+96\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -10\,i\cos \left ( dx+c \right ) +70\,\sin \left ( dx+c \right ) }{315\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\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 [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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Fricas [A] time = 2.23593, size = 342, normalized size = 3.11 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (2016 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1152 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 256 i\right )} e^{\left (i \, d x + i \, c\right )}}{315 \,{\left (d e^{\left (9 i \, d x + 9 i \, c\right )} + 4 \, d e^{\left (7 i \, d x + 7 i \, c\right )} + 6 \, d e^{\left (5 i \, d x + 5 i \, c\right )} + 4 \, d e^{\left (3 i \, d x + 3 i \, c\right )} + 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 \sqrt{i \, a \tan \left (d x + c\right ) + a} \sec \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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