Optimal. Leaf size=110 \[ \frac{16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}} \]
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Rubi [A] time = 0.187032, 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} \[ \frac{16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac{64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3494
Rule 3493
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{\sqrt{a+i a \tan (c+d x)}} \, dx &=\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}+\frac{1}{11} (8 a) \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac{16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}+\frac{1}{99} \left (32 a^2\right ) \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx\\ &=\frac{64 i a^3 \sec ^7(c+d x)}{693 d (a+i a \tan (c+d x))^{7/2}}+\frac{16 i a^2 \sec ^7(c+d x)}{99 d (a+i a \tan (c+d x))^{5/2}}+\frac{2 i a \sec ^7(c+d x)}{11 d (a+i a \tan (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.378569, size = 77, normalized size = 0.7 \[ \frac{2 \sec ^6(c+d x) (91 i \sin (2 (c+d x))+107 \cos (2 (c+d x))+44) (\sin (3 (c+d x))+i \cos (3 (c+d x)))}{693 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.388, size = 127, normalized size = 1.2 \begin{align*}{\frac{512\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+512\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) -64\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+192\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -20\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+140\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -126\,i}{693\,ad \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 = 1.90074, size = 640, normalized size = 5.82 \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 = 2.02557, size = 398, normalized size = 3.62 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (6336 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 2816 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 512 i\right )} e^{\left (i \, d x + i \, c\right )}}{693 \,{\left (a d e^{\left (11 i \, d x + 11 i \, c\right )} + 5 \, a d e^{\left (9 i \, d x + 9 i \, c\right )} + 10 \, a d e^{\left (7 i \, d x + 7 i \, c\right )} + 10 \, a d e^{\left (5 i \, d x + 5 i \, c\right )} + 5 \, a d e^{\left (3 i \, d x + 3 i \, c\right )} + a 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 )^{7}}{\sqrt{i \, a \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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