Optimal. Leaf size=35 \[ \frac{2 i a \sec ^7(c+d x)}{7 d (a+i a \tan (c+d x))^{7/2}} \]
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Rubi [A] time = 0.0629306, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {3493} \[ \frac{2 i a \sec ^7(c+d x)}{7 d (a+i a \tan (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 3493
Rubi steps
\begin{align*} \int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{2 i a \sec ^7(c+d x)}{7 d (a+i a \tan (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.255662, size = 57, normalized size = 1.63 \[ -\frac{2 (\tan (c+d x)+i) \sec ^5(c+d x)}{7 a^2 d (\tan (c+d x)-i)^2 \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.284, size = 100, normalized size = 2.9 \begin{align*}{\frac{16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+16\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) -16\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}-8\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\,i}{7\,d{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\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 = 1.7146, size = 659, normalized size = 18.83 \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 = 2.07692, size = 240, normalized size = 6.86 \begin{align*} \frac{16 i \, \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}}{7 \,{\left (a^{3} d e^{\left (7 i \, d x + 7 i \, c\right )} + 3 \, a^{3} d e^{\left (5 i \, d x + 5 i \, c\right )} + 3 \, 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 )^{7}}{{\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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