Optimal. Leaf size=156 \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a^8 d}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac{2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7} \]
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Rubi [A] time = 0.216498, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3500, 3770} \[ \frac{\tanh ^{-1}(\sin (c+d x))}{a^8 d}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}-\frac{2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 3770
Rubi steps
\begin{align*} \int \frac{\sec ^9(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac{\int \frac{\sec ^7(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{a^2}\\ &=\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{\int \frac{\sec ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{a^4}\\ &=\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6}\\ &=\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac{2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{\int \sec (c+d x) \, dx}{a^8}\\ &=\frac{\tanh ^{-1}(\sin (c+d x))}{a^8 d}+\frac{2 i \sec ^7(c+d x)}{7 a d (a+i a \tan (c+d x))^7}-\frac{2 i \sec ^5(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac{2 i \sec ^3(c+d x)}{3 a^5 d (a+i a \tan (c+d x))^3}-\frac{2 i \sec (c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 1.03443, size = 304, normalized size = 1.95 \[ \frac{\sec ^8(c+d x) \left (\cos \left (\frac{9}{2} (c+d x)\right )+i \sin \left (\frac{9}{2} (c+d x)\right )\right ) \left (-70 \sin \left (\frac{1}{2} (c+d x)\right )-42 \sin \left (\frac{3}{2} (c+d x)\right )+210 \sin \left (\frac{5}{2} (c+d x)\right )+30 \sin \left (\frac{7}{2} (c+d x)\right )+70 i \cos \left (\frac{1}{2} (c+d x)\right )-42 i \cos \left (\frac{3}{2} (c+d x)\right )-210 i \cos \left (\frac{5}{2} (c+d x)\right )+30 i \cos \left (\frac{7}{2} (c+d x)\right )-105 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-105 i \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+105 i \sin \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{105 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.11, size = 176, normalized size = 1.1 \begin{align*}{\frac{1}{d{a}^{8}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{128\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-6}}+{\frac{16\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-2}}-{\frac{128\,i}{d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-4}}-{\frac{256}{7\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-7}}+{\frac{896}{5\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-5}}-{\frac{160}{3\,d{a}^{8}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -i \right ) ^{-3}}-{\frac{1}{d{a}^{8}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.85278, size = 250, normalized size = 1.6 \begin{align*} \frac{-210 i \, \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - 210 i \, \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) + 60 i \, \cos \left (7 \, d x + 7 \, c\right ) - 84 i \, \cos \left (5 \, d x + 5 \, c\right ) + 140 i \, \cos \left (3 \, d x + 3 \, c\right ) - 420 i \, \cos \left (d x + c\right ) + 105 \, \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - 105 \, \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 60 \, \sin \left (7 \, d x + 7 \, c\right ) - 84 \, \sin \left (5 \, d x + 5 \, c\right ) + 140 \, \sin \left (3 \, d x + 3 \, c\right ) - 420 \, \sin \left (d x + c\right )}{210 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.65172, size = 306, normalized size = 1.96 \begin{align*} \frac{{\left (105 \, e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, e^{\left (7 i \, d x + 7 i \, c\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 210 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 42 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 30 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{105 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23051, size = 169, normalized size = 1.08 \begin{align*} \frac{\frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{8}} - \frac{105 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{8}} - \frac{2 \,{\left (-840 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1400 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 3920 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2352 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1064 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 152\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{7}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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