Optimal. Leaf size=213 \[ \frac{8 i \sec ^3(c+d x)}{9009 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.275823, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3502, 3488} \[ \frac{8 i \sec ^3(c+d x)}{9009 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{13 a}\\ &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{20 \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{143 a^2}\\ &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{20 \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{429 a^3}\\ &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{40 \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{3003 a^4}\\ &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{8 \int \frac{\sec ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{3003 a^5}\\ &=\frac{i \sec ^3(c+d x)}{13 d (a+i a \tan (c+d x))^8}+\frac{5 i \sec ^3(c+d x)}{143 a d (a+i a \tan (c+d x))^7}+\frac{20 i \sec ^3(c+d x)}{1287 a^2 d (a+i a \tan (c+d x))^6}+\frac{20 i \sec ^3(c+d x)}{3003 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec ^3(c+d x)}{9009 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec ^3(c+d x)}{3003 d \left (a^2+i a^2 \tan (c+d x)\right )^4}\\ \end{align*}
Mathematica [A] time = 0.249824, size = 95, normalized size = 0.45 \[ \frac{i \sec ^8(c+d x) (1430 i \sin (c+d x)+2457 i \sin (3 (c+d x))+1155 i \sin (5 (c+d x))+11440 \cos (c+d x)+6552 \cos (3 (c+d x))+1848 \cos (5 (c+d x)))}{144144 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 222, normalized size = 1. \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{5840}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{4528}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{128}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}+{\frac{432\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-{\frac{2272}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}-{\frac{736\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}-{\frac{94}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}-{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}+240\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-5}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}-{\frac{100\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{{\frac{1336\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{7\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.24318, size = 190, normalized size = 0.89 \begin{align*} \frac{693 i \, \cos \left (13 \, d x + 13 \, c\right ) + 4095 i \, \cos \left (11 \, d x + 11 \, c\right ) + 10010 i \, \cos \left (9 \, d x + 9 \, c\right ) + 12870 i \, \cos \left (7 \, d x + 7 \, c\right ) + 9009 i \, \cos \left (5 \, d x + 5 \, c\right ) + 3003 i \, \cos \left (3 \, d x + 3 \, c\right ) + 693 \, \sin \left (13 \, d x + 13 \, c\right ) + 4095 \, \sin \left (11 \, d x + 11 \, c\right ) + 10010 \, \sin \left (9 \, d x + 9 \, c\right ) + 12870 \, \sin \left (7 \, d x + 7 \, c\right ) + 9009 \, \sin \left (5 \, d x + 5 \, c\right ) + 3003 \, \sin \left (3 \, d x + 3 \, c\right )}{288288 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31983, size = 267, normalized size = 1.25 \begin{align*} \frac{{\left (3003 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 9009 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 12870 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 10010 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 4095 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 693 i\right )} e^{\left (-13 i \, d x - 13 i \, c\right )}}{288288 \, 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 = 1.19599, size = 239, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (9009 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 45045 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 183183 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 435435 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 810810 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 1051050 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 1076790 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 785070 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 451165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 171457 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 51675 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 7111 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1240\right )}}{9009 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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