Optimal. Leaf size=269 \[ \frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.259754, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3502, 3488} \[ \frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3488
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{15 a}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{65 a^2}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{143 a^3}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{56 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{1287 a^4}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{8 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{429 a^5}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2145 a^6}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{16 \int \frac{\sec (c+d x)}{a+i a \tan (c+d x)} \, dx}{6435 a^7}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.428746, size = 117, normalized size = 0.43 \[ \frac{i \sec ^8(c+d x) (3575 i \sin (c+d x)+7371 i \sin (3 (c+d x))+5775 i \sin (5 (c+d x))+3003 i \sin (7 (c+d x))+28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x)))}{411840 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 255, normalized size = 1. \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{-{\frac{1792\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}+{\frac{{\frac{1876\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{1472\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}+{\frac{14896}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{112\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}-{\frac{11872}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+{\frac{7\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{{\frac{7504\,i}{5}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-1064\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-7}-{\frac{98}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{3136}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}+{\frac{1484}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{128}{15\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.25304, size = 242, normalized size = 0.9 \begin{align*} \frac{429 i \, \cos \left (15 \, d x + 15 \, c\right ) + 3465 i \, \cos \left (13 \, d x + 13 \, c\right ) + 12285 i \, \cos \left (11 \, d x + 11 \, c\right ) + 25025 i \, \cos \left (9 \, d x + 9 \, c\right ) + 32175 i \, \cos \left (7 \, d x + 7 \, c\right ) + 27027 i \, \cos \left (5 \, d x + 5 \, c\right ) + 15015 i \, \cos \left (3 \, d x + 3 \, c\right ) + 6435 i \, \cos \left (d x + c\right ) + 429 \, \sin \left (15 \, d x + 15 \, c\right ) + 3465 \, \sin \left (13 \, d x + 13 \, c\right ) + 12285 \, \sin \left (11 \, d x + 11 \, c\right ) + 25025 \, \sin \left (9 \, d x + 9 \, c\right ) + 32175 \, \sin \left (7 \, d x + 7 \, c\right ) + 27027 \, \sin \left (5 \, d x + 5 \, c\right ) + 15015 \, \sin \left (3 \, d x + 3 \, c\right ) + 6435 \, \sin \left (d x + c\right )}{823680 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53364, size = 355, normalized size = 1.32 \begin{align*} \frac{{\left (6435 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 15015 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27027 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 32175 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 25025 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12285 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3465 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 429 i\right )} e^{\left (-15 i \, d x - 15 i \, c\right )}}{823680 \, 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 = 1.16317, size = 274, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (6435 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 45045 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 210210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 630630 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1414413 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 2357355 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 3063060 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 3063060 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 2407405 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1444443 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 668850 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 222950 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 54915 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7845 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 952\right )}}{6435 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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