Optimal. Leaf size=138 \[ \frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.18, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3502, 3488} \[ \frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {3 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{11 a}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{33 a^2}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{231 a^3}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 73, normalized size = 0.53 \[ \frac {i \sec ^8(c+d x) (55 i \sin (c+d x)+63 i \sin (3 (c+d x))+440 \cos (c+d x)+168 \cos (3 (c+d x)))}{4620 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 52, normalized size = 0.38 \[ \frac {{\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 495 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 385 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{9240 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.87, size = 151, normalized size = 1.09 \[ \frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3465 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 23100 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 32802 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11220 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 517 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 152\right )}}{1155 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 189, normalized size = 1.37 \[ \frac {-\frac {4752}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}+\frac {14 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {176 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}-\frac {256}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{11}}+\frac {584 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}+\frac {1864}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}-\frac {576 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{8}}+\frac {1024}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{9}}+\frac {128 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{10}}-\frac {60}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}}{a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 97, normalized size = 0.70 \[ \frac {105 i \, \cos \left (11 \, d x + 11 \, c\right ) + 385 i \, \cos \left (9 \, d x + 9 \, c\right ) + 495 i \, \cos \left (7 \, d x + 7 \, c\right ) + 231 i \, \cos \left (5 \, d x + 5 \, c\right ) + 105 \, \sin \left (11 \, d x + 11 \, c\right ) + 385 \, \sin \left (9 \, d x + 9 \, c\right ) + 495 \, \sin \left (7 \, d x + 7 \, c\right ) + 231 \, \sin \left (5 \, d x + 5 \, c\right )}{9240 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.91, size = 64, normalized size = 0.46 \[ \frac {\frac {{\mathrm {e}}^{-c\,5{}\mathrm {i}-d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{-c\,11{}\mathrm {i}-d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}}{a^8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.83, size = 620, normalized size = 4.49 \[ \begin {cases} \frac {2 \tan ^{3}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {16 i \tan ^{2}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {61 \tan {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} + \frac {152 i \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{5}{\relax (c )}}{\left (i a \tan {\relax (c )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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