Optimal. Leaf size=68 \[ \frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3502, 3488} \[ \frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
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Rule 3488
Rule 3502
Rubi steps
\begin {align*} \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {\int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{9 a}\\ &=\frac {i \sec ^7(c+d x)}{9 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^7(c+d x)}{63 a d (a+i a \tan (c+d x))^7}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 40, normalized size = 0.59 \[ -\frac {(\tan (c+d x)-8 i) \sec ^7(c+d x)}{63 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 30, normalized size = 0.44 \[ \frac {{\left (9 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{126 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.55, size = 125, normalized size = 1.84 \[ \frac {2 \, {\left (63 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 63 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 483 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 315 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 693 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 189 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 225 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 8\right )}}{63 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 156, normalized size = 2.29 \[ \frac {-\frac {172}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {256}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{9}}-\frac {128 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{8}}+\frac {272}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}-\frac {152 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {14 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}+\frac {992 i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}-\frac {1856}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}}{a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 53, normalized size = 0.78 \[ \frac {7 i \, \cos \left (9 \, d x + 9 \, c\right ) + 9 i \, \cos \left (7 \, d x + 7 \, c\right ) + 7 \, \sin \left (9 \, d x + 9 \, c\right ) + 9 \, \sin \left (7 \, d x + 7 \, c\right )}{126 \, a^{8} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.74, size = 37, normalized size = 0.54 \[ \frac {2\,\left (\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,9{}\mathrm {i}}{4}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,7{}\mathrm {i}}{4}\right )}{63\,a^8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.77, size = 311, normalized size = 4.57 \[ \begin {cases} - \frac {\tan {\left (c + d x \right )} \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} + \frac {8 i \sec ^{7}{\left (c + d x \right )}}{63 a^{8} d \tan ^{8}{\left (c + d x \right )} - 504 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{6}{\left (c + d x \right )} + 3528 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 4410 a^{8} d \tan ^{4}{\left (c + d x \right )} - 3528 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 1764 a^{8} d \tan ^{2}{\left (c + d x \right )} + 504 i a^{8} d \tan {\left (c + d x \right )} + 63 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{7}{\relax (c )}}{\left (i a \tan {\relax (c )} + a\right )^{8}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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