Integrand size = 24, antiderivative size = 27 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{7 a d (a+i a \tan (c+d x))^7} \]
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Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 32} \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{7 a d (a+i a \tan (c+d x))^7} \]
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Rule 32
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int \frac {1}{(a+x)^8} \, dx,x,i a \tan (c+d x)\right )}{a d} \\ & = \frac {i}{7 a d (a+i a \tan (c+d x))^7} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {1}{7 a^8 d (-i+\tan (c+d x))^7} \]
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Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {i}{7 a d \left (a +i a \tan \left (d x +c \right )\right )^{7}}\) | \(24\) |
default | \(\frac {i}{7 a d \left (a +i a \tan \left (d x +c \right )\right )^{7}}\) | \(24\) |
risch | \(\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{8} d}+\frac {3 i {\mathrm e}^{-4 i \left (d x +c \right )}}{128 a^{8} d}+\frac {5 i {\mathrm e}^{-6 i \left (d x +c \right )}}{128 a^{8} d}+\frac {5 i {\mathrm e}^{-8 i \left (d x +c \right )}}{128 a^{8} d}+\frac {3 i {\mathrm e}^{-10 i \left (d x +c \right )}}{128 a^{8} d}+\frac {i {\mathrm e}^{-12 i \left (d x +c \right )}}{128 a^{8} d}+\frac {i {\mathrm e}^{-14 i \left (d x +c \right )}}{896 a^{8} d}\) | \(128\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (21) = 42\).
Time = 0.23 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.15 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {{\left (7 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 21 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 35 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 35 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 21 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )} e^{\left (-14 i \, d x - 14 i \, c\right )}}{896 \, a^{8} d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1081 vs. \(2 (19) = 38\).
Time = 8.89 (sec) , antiderivative size = 1081, normalized size of antiderivative = 40.04 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\text {Too large to display} \]
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none
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=\frac {i}{7 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{7} a d} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (21) = 42\).
Time = 1.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 7.00 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {2 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 42 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 182 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 490 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 1001 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1484 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1716 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1484 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1001 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 490 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 182 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 42 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{7 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{14}} \]
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Time = 4.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \frac {\sec ^2(c+d x)}{(a+i a \tan (c+d x))^8} \, dx=-\frac {1}{7\,a^8\,d\,{\left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}^7} \]
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