Integrand size = 24, antiderivative size = 55 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac {i (a-i a \tan (c+d x))^4}{4 a^5 d} \]
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Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac {i (a-i a \tan (c+d x))^4}{4 a^5 d} \]
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Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^2 (a+x) \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = -\frac {i \text {Subst}\left (\int \left (2 a (a-x)^2-(a-x)^3\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = \frac {2 i (a-i a \tan (c+d x))^3}{3 a^4 d}-\frac {i (a-i a \tan (c+d x))^4}{4 a^5 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.91 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\tan (c+d x) \left (12-6 i \tan (c+d x)+4 \tan ^2(c+d x)-3 i \tan ^3(c+d x)\right )}{12 a d} \]
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Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {4 i \left (4 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{3 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}\) | \(36\) |
derivativedivides | \(-\frac {-\tan \left (d x +c \right )+\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{a d}\) | \(50\) |
default | \(-\frac {-\tan \left (d x +c \right )+\frac {i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{a d}\) | \(50\) |
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Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {4 \, {\left (-4 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}}{3 \, {\left (a d e^{\left (8 i \, d x + 8 i \, c\right )} + 4 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 6 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 4 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]
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\[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\sec ^{6}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]
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Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {-3 i \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{3} - 6 i \, \tan \left (d x + c\right )^{2} + 12 \, \tan \left (d x + c\right )}{12 \, a d} \]
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Time = 0.38 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {3 i \, \tan \left (d x + c\right )^{4} - 4 \, \tan \left (d x + c\right )^{3} + 6 i \, \tan \left (d x + c\right )^{2} - 12 \, \tan \left (d x + c\right )}{12 \, a d} \]
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Time = 4.52 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.40 \[ \int \frac {\sec ^6(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {\sin \left (c+d\,x\right )\,\left (12\,{\cos \left (c+d\,x\right )}^3-{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,6{}\mathrm {i}+4\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^2-{\sin \left (c+d\,x\right )}^3\,3{}\mathrm {i}\right )}{12\,a\,d\,{\cos \left (c+d\,x\right )}^4} \]
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