Integrand size = 9, antiderivative size = 16 \[ \int \cos (5 x) \sec ^5(x) \, dx=16 x-15 \tan (x)+\frac {5 \tan ^3(x)}{3} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1167, 209} \[ \int \cos (5 x) \sec ^5(x) \, dx=16 x+\frac {5 \tan ^3(x)}{3}-15 \tan (x) \]
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Rule 209
Rule 1167
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-10 x^2+5 x^4}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (-15+5 x^2+\frac {16}{1+x^2}\right ) \, dx,x,\tan (x)\right ) \\ & = -15 \tan (x)+\frac {5 \tan ^3(x)}{3}+16 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (x)\right ) \\ & = 16 x-15 \tan (x)+\frac {5 \tan ^3(x)}{3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25 \[ \int \cos (5 x) \sec ^5(x) \, dx=16 x-\frac {50 \tan (x)}{3}+\frac {5}{3} \sec ^2(x) \tan (x) \]
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Time = 33.88 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31
method | result | size |
default | \(16 x -5 \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (x \right )\right )}{3}\right ) \tan \left (x \right )-20 \tan \left (x \right )\) | \(21\) |
risch | \(16 x -\frac {20 i \left (6 \,{\mathrm e}^{4 i x}+9 \,{\mathrm e}^{2 i x}+5\right )}{3 \left ({\mathrm e}^{2 i x}+1\right )^{3}}\) | \(33\) |
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Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \cos (5 x) \sec ^5(x) \, dx=\frac {48 \, x \cos \left (x\right )^{3} - 5 \, {\left (10 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )}{3 \, \cos \left (x\right )^{3}} \]
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Time = 7.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \cos (5 x) \sec ^5(x) \, dx=16 x - \frac {20 \sin {\left (x \right )}}{\cos {\left (x \right )}} + \frac {5 \tan ^{3}{\left (x \right )}}{3} + 5 \tan {\left (x \right )} \]
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Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \cos (5 x) \sec ^5(x) \, dx=\frac {5}{3} \, \tan \left (x\right )^{3} + 16 \, x - 15 \, \tan \left (x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \cos (5 x) \sec ^5(x) \, dx=\frac {5}{3} \, \tan \left (x\right )^{3} + 16 \, x - 15 \, \tan \left (x\right ) \]
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Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62 \[ \int \cos (5 x) \sec ^5(x) \, dx=\frac {48\,x\,{\cos \left (x\right )}^3-50\,\sin \left (x\right )\,{\cos \left (x\right )}^2+5\,\sin \left (x\right )}{3\,{\cos \left (x\right )}^3} \]
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