Integrand size = 7, antiderivative size = 12 \[ \int \cos (4 x) \sec (x) \, dx=\text {arctanh}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
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Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4449, 1167, 212} \[ \int \cos (4 x) \sec (x) \, dx=\text {arctanh}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
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Rule 212
Rule 1167
Rule 4449
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1-8 x^2+8 x^4}{1-x^2} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (-8 x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (x)\right ) \\ & = -\frac {8}{3} \sin ^3(x)+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right ) \\ & = \text {arctanh}(\sin (x))-\frac {8 \sin ^3(x)}{3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \cos (4 x) \sec (x) \, dx=\text {arctanh}(\sin (x))-\frac {8 \sin ^3(x)}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).
Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.83
method | result | size |
default | \(\ln \left (\sec \left (x \right )+\tan \left (x \right )\right )+\frac {8 \left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}-8 \sin \left (x \right )\) | \(22\) |
risch | \(i {\mathrm e}^{i x}-i {\mathrm e}^{-i x}+\ln \left (i+{\mathrm e}^{i x}\right )-\ln \left ({\mathrm e}^{i x}-i\right )+\frac {2 \sin \left (3 x \right )}{3}\) | \(44\) |
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Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \cos (4 x) \sec (x) \, dx=\frac {8}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.00 \[ \int \cos (4 x) \sec (x) \, dx=- \frac {\log {\left (\sin {\left (x \right )} - 1 \right )}}{2} + \frac {\log {\left (\sin {\left (x \right )} + 1 \right )}}{2} - \frac {8 \sin ^{3}{\left (x \right )}}{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (10) = 20\).
Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.75 \[ \int \cos (4 x) \sec (x) \, dx=-\frac {8}{3} \, \sin \left (x\right )^{3} + \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (\sin \left (x\right ) - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 23 vs. \(2 (10) = 20\).
Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.92 \[ \int \cos (4 x) \sec (x) \, dx=-\frac {8}{3} \, \sin \left (x\right )^{3} + \frac {1}{2} \, \log \left (\sin \left (x\right ) + 1\right ) - \frac {1}{2} \, \log \left (-\sin \left (x\right ) + 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \cos (4 x) \sec (x) \, dx=\mathrm {atanh}\left (\sin \left (x\right )\right )-\frac {8\,{\sin \left (x\right )}^3}{3} \]
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