Integrand size = 7, antiderivative size = 21 \[ \int \sec (3 x) \sin (x) \, dx=\frac {1}{3} \log (\cos (x))-\frac {1}{6} \log \left (3-4 \cos ^2(x)\right ) \]
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Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4442, 272, 36, 29, 31} \[ \int \sec (3 x) \sin (x) \, dx=\frac {1}{3} \log (\cos (x))-\frac {1}{6} \log \left (3-4 \cos ^2(x)\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 4442
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x \left (-3+4 x^2\right )} \, dx,x,\cos (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (-3+4 x)} \, dx,x,\cos ^2(x)\right )\right ) \\ & = \frac {1}{6} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\cos ^2(x)\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3+4 x} \, dx,x,\cos ^2(x)\right ) \\ & = \frac {1}{3} \log (\cos (x))-\frac {1}{6} \log \left (3-4 \cos ^2(x)\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \sec (3 x) \sin (x) \, dx=-\frac {1}{3} \text {arctanh}\left (\frac {1}{3} \left (-5+8 \sin ^2(x)\right )\right ) \]
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Time = 1.55 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\ln \left (\cos \left (x \right )\right )}{3}-\frac {\ln \left (4 \cos \left (x \right )^{2}-3\right )}{6}\) | \(18\) |
risch | \(\frac {\ln \left ({\mathrm e}^{2 i x}+1\right )}{3}-\frac {\ln \left ({\mathrm e}^{4 i x}-{\mathrm e}^{2 i x}+1\right )}{6}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \sec (3 x) \sin (x) \, dx=-\frac {1}{6} \, \log \left (4 \, \cos \left (x\right )^{2} - 3\right ) + \frac {1}{3} \, \log \left (-\cos \left (x\right )\right ) \]
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\[ \int \sec (3 x) \sin (x) \, dx=\int \sin {\left (x \right )} \sec {\left (3 x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (17) = 34\).
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.86 \[ \int \sec (3 x) \sin (x) \, dx=-\frac {1}{12} \, \log \left (-2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{6} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \]
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Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \sec (3 x) \sin (x) \, dx=\frac {1}{6} \, \log \left (-\sin \left (x\right )^{2} + 1\right ) - \frac {1}{6} \, \log \left ({\left | 4 \, \sin \left (x\right )^{2} - 1 \right |}\right ) \]
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Time = 27.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \sec (3 x) \sin (x) \, dx=\frac {\ln \left (\cos \left (x\right )\right )}{3}-\frac {\ln \left ({\cos \left (x\right )}^2-\frac {3}{4}\right )}{6} \]
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