Integrand size = 22, antiderivative size = 44 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \]
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Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3191} \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)-5}} \]
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Rule 3191
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x))}{e \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.70 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\frac {2 \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right ) \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}}{e \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )} \]
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Time = 0.65 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {10 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right )}{\cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\) | \(50\) |
risch | \(\frac {5 i \sqrt {2}\, \sqrt {-10+8 \cos \left (e x +d \right )+6 \sin \left (e x +d \right )}\, \sqrt {\left (4-3 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}-30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}\, \left (5 \,{\mathrm e}^{i \left (e x +d \right )}+4+3 i\right ) \left (5 \,{\mathrm e}^{i \left (e x +d \right )}-4-3 i\right )}{\left (30 i {\mathrm e}^{i \left (e x +d \right )}-25 \,{\mathrm e}^{2 i \left (e x +d \right )}-7-24 i+40 \,{\mathrm e}^{i \left (e x +d \right )}\right ) e \sqrt {\left (100-75 i\right ) \left (25 \,{\mathrm e}^{3 i \left (e x +d \right )}-30 i {\mathrm e}^{2 i \left (e x +d \right )}+7 \,{\mathrm e}^{i \left (e x +d \right )}+24 i {\mathrm e}^{i \left (e x +d \right )}-40 \,{\mathrm e}^{2 i \left (e x +d \right )}\right )}}\) | \(228\) |
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Time = 0.24 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\frac {2 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} {\left (3 \, \cos \left (e x + d\right ) + \sin \left (e x + d\right ) + 3\right )}}{e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + e} \]
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\[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int \sqrt {3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5}\, dx \]
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\[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int { \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} \,d x } \]
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\[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=\int { \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} \,d x } \]
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Time = 0.55 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx=-\frac {2\,\sqrt {5}\,\left (3\,\cos \left (d+e\,x\right )-4\,\sin \left (d+e\,x\right )\right )}{5\,e\,\sqrt {\cos \left (d-\mathrm {atan}\left (\frac {3}{4}\right )+e\,x\right )-1}} \]
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