Integrand size = 10, antiderivative size = 16 \[ \int \sqrt {\cos (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
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Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2719} \[ \int \sqrt {\cos (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
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Rule 2719
Rubi steps \begin{align*} \text {integral}& = \frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {\cos (a+b x)} \, dx=\frac {2 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{b} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(42)=84\).
Time = 1.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 8.31
method | result | size |
default | \(\frac {2 \sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, b}\) | \(133\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}}{b}-\frac {i \left (-\frac {2 \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}{\sqrt {\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{i \left (b x +a \right )}}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (b x +a \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (b x +a \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (b x +a \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 i \left (b x +a \right )}+{\mathrm e}^{i \left (b x +a \right )}}}\right ) \sqrt {2}\, \sqrt {\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{-i \left (b x +a \right )}}\, \sqrt {\left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right ) {\mathrm e}^{i \left (b x +a \right )}}}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\) | \(285\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.56 \[ \int \sqrt {\cos (a+b x)} \, dx=\frac {i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) - i \, \sqrt {2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )}{b} \]
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\[ \int \sqrt {\cos (a+b x)} \, dx=\int \sqrt {\cos {\left (a + b x \right )}}\, dx \]
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\[ \int \sqrt {\cos (a+b x)} \, dx=\int { \sqrt {\cos \left (b x + a\right )} \,d x } \]
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\[ \int \sqrt {\cos (a+b x)} \, dx=\int { \sqrt {\cos \left (b x + a\right )} \,d x } \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \sqrt {\cos (a+b x)} \, dx=\frac {2\,\mathrm {E}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{b} \]
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