Integrand size = 11, antiderivative size = 27 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {x}{2}+\frac {1}{4} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3531, 3527, 3433} \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 x+1}{\sqrt {\pi }}\right )+\frac {x}{2} \]
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Rule 3433
Rule 3527
Rule 3531
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (\frac {1}{2}+2 x+2 x^2\right )\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \int \cos \left (\frac {1}{2}+2 x+2 x^2\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \int \cos \left (\frac {1}{8} (2+4 x)^2\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{4} \sqrt {\pi } \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \left (2 x+\sqrt {\pi } \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {\pi }}\right )\right ) \]
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Time = 1.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74
method | result | size |
default | \(\frac {x}{2}+\frac {\operatorname {C}\left (\frac {1+2 x}{\sqrt {\pi }}\right ) \sqrt {\pi }}{4}\) | \(20\) |
risch | \(\frac {x}{2}-\frac {\sqrt {\pi }\, \sqrt {2}\, \left (-1\right )^{\frac {3}{4}} \operatorname {erf}\left (\sqrt {2}\, \left (-1\right )^{\frac {1}{4}} x +\frac {\sqrt {2}\, \left (-1\right )^{\frac {1}{4}}}{2}\right )}{16}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-2 i}\, x -\frac {i}{\sqrt {-2 i}}\right )}{8 \sqrt {-2 i}}\) | \(58\) |
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none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{4} \, \sqrt {\pi } \operatorname {C}\left (\frac {2 \, x + 1}{\sqrt {\pi }}\right ) + \frac {1}{2} \, x \]
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Time = 0.37 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {x}{2} + \frac {\sqrt {\pi } C\left (\frac {4 x + 2}{2 \sqrt {\pi }}\right )}{4} \]
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Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {1}{16} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {2 i}}\right ) + \left (i + 1\right ) \, \operatorname {erf}\left (\frac {2 i \, x + i}{\sqrt {-2 i}}\right )\right )} + \frac {1}{2} \, x \]
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Result contains complex when optimal does not.
Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=-\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, x + \frac {1}{2} i - \frac {1}{2}\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, x - \frac {1}{2} i - \frac {1}{2}\right ) + \frac {1}{2} \, x \]
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Timed out. \[ \int \cos ^2\left (\frac {1}{4}+x+x^2\right ) \, dx=\int {\cos \left (x^2+x+\frac {1}{4}\right )}^2 \,d x \]
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