Integrand size = 9, antiderivative size = 24 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3527, 3433} \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {2 x+1}{\sqrt {2 \pi }}\right ) \]
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Rule 3433
Rule 3527
Rubi steps \begin{align*} \text {integral}& = \int \cos \left (\frac {1}{4} (1+2 x)^2\right ) \, dx \\ & = \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right ) \]
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Time = 0.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \operatorname {C}\left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{2}\) | \(20\) |
risch | \(-\frac {\sqrt {\pi }\, \left (-1\right )^{\frac {3}{4}} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} x +\frac {\left (-1\right )^{\frac {1}{4}}}{2}\right )}{4}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-i}\, x -\frac {i}{2 \sqrt {-i}}\right )}{4 \sqrt {-i}}\) | \(46\) |
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none
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, x + 1\right )}}{2 \, \sqrt {\pi }}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \cdot \left (2 x + 1\right )}{2 \sqrt {\pi }}\right )}{2} \]
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Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.92 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=-\frac {1}{16} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (-\frac {1}{2} \, \left (-1\right )^{\frac {3}{4}} {\left (2 i \, x + i\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (-\left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} {\left (2 i \, x + i\right )}\right ) - \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} {\left (2 i \, x + i\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\frac {2 i \, x + i}{2 \, \sqrt {-i}}\right )\right )} \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=-\left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} {\left (2 \, x + 1\right )}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx=\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (x+\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{2} \]
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