Optimal. Leaf size=24 \[ \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3527, 3433}
\begin {gather*} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {2 x+1}{\sqrt {2 \pi }}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3433
Rule 3527
Rubi steps
\begin {align*} \int \cos \left (\frac {1}{4}+x+x^2\right ) \, dx &=\int \cos \left (\frac {1}{4} (1+2 x)^2\right ) \, dx\\ &=\sqrt {\frac {\pi }{2}} C\left (\frac {1+2 x}{\sqrt {2 \pi }}\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 24, normalized size = 1.00 \begin {gather*} \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {1+2 x}{\sqrt {2 \pi }}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 20, normalized size = 0.83
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \left (x +\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{2}\) | \(20\) |
risch | \(-\frac {\sqrt {\pi }\, \left (-1\right )^{\frac {3}{4}} \erf \left (\left (-1\right )^{\frac {1}{4}} x +\frac {\left (-1\right )^{\frac {1}{4}}}{2}\right )}{4}+\frac {\sqrt {\pi }\, \erf \left (\sqrt {-i}\, x -\frac {i}{2 \sqrt {-i}}\right )}{4 \sqrt {-i}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.54, size = 70, normalized size = 2.92 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} {\left (2 \, x + 1\right )}}{2 \, \sqrt {\pi }}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 29, normalized size = 1.21 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \cdot \left (2 x + 1\right )}{2 \sqrt {\pi }}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.48, size = 39, normalized size = 1.62 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 19, normalized size = 0.79 \begin {gather*} \frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\left (x+\frac {1}{2}\right )}{\sqrt {\pi }}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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