Optimal. Leaf size=24 \[ \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3385, 3433}
\begin {gather*} \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rubi steps
\begin {align*} \int \frac {\cos (x)}{\sqrt {x}} \, dx &=2 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=\sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {x}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.01, size = 51, normalized size = 2.12 \begin {gather*} -\frac {i \left (\sqrt {-i x} \text {Gamma}\left (\frac {1}{2},-i x\right )-\sqrt {i x} \text {Gamma}\left (\frac {1}{2},i x\right )\right )}{2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 19, normalized size = 0.79
method | result | size |
derivativedivides | \(\FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\) | \(19\) |
default | \(\FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\) | \(19\) |
meijerg | \(\FresnelC \left (\frac {\sqrt {2}\, \sqrt {x}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi }\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 60, normalized size = 2.50 \begin {gather*} -\frac {1}{8} \, \sqrt {\pi } {\left (\left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) - \left (i + 1\right ) \, \sqrt {2} \operatorname {erf}\left (\sqrt {-i} \sqrt {x}\right ) + \left (i - 1\right ) \, \sqrt {2} \operatorname {erf}\left (\left (-1\right )^{\frac {1}{4}} \sqrt {x}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 18, normalized size = 0.75 \begin {gather*} \sqrt {2} \sqrt {\pi } \operatorname {C}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 37, normalized size = 1.54 \begin {gather*} \frac {\sqrt {2} \sqrt {\pi } C\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {\pi }}\right ) \Gamma \left (\frac {1}{4}\right )}{4 \Gamma \left (\frac {5}{4}\right )} \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 = 0.41, size = 35, normalized size = 1.46 \begin {gather*} -\left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) + \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 18, normalized size = 0.75 \begin {gather*} \sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {x}}{\sqrt {\pi }}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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