Optimal. Leaf size=17 \[ \frac {C(b x)^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6441, 30} \[ \frac {\text {FresnelC}(b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6441
Rubi steps
\begin {align*} \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)^n \, dx &=\frac {\operatorname {Subst}\left (\int x^n \, dx,x,C(b x)\right )}{b}\\ &=\frac {C(b x)^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 1.00 \[ \frac {C(b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 18, normalized size = 1.06 \[ \frac {\FresnelC \left (b x \right )^{1+n}}{b \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int {\mathrm {FresnelC}\left (b\,x\right )}^n\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.59, size = 34, normalized size = 2.00 \[ \begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge n = -1 \\0^{n} x & \text {for}\: b = 0 \\\frac {\log {\left (C\left (b x\right ) \right )}}{b} & \text {for}\: n = -1 \\\frac {C\left (b x\right ) C^{n}\left (b x\right )}{b n + b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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