Optimal. Leaf size=17 \[ \frac{\text{FresnelC}(b x)^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.0187874, antiderivative size = 17, normalized size of antiderivative = 1., 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 6441
Rule 30
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*}
Mathematica [A] time = 0.0102103, size = 17, normalized size = 1. \[ \frac{\text{FresnelC}(b x)^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 18, normalized size = 1.1 \begin{align*}{\frac{ \left ({\it FresnelC} \left ( bx \right ) \right ) ^{1+n}}{b \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.59086, size = 34, normalized size = 2. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnelc}\left (b x\right )^{n} \cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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