Optimal. Leaf size=11 \[ -\frac{1}{b \text{FresnelC}(b x)} \]
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Rubi [A] time = 0.0149059, antiderivative size = 11, 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} \[ -\frac{1}{b \text{FresnelC}(b x)} \]
Antiderivative was successfully verified.
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Rule 6441
Rule 30
Rubi steps
\begin{align*} \int \frac{\cos \left (\frac{1}{2} b^2 \pi x^2\right )}{C(b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,C(b x)\right )}{b}\\ &=-\frac{1}{b C(b x)}\\ \end{align*}
Mathematica [A] time = 0.0055501, size = 11, normalized size = 1. \[ -\frac{1}{b \text{FresnelC}(b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 12, normalized size = 1.1 \begin{align*} -{\frac{1}{b{\it FresnelC} \left ( bx \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 \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}\,{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 (\frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.797724, size = 12, normalized size = 1.09 \begin{align*} \begin{cases} - \frac{1}{b C\left (b x\right )} & \text{for}\: b \neq 0 \\\tilde{\infty } x & \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 \frac{\cos \left (\frac{1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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