Optimal. Leaf size=13 \[ -\frac {1}{2 b C(b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 13, 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} \[ -\frac {1}{2 b \text {FresnelC}(b x)^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 6441
Rubi steps
\begin {align*} \int \frac {\cos \left (\frac {1}{2} b^2 \pi x^2\right )}{C(b x)^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,C(b x)\right )}{b}\\ &=-\frac {1}{2 b C(b x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ -\frac {1}{2 b C(b x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{3}}, x\right ) \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 0.92 \[ -\frac {1}{2 b \FresnelC \left (b x \right )^{2}} \]
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 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{{\mathrm {FresnelC}\left (b\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.66, size = 15, normalized size = 1.15 \[ \begin {cases} - \frac {1}{2 b C^{2}\left (b x\right )} & \text {for}\: b \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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