3.177 \(\int \frac {\cos (\frac {1}{2} b^2 \pi x^2)}{C(b x)^2} \, dx\)

Optimal. Leaf size=11 \[ -\frac {1}{b C(b x)} \]

[Out]

-1/b/FresnelC(b*x)

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Rubi [A]  time = 0.01, antiderivative size = 11, 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} \[ -\frac {1}{b \text {FresnelC}(b x)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[(b^2*Pi*x^2)/2]/FresnelC[b*x]^2,x]

[Out]

-(1/(b*FresnelC[b*x]))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6441

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(Pi*b)/(2*d), Subst[Int[x^n, x], x, Fresne
lC[b*x]], x] /; FreeQ[{b, d, n}, x] && EqQ[d^2, (Pi^2*b^4)/4]

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*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 1.00 \[ -\frac {1}{b C(b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[(b^2*Pi*x^2)/2]/FresnelC[b*x]^2,x]

[Out]

-(1/(b*FresnelC[b*x]))

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)/fresnelc(b*x)^2,x, algorithm="fricas")

[Out]

integral(cos(1/2*pi*b^2*x^2)/fresnelc(b*x)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)/fresnelc(b*x)^2,x, algorithm="giac")

[Out]

integrate(cos(1/2*pi*b^2*x^2)/fresnelc(b*x)^2, x)

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maple [A]  time = 0.02, size = 12, normalized size = 1.09 \[ -\frac {1}{b \FresnelC \left (b x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)/FresnelC(b*x)^2,x)

[Out]

-1/b/FresnelC(b*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{{\rm fresnelc}\left (b x\right )^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b^2*pi*x^2)/fresnelc(b*x)^2,x, algorithm="maxima")

[Out]

integrate(cos(1/2*pi*b^2*x^2)/fresnelc(b*x)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.09 \[ \int \frac {\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right )}{{\mathrm {FresnelC}\left (b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos((Pi*b^2*x^2)/2)/FresnelC(b*x)^2,x)

[Out]

int(cos((Pi*b^2*x^2)/2)/FresnelC(b*x)^2, x)

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sympy [A]  time = 0.75, size = 12, normalized size = 1.09 \[ \begin {cases} - \frac {1}{b C\left (b x\right )} & \text {for}\: b \neq 0 \\\tilde {\infty } x & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(1/2*b**2*pi*x**2)/fresnelc(b*x)**2,x)

[Out]

Piecewise((-1/(b*fresnelc(b*x)), Ne(b, 0)), (zoo*x, True))

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