3.2.79 \(\int \cos (\frac {1}{2} b^2 \pi x^2) \text {FresnelC}(b x)^n \, dx\) [179]

Optimal. Leaf size=17 \[ \frac {\text {FresnelC}(b x)^{1+n}}{b (1+n)} \]

[Out]

FresnelC(b*x)^(1+n)/b/(1+n)

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Rubi [A]
time = 0.01, 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 = {6576, 30} \begin {gather*} \frac {\text {FresnelC}(b x)^{n+1}}{b (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

Rule 30

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

Rule 6576

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^2/4)*b^4]

Rubi steps

\begin {align*} \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) C(b x)^n \, dx &=\frac {\text {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 \begin {gather*} \frac {\text {FresnelC}(b x)^{1+n}}{b (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

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Maple [A]
time = 0.19, size = 18, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\FresnelC \left (b x \right )^{1+n}}{b \left (1+n \right )}\)	\(18\)
default	\(\frac {\FresnelC \left (b x \right )^{1+n}}{b \left (1+n \right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)^n,x,method=_RETURNVERBOSE)

[Out]

FresnelC(b*x)^(1+n)/b/(1+n)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(n>0)', see `assume?` for more
details)Is n

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Fricas [A]
time = 0.34, size = 18, normalized size = 1.06 \begin {gather*} \frac {\operatorname {C}\left (b x\right )^{n} \operatorname {C}\left (b x\right )}{b n + b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

fresnel_cos(b*x)^n*fresnel_cos(b*x)/(b*n + b)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
time = 0.61, size = 34, normalized size = 2.00 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(b, 0) & Eq(n, -1)), (0**n*x, Eq(b, 0)), (log(fresnelc(b*x))/b, Eq(n, -1)), (fresnelc(b*x)
*fresnelc(b*x)**n/(b*n + b), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int {\mathrm {FresnelC}\left (b\,x\right )}^n\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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