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

Optimal result

Integrand size = 19, antiderivative size = 17 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6576, 30} \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{n+1}}{b (n+1)} \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int x^n \, dx,x,\operatorname {FresnelC}(b x)\right )}{b} \\ & = \frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {FresnelC}(b x)^{1+n}}{b (1+n)} \]

[In]

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

[Out]

FresnelC[b*x]^(1 + n)/(b*(1 + n))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {C}\left (b x\right )^{n} \operatorname {C}\left (b x\right )}{b n + b} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).

Time = 0.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: b = 0 \wedge n = -1 \\0^{n} x & \text {for}\: b = 0 \\\frac {\log {\left (C\left (b x\right ) \right )}}{b} & \text {for}\: n = -1 \\\frac {C\left (b x\right ) C^{n}\left (b x\right )}{b n + b} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\frac {\operatorname {C}\left (b x\right )^{n + 1}}{b {\left (n + 1\right )}} \]

[In]

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

[Out]

fresnel_cos(b*x)^(n + 1)/(b*(n + 1))

Giac [F]

\[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\int { \operatorname {C}\left (b x\right )^{n} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)^n \, dx=\int {\mathrm {FresnelC}\left (b\,x\right )}^n\,\cos \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \]

[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)