\(\int x^3 \operatorname {FresnelC}(b x) \sin (\frac {1}{2} b^2 \pi x^2) \, dx\) [205]

Optimal result

Integrand size = 20, antiderivative size = 109 \[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {x^3}{6 b \pi }-\frac {x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{b^2 \pi }-\frac {5 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{4 \sqrt {2} b^4 \pi ^2}+\frac {2 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{b^4 \pi ^2}+\frac {x \sin \left (b^2 \pi x^2\right )}{4 b^3 \pi ^2} \] Output:

1/6*x^3/b/Pi-x^2*cos(1/2*b^2*Pi*x^2)*FresnelC(b*x)/b^2/Pi-5/8*FresnelS(2^( 
1/2)*b*x)*2^(1/2)/b^4/Pi^2+2*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^4/Pi^2+1/ 
4*x*sin(b^2*Pi*x^2)/b^3/Pi^2
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {4 b^3 \pi x^3-15 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )-24 \operatorname {FresnelC}(b x) \left (b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )-2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+6 b x \sin \left (b^2 \pi x^2\right )}{24 b^4 \pi ^2} \] Input:

Integrate[x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
 

Output:

(4*b^3*Pi*x^3 - 15*Sqrt[2]*FresnelS[Sqrt[2]*b*x] - 24*FresnelC[b*x]*(b^2*P 
i*x^2*Cos[(b^2*Pi*x^2)/2] - 2*Sin[(b^2*Pi*x^2)/2]) + 6*b*x*Sin[b^2*Pi*x^2] 
)/(24*b^4*Pi^2)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.36, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {7017, 3873, 15, 3867, 3832, 7007, 3832}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right ) \, dx\)

\(\Big \downarrow \) 7017

\(\displaystyle \frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3873

\(\displaystyle \frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx+\frac {\int x^2dx}{2}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx+\frac {x^3}{6}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\)

\(\Big \downarrow \) 7007

\(\displaystyle \frac {2 \left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {2 \left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\)

Input:

Int[x^3*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2],x]
 

Output:

-((x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi)) + (2*(-1/2*FresnelS[Sq 
rt[2]*b*x]/(Sqrt[2]*b^2*Pi) + (FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi) 
))/(b^2*Pi) + (x^3/6 + (-1/2*FresnelS[Sqrt[2]*b*x]/(Sqrt[2]*b^3*Pi) + (x*S 
in[b^2*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)
 

Defintions of rubi rules used

rule 15

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

rule 3832

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 3867

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n 
 - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ 
(d*n))   Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && 
 IGtQ[n, 0] && LtQ[n, m + 1]
 

rule 3873

Int[Cos[(a_.) + ((b_.)*(x_)^(n_))/2]^2*(x_)^(m_.), x_Symbol] :> Simp[1/2 
Int[x^m, x], x] + Simp[1/2   Int[x^m*Cos[2*a + b*x^n], x], x] /; FreeQ[{a, 
b, m, n}, x]
 

rule 7007

Int[Cos[(d_.)*(x_)^2]*FresnelC[(b_.)*(x_)]*(x_), x_Symbol] :> Simp[Sin[d*x^ 
2]*(FresnelC[b*x]/(2*d)), x] - Simp[b/(4*d)   Int[Sin[2*d*x^2], x], x] /; F 
reeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4]
 

rule 7017

Int[FresnelC[(b_.)*(x_)]*(x_)^(m_)*Sin[(d_.)*(x_)^2], x_Symbol] :> Simp[(-x 
^(m - 1))*Cos[d*x^2]*(FresnelC[b*x]/(2*d)), x] + (Simp[(m - 1)/(2*d)   Int[ 
x^(m - 2)*Cos[d*x^2]*FresnelC[b*x], x], x] + Simp[b/(2*d)   Int[x^(m - 1)*C 
os[d*x^2]^2, x], x]) /; FreeQ[{b, d}, x] && EqQ[d^2, (Pi^2/4)*b^4] && IGtQ[ 
m, 1]
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{b^{3}}-\frac {\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{2 \pi ^{2}}-\frac {b^{3} x^{3}}{6 \pi }-\frac {\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }}{2 \pi }}{b^{3}}}{b}\)	\(120\)

Input:

int(x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x,method=_RETURNVERBOSE)
 

Output:

(FresnelC(b*x)/b^3*(-1/Pi*b^2*x^2*cos(1/2*b^2*Pi*x^2)+2/Pi^2*sin(1/2*b^2*P 
i*x^2))-1/b^3*(1/2/Pi^2*2^(1/2)*FresnelS(2^(1/2)*b*x)-1/6/Pi*b^3*x^3-1/2/P 
i*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(2^(1/2)*b*x))))/b
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.89 \[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\frac {4 \, \pi b^{4} x^{3} - 24 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 15 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 12 \, {\left (b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 4 \, b \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{24 \, \pi ^{2} b^{5}} \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

1/24*(4*pi*b^4*x^3 - 24*pi*b^3*x^2*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x) - 
15*sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x) + 12*(b^2*x*cos(1/2* 
pi*b^2*x^2) + 4*b*fresnel_cos(b*x))*sin(1/2*pi*b^2*x^2))/(pi^2*b^5)
 

Sympy [F]

\[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} C\left (b x\right )\, dx \] Input:

integrate(x**3*fresnelc(b*x)*sin(1/2*b**2*pi*x**2),x)
 

Output:

Integral(x**3*sin(pi*b**2*x**2/2)*fresnelc(b*x), x)
 

Maxima [F]

\[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="maxima")
 

Output:

integrate(x^3*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
 

Giac [F]

\[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \,d x } \] Input:

integrate(x^3*fresnel_cos(b*x)*sin(1/2*b^2*pi*x^2),x, algorithm="giac")
 

Output:

integrate(x^3*fresnel_cos(b*x)*sin(1/2*pi*b^2*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^3\,\mathrm {FresnelC}\left (b\,x\right )\,\sin \left (\frac {\Pi \,b^2\,x^2}{2}\right ) \,d x \] Input:

int(x^3*FresnelC(b*x)*sin((Pi*b^2*x^2)/2),x)
 

Output:

int(x^3*FresnelC(b*x)*sin((Pi*b^2*x^2)/2), x)
 

Reduce [F]

\[ \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right ) \, dx=\int x^{3} \mathrm {FresnelC}\left (b x \right ) \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )d x \] Input:

int(x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)
 

Output:

int(x^3*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2),x)