\(\int x^4 \operatorname {FresnelC}(b x)^2 \, dx\) [143]

Optimal result

Integrand size = 10, antiderivative size = 177 \[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\frac {4 x^3}{15 b^2 \pi ^2}-\frac {x^3 \cos \left (b^2 \pi x^2\right )}{10 b^2 \pi ^2}-\frac {8 x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{5 b^3 \pi ^2}+\frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {43 \operatorname {FresnelS}\left (\sqrt {2} b x\right )}{20 \sqrt {2} b^5 \pi ^3}+\frac {16 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b^5 \pi ^3}-\frac {2 x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{5 b \pi }+\frac {11 x \sin \left (b^2 \pi x^2\right )}{20 b^4 \pi ^3} \] Output:

4/15*x^3/b^2/Pi^2-1/10*x^3*cos(b^2*Pi*x^2)/b^2/Pi^2-8/5*x^2*cos(1/2*b^2*Pi 
*x^2)*FresnelC(b*x)/b^3/Pi^2+1/5*x^5*FresnelC(b*x)^2-43/40*FresnelS(2^(1/2 
)*b*x)*2^(1/2)/b^5/Pi^3+16/5*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b^5/Pi^3-2/ 
5*x^4*FresnelC(b*x)*sin(1/2*b^2*Pi*x^2)/b/Pi+11/20*x*sin(b^2*Pi*x^2)/b^4/P 
i^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.77 \[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\frac {32 b^3 \pi x^3-12 b^3 \pi x^3 \cos \left (b^2 \pi x^2\right )+24 b^5 \pi ^3 x^5 \operatorname {FresnelC}(b x)^2-129 \sqrt {2} \operatorname {FresnelS}\left (\sqrt {2} b x\right )-48 \operatorname {FresnelC}(b x) \left (4 b^2 \pi x^2 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+\left (-8+b^4 \pi ^2 x^4\right ) \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+66 b x \sin \left (b^2 \pi x^2\right )}{120 b^5 \pi ^3} \] Input:

Integrate[x^4*FresnelC[b*x]^2,x]
 

Output:

(32*b^3*Pi*x^3 - 12*b^3*Pi*x^3*Cos[b^2*Pi*x^2] + 24*b^5*Pi^3*x^5*FresnelC[ 
b*x]^2 - 129*Sqrt[2]*FresnelS[Sqrt[2]*b*x] - 48*FresnelC[b*x]*(4*b^2*Pi*x^ 
2*Cos[(b^2*Pi*x^2)/2] + (-8 + b^4*Pi^2*x^4)*Sin[(b^2*Pi*x^2)/2]) + 66*b*x* 
Sin[b^2*Pi*x^2])/(120*b^5*Pi^3)
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.64, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {6985, 7009, 3866, 3867, 3832, 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.

\(\displaystyle \int x^4 \operatorname {FresnelC}(b x)^2 \, dx\)

\(\Big \downarrow \) 6985

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \int x^5 \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx\)

\(\Big \downarrow \) 7009

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\int x^4 \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3866

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \int x^2 \cos \left (b^2 \pi x^2\right )dx}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \int x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 7017

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\int x^2 \cos ^2\left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3873

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx+\frac {\int x^2dx}{2}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \int x^2 \cos \left (b^2 \pi x^2\right )dx+\frac {x^3}{6}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3867

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b^2}\right )+\frac {x^3}{6}}{\pi b}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \int x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)dx}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 7007

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\int \sin \left (b^2 \pi x^2\right )dx}{2 \pi b}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}+\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {1}{5} x^5 \operatorname {FresnelC}(b x)^2-\frac {2}{5} b \left (\frac {x^4 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {4 \left (\frac {2 \left (\frac {\operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^2}\right )}{\pi b^2}-\frac {x^2 \operatorname {FresnelC}(b x) \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}+\frac {\frac {1}{2} \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )+\frac {x^3}{6}}{\pi b}\right )}{\pi b^2}-\frac {\frac {3 \left (\frac {x \sin \left (\pi b^2 x^2\right )}{2 \pi b^2}-\frac {\operatorname {FresnelS}\left (\sqrt {2} b x\right )}{2 \sqrt {2} \pi b^3}\right )}{2 \pi b^2}-\frac {x^3 \cos \left (\pi b^2 x^2\right )}{2 \pi b^2}}{2 \pi b}\right )\)

Input:

Int[x^4*FresnelC[b*x]^2,x]
 

Output:

(x^5*FresnelC[b*x]^2)/5 - (2*b*((x^4*FresnelC[b*x]*Sin[(b^2*Pi*x^2)/2])/(b 
^2*Pi) - (-1/2*(x^3*Cos[b^2*Pi*x^2])/(b^2*Pi) + (3*(-1/2*FresnelS[Sqrt[2]* 
b*x]/(Sqrt[2]*b^3*Pi) + (x*Sin[b^2*Pi*x^2])/(2*b^2*Pi)))/(2*b^2*Pi))/(2*b* 
Pi) - (4*(-((x^2*Cos[(b^2*Pi*x^2)/2]*FresnelC[b*x])/(b^2*Pi)) + (2*(-1/2*F 
resnelS[Sqrt[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*Sin[b^2*Pi*x^2])/(2*b^2*Pi))/2)/(b*Pi)))/(b^2*Pi)))/5
 

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 3866

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

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 6985

Int[FresnelC[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Fresnel 
C[b*x]^2/(m + 1)), x] - Simp[2*(b/(m + 1))   Int[x^(m + 1)*Cos[(Pi/2)*b^2*x 
^2]*FresnelC[b*x], x], x] /; FreeQ[b, x] && IntegerQ[m] && NeQ[m, -1]
 

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 7009

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

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 = 0.77 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.18

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{5 \pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}+\frac {\frac {2 b x \sin \left (b^{2} \pi \,x^{2}\right )}{5 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi }}{\pi ^{2}}+\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi ^{3}}}{b^{5}}\)	\(209\)
default	\(\frac {\frac {\operatorname {FresnelC}\left (b x \right )^{2} b^{5} x^{5}}{5}-2 \,\operatorname {FresnelC}\left (b x \right ) \left (\frac {b^{4} x^{4} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{5 \pi }-\frac {4 \left (-\frac {b^{2} x^{2} \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi }+\frac {2 \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{\pi ^{2}}\right )}{5 \pi }\right )+\frac {4 b^{3} x^{3}}{15 \pi ^{2}}+\frac {\frac {2 b x \sin \left (b^{2} \pi \,x^{2}\right )}{5 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi }}{\pi ^{2}}+\frac {-\frac {\pi \,b^{3} x^{3} \cos \left (b^{2} \pi \,x^{2}\right )}{2}+\frac {3 \pi \left (\frac {b x \sin \left (b^{2} \pi \,x^{2}\right )}{2 \pi }-\frac {\sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{4 \pi }\right )}{2}-4 \sqrt {2}\, \operatorname {FresnelS}\left (\sqrt {2}\, b x \right )}{5 \pi ^{3}}}{b^{5}}\)	\(209\)

Input:

int(x^4*FresnelC(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/b^5*(1/5*FresnelC(b*x)^2*b^5*x^5-2*FresnelC(b*x)*(1/5/Pi*b^4*x^4*sin(1/2 
*b^2*Pi*x^2)-4/5/Pi*(-1/Pi*b^2*x^2*cos(1/2*b^2*Pi*x^2)+2/Pi^2*sin(1/2*b^2* 
Pi*x^2)))+4/15/Pi^2*b^3*x^3+4/5/Pi^2*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^ 
(1/2)*FresnelS(2^(1/2)*b*x))+1/5/Pi^3*(-1/2*Pi*b^3*x^3*cos(b^2*Pi*x^2)+3/2 
*Pi*(1/2/Pi*b*x*sin(b^2*Pi*x^2)-1/4/Pi*2^(1/2)*FresnelS(2^(1/2)*b*x))-4*2^ 
(1/2)*FresnelS(2^(1/2)*b*x)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.84 \[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\frac {24 \, \pi ^{3} b^{6} x^{5} \operatorname {C}\left (b x\right )^{2} - 24 \, \pi b^{4} x^{3} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} + 44 \, \pi b^{4} x^{3} - 192 \, \pi b^{3} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - 129 \, \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 12 \, {\left (11 \, b^{2} x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - 4 \, {\left (\pi ^{2} b^{5} x^{4} - 8 \, b\right )} \operatorname {C}\left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{120 \, \pi ^{3} b^{6}} \] Input:

integrate(x^4*fresnel_cos(b*x)^2,x, algorithm="fricas")
 

Output:

1/120*(24*pi^3*b^6*x^5*fresnel_cos(b*x)^2 - 24*pi*b^4*x^3*cos(1/2*pi*b^2*x 
^2)^2 + 44*pi*b^4*x^3 - 192*pi*b^3*x^2*cos(1/2*pi*b^2*x^2)*fresnel_cos(b*x 
) - 129*sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x) + 12*(11*b^2*x* 
cos(1/2*pi*b^2*x^2) - 4*(pi^2*b^5*x^4 - 8*b)*fresnel_cos(b*x))*sin(1/2*pi* 
b^2*x^2))/(pi^3*b^6)
 

Sympy [F]

\[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{4} C^{2}\left (b x\right )\, dx \] Input:

integrate(x**4*fresnelc(b*x)**2,x)
 

Output:

Integral(x**4*fresnelc(b*x)**2, x)
 

Maxima [F]

\[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{4} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^4*fresnel_cos(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^4*fresnel_cos(b*x)^2, x)
 

Giac [F]

\[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{4} \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^4*fresnel_cos(b*x)^2,x, algorithm="giac")
 

Output:

integrate(x^4*fresnel_cos(b*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\int x^4\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \] Input:

int(x^4*FresnelC(b*x)^2,x)
 

Output:

int(x^4*FresnelC(b*x)^2, x)
 

Reduce [F]

\[ \int x^4 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{4} \mathrm {FresnelC}\left (b x \right )^{2}d x \] Input:

int(x^4*FresnelC(b*x)^2,x)
 

Output:

int(x^4*FresnelC(b*x)^2,x)