3.2.44 \(\int x^3 \operatorname {FresnelC}(b x)^2 \, dx\) [144]

3.2.44.1 Optimal result

Integrand size = 10, antiderivative size = 140 \[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\frac {3 x^2}{8 b^2 \pi ^2}-\frac {x^2 \cos \left (b^2 \pi x^2\right )}{8 b^2 \pi ^2}-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right ) \operatorname {FresnelC}(b x)}{2 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)^2}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)^2-\frac {x^3 \operatorname {FresnelC}(b x) \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{2 b \pi }+\frac {\sin \left (b^2 \pi x^2\right )}{2 b^4 \pi ^3} \]

output

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

3.2.44.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81 \[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\frac {3 b^2 \pi x^2-b^2 \pi x^2 \cos \left (b^2 \pi x^2\right )+2 \pi \left (3+b^4 \pi ^2 x^4\right ) \operatorname {FresnelC}(b x)^2-4 b \pi x \operatorname {FresnelC}(b x) \left (3 \cos \left (\frac {1}{2} b^2 \pi x^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )\right )+4 \sin \left (b^2 \pi x^2\right )}{8 b^4 \pi ^3} \]

input

Integrate[x^3*FresnelC[b*x]^2,x]
 

output

(3*b^2*Pi*x^2 - b^2*Pi*x^2*Cos[b^2*Pi*x^2] + 2*Pi*(3 + b^4*Pi^2*x^4)*Fresn 
elC[b*x]^2 - 4*b*Pi*x*FresnelC[b*x]*(3*Cos[(b^2*Pi*x^2)/2] + b^2*Pi*x^2*Si 
n[(b^2*Pi*x^2)/2]) + 4*Sin[b^2*Pi*x^2])/(8*b^4*Pi^3)
 

3.2.44.3 Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {6985, 7009, 3860, 3042, 3777, 3042, 3117, 7017, 3861, 3042, 3114, 6995, 15}

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.

\(\displaystyle \int x^3 \operatorname {FresnelC}(b x)^2 \, dx\)

\(\Big \downarrow \) 6985

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

\(\Big \downarrow \) 7009

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

\(\Big \downarrow \) 3860

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 7017

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

\(\Big \downarrow \) 3861

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3114

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

\(\Big \downarrow \) 6995

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

\(\Big \downarrow \) 15

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

input

Int[x^3*FresnelC[b*x]^2,x]
 

output

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

3.2.44.3.1 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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3114

Int[sin[(c_.) + ((d_.)*(x_))/2]^2, x_Symbol] :> Simp[x/2, x] - Simp[Sin[2*c 
 + d*x]/(2*d), x] /; FreeQ[{c, d}, x]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3860

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

rule 3861

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^ 
p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ 
(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ 
(m + 1)/n], 0]))
 

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 6995

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

3.2.44.4 Maple [F]

\[\int x^{3} \operatorname {FresnelC}\left (b x \right )^{2}d x\]

input

int(x^3*FresnelC(b*x)^2,x)
 

output

int(x^3*FresnelC(b*x)^2,x)
 

3.2.44.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=-\frac {\pi b^{2} x^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )^{2} - 2 \, \pi b^{2} x^{2} + 6 \, \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \operatorname {C}\left (b x\right ) - {\left (3 \, \pi + \pi ^{3} b^{4} x^{4}\right )} \operatorname {C}\left (b x\right )^{2} + 2 \, {\left (\pi ^{2} b^{3} x^{3} \operatorname {C}\left (b x\right ) - 2 \, \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{4 \, \pi ^{3} b^{4}} \]

input

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="fricas")
 

output

-1/4*(pi*b^2*x^2*cos(1/2*pi*b^2*x^2)^2 - 2*pi*b^2*x^2 + 6*pi*b*x*cos(1/2*p 
i*b^2*x^2)*fresnel_cos(b*x) - (3*pi + pi^3*b^4*x^4)*fresnel_cos(b*x)^2 + 2 
*(pi^2*b^3*x^3*fresnel_cos(b*x) - 2*cos(1/2*pi*b^2*x^2))*sin(1/2*pi*b^2*x^ 
2))/(pi^3*b^4)
 

3.2.44.6 Sympy [F]

\[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\int x^{3} C^{2}\left (b x\right )\, dx \]

input

integrate(x**3*fresnelc(b*x)**2,x)
 

output

Integral(x**3*fresnelc(b*x)**2, x)
 

3.2.44.7 Maxima [F]

\[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{3} \operatorname {C}\left (b x\right )^{2} \,d x } \]

input

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="maxima")
 

output

integrate(x^3*fresnel_cos(b*x)^2, x)
 

3.2.44.8 Giac [F]

\[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\int { x^{3} \operatorname {C}\left (b x\right )^{2} \,d x } \]

input

integrate(x^3*fresnel_cos(b*x)^2,x, algorithm="giac")
 

output

integrate(x^3*fresnel_cos(b*x)^2, x)
 

3.2.44.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \operatorname {FresnelC}(b x)^2 \, dx=\int x^3\,{\mathrm {FresnelC}\left (b\,x\right )}^2 \,d x \]

input

int(x^3*FresnelC(b*x)^2,x)
 

output

int(x^3*FresnelC(b*x)^2, x)