3.1.49 \(\int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx\) [49]

3.1.49.1 Optimal result

Integrand size = 16, antiderivative size = 497 \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\frac {2 d^2 x}{3 b^2 \pi ^2}+\frac {d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 b^3 \pi ^2}+\frac {d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 b^3 \pi ^2}-\frac {5 d^2 \operatorname {FresnelC}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2} b^3 \pi ^2}+\frac {2 (b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^3 \pi }+\frac {2 d (b c-a d) (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{b^3 \pi }+\frac {2 d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) \operatorname {FresnelS}(a+b x)}{3 b^3 \pi }-\frac {d (b c-a d) \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{b^3 \pi }+\frac {(b c-a d)^2 (a+b x) \operatorname {FresnelS}(a+b x)^2}{b^3}+\frac {d (b c-a d) (a+b x)^2 \operatorname {FresnelS}(a+b x)^2}{b^3}+\frac {d^2 (a+b x)^3 \operatorname {FresnelS}(a+b x)^2}{3 b^3}-\frac {(b c-a d)^2 \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} b^3 \pi }+\frac {i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac {i d (b c-a d) (a+b x)^2 \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{4 b^3 \pi }-\frac {4 d^2 \operatorname {FresnelS}(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]

output

2/3*d^2*x/b^2/Pi^2+1/2*d*(-a*d+b*c)*cos(Pi*(b*x+a)^2)/b^3/Pi^2+1/6*d^2*(b* 
x+a)*cos(Pi*(b*x+a)^2)/b^3/Pi^2+2*(-a*d+b*c)^2*cos(1/2*Pi*(b*x+a)^2)*Fresn 
elS(b*x+a)/b^3/Pi+2*d*(-a*d+b*c)*(b*x+a)*cos(1/2*Pi*(b*x+a)^2)*FresnelS(b* 
x+a)/b^3/Pi+2/3*d^2*(b*x+a)^2*cos(1/2*Pi*(b*x+a)^2)*FresnelS(b*x+a)/b^3/Pi 
-d*(-a*d+b*c)*FresnelC(b*x+a)*FresnelS(b*x+a)/b^3/Pi+(-a*d+b*c)^2*(b*x+a)* 
FresnelS(b*x+a)^2/b^3+d*(-a*d+b*c)*(b*x+a)^2*FresnelS(b*x+a)^2/b^3+1/3*d^2 
*(b*x+a)^3*FresnelS(b*x+a)^2/b^3+1/4*I*d*(-a*d+b*c)*(b*x+a)^2*hypergeom([1 
, 1],[3/2, 2],-1/2*I*Pi*(b*x+a)^2)/b^3/Pi-1/4*I*d*(-a*d+b*c)*(b*x+a)^2*hyp 
ergeom([1, 1],[3/2, 2],1/2*I*Pi*(b*x+a)^2)/b^3/Pi-4/3*d^2*FresnelS(b*x+a)* 
sin(1/2*Pi*(b*x+a)^2)/b^3/Pi^2-5/12*d^2*FresnelC((b*x+a)*2^(1/2))/b^3/Pi^2 
*2^(1/2)-1/2*(-a*d+b*c)^2*FresnelS((b*x+a)*2^(1/2))/b^3/Pi*2^(1/2)
 

3.1.49.2 Mathematica [F]

\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx \]

input

Integrate[(c + d*x)^2*FresnelS[a + b*x]^2,x]
 

output

Integrate[(c + d*x)^2*FresnelS[a + b*x]^2, x]
 

3.1.49.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6986, 2009}

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.

\(\displaystyle \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx\)

\(\Big \downarrow \) 6986

\(\displaystyle \frac {\int \left ((b c-a d)^2 \operatorname {FresnelS}(a+b x)^2+d^2 (a+b x)^2 \operatorname {FresnelS}(a+b x)^2+2 d (b c-a d) (a+b x) \operatorname {FresnelS}(a+b x)^2\right )d(a+b x)}{b^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {i d (a+b x)^2 (b c-a d) \, _2F_2\left (1,1;\frac {3}{2},2;-\frac {1}{2} i \pi (a+b x)^2\right )}{4 \pi }-\frac {i d (a+b x)^2 (b c-a d) \, _2F_2\left (1,1;\frac {3}{2},2;\frac {1}{2} i \pi (a+b x)^2\right )}{4 \pi }-\frac {d (b c-a d) \operatorname {FresnelC}(a+b x) \operatorname {FresnelS}(a+b x)}{\pi }+d (a+b x)^2 (b c-a d) \operatorname {FresnelS}(a+b x)^2+(a+b x) (b c-a d)^2 \operatorname {FresnelS}(a+b x)^2-\frac {(b c-a d)^2 \operatorname {FresnelS}\left (\sqrt {2} (a+b x)\right )}{\sqrt {2} \pi }+\frac {2 d (a+b x) (b c-a d) \operatorname {FresnelS}(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {2 (b c-a d)^2 \operatorname {FresnelS}(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {d (b c-a d) \cos \left (\pi (a+b x)^2\right )}{2 \pi ^2}-\frac {5 d^2 \operatorname {FresnelC}\left (\sqrt {2} (a+b x)\right )}{6 \sqrt {2} \pi ^2}+\frac {1}{3} d^2 (a+b x)^3 \operatorname {FresnelS}(a+b x)^2-\frac {4 d^2 \operatorname {FresnelS}(a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi ^2}+\frac {2 d^2 (a+b x)^2 \operatorname {FresnelS}(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 \pi }+\frac {2 d^2 (a+b x)}{3 \pi ^2}+\frac {d^2 (a+b x) \cos \left (\pi (a+b x)^2\right )}{6 \pi ^2}}{b^3}\)

input

Int[(c + d*x)^2*FresnelS[a + b*x]^2,x]
 

output

((2*d^2*(a + b*x))/(3*Pi^2) + (d*(b*c - a*d)*Cos[Pi*(a + b*x)^2])/(2*Pi^2) 
 + (d^2*(a + b*x)*Cos[Pi*(a + b*x)^2])/(6*Pi^2) - (5*d^2*FresnelC[Sqrt[2]* 
(a + b*x)])/(6*Sqrt[2]*Pi^2) + (2*(b*c - a*d)^2*Cos[(Pi*(a + b*x)^2)/2]*Fr 
esnelS[a + b*x])/Pi + (2*d*(b*c - a*d)*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2]*F 
resnelS[a + b*x])/Pi + (2*d^2*(a + b*x)^2*Cos[(Pi*(a + b*x)^2)/2]*FresnelS 
[a + b*x])/(3*Pi) - (d*(b*c - a*d)*FresnelC[a + b*x]*FresnelS[a + b*x])/Pi 
 + (b*c - a*d)^2*(a + b*x)*FresnelS[a + b*x]^2 + d*(b*c - a*d)*(a + b*x)^2 
*FresnelS[a + b*x]^2 + (d^2*(a + b*x)^3*FresnelS[a + b*x]^2)/3 - ((b*c - a 
*d)^2*FresnelS[Sqrt[2]*(a + b*x)])/(Sqrt[2]*Pi) + ((I/4)*d*(b*c - a*d)*(a 
+ b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (-1/2*I)*Pi*(a + b*x)^2])/Pi 
- ((I/4)*d*(b*c - a*d)*(a + b*x)^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, (I/ 
2)*Pi*(a + b*x)^2])/Pi - (4*d^2*FresnelS[a + b*x]*Sin[(Pi*(a + b*x)^2)/2]) 
/(3*Pi^2))/b^3
 

3.1.49.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6986

Int[FresnelS[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[1/b^(m + 1)   Subst[Int[ExpandIntegrand[FresnelS[x]^2, (b*c - a*d + d* 
x)^m, x], x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
 

3.1.49.4 Maple [F]

\[\int \left (d x +c \right )^{2} \operatorname {FresnelS}\left (b x +a \right )^{2}d x\]

input

int((d*x+c)^2*FresnelS(b*x+a)^2,x)
 

output

int((d*x+c)^2*FresnelS(b*x+a)^2,x)
 

3.1.49.5 Fricas [F]

\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]

input

integrate((d*x+c)^2*fresnel_sin(b*x+a)^2,x, algorithm="fricas")
 

output

integral((d^2*x^2 + 2*c*d*x + c^2)*fresnel_sin(b*x + a)^2, x)
 

3.1.49.6 Sympy [F]

\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int \left (c + d x\right )^{2} S^{2}\left (a + b x\right )\, dx \]

input

integrate((d*x+c)**2*fresnels(b*x+a)**2,x)
 

output

Integral((c + d*x)**2*fresnels(a + b*x)**2, x)
 

3.1.49.7 Maxima [F]

\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]

input

integrate((d*x+c)^2*fresnel_sin(b*x+a)^2,x, algorithm="maxima")
 

output

integrate((d*x + c)^2*fresnel_sin(b*x + a)^2, x)
 

3.1.49.8 Giac [F]

\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {S}\left (b x + a\right )^{2} \,d x } \]

input

integrate((d*x+c)^2*fresnel_sin(b*x+a)^2,x, algorithm="giac")
 

output

integrate((d*x + c)^2*fresnel_sin(b*x + a)^2, x)
 

3.1.49.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x)^2 \, dx=\int {\mathrm {FresnelS}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2 \,d x \]

input

int(FresnelS(a + b*x)^2*(c + d*x)^2,x)
 

output

int(FresnelS(a + b*x)^2*(c + d*x)^2, x)