\(\int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx\) [129]

Optimal result

Integrand size = 14, antiderivative size = 194 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}-\frac {(b c-a d)^3 \operatorname {FresnelC}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelC}(a+b x)}{3 d}+\frac {d (b c-a d) \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {(b c-a d)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d (b c-a d) (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \] Output:

-2/3*d^2*cos(1/2*Pi*(b*x+a)^2)/b^3/Pi^2-1/3*(-a*d+b*c)^3*FresnelC(b*x+a)/b 
^3/d+1/3*(d*x+c)^3*FresnelC(b*x+a)/d+d*(-a*d+b*c)*FresnelS(b*x+a)/b^3/Pi-( 
-a*d+b*c)^2*sin(1/2*Pi*(b*x+a)^2)/b^3/Pi-d*(-a*d+b*c)*(b*x+a)*sin(1/2*Pi*( 
b*x+a)^2)/b^3/Pi-1/3*d^2*(b*x+a)^2*sin(1/2*Pi*(b*x+a)^2)/b^3/Pi
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {-2 d^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi ^2 \left (3 a b^2 c^2-3 a^2 b c d+a^3 d^2+b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \operatorname {FresnelC}(a+b x)+3 d (b c-a d) \pi \operatorname {FresnelS}(a+b x)-3 b^2 c^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+3 a b c d \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-a^2 d^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-3 b^2 c d \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+a b d^2 \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-b^2 d^2 \pi x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \] Input:

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

Output:

(-2*d^2*Cos[(Pi*(a + b*x)^2)/2] + Pi^2*(3*a*b^2*c^2 - 3*a^2*b*c*d + a^3*d^ 
2 + b^3*x*(3*c^2 + 3*c*d*x + d^2*x^2))*FresnelC[a + b*x] + 3*d*(b*c - a*d) 
*Pi*FresnelS[a + b*x] - 3*b^2*c^2*Pi*Sin[(Pi*(a + b*x)^2)/2] + 3*a*b*c*d*P 
i*Sin[(Pi*(a + b*x)^2)/2] - a^2*d^2*Pi*Sin[(Pi*(a + b*x)^2)/2] - 3*b^2*c*d 
*Pi*x*Sin[(Pi*(a + b*x)^2)/2] + a*b*d^2*Pi*x*Sin[(Pi*(a + b*x)^2)/2] - b^2 
*d^2*Pi*x^2*Sin[(Pi*(a + b*x)^2)/2])/(3*b^3*Pi^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6983, 3915, 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.

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

\(\Big \downarrow \) 6983

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

\(\Big \downarrow \) 3915

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((c + d*x)^3*FresnelC[a + b*x])/(3*d) - ((2*d^3*Cos[(Pi*(a + b*x)^2)/2])/P 
i^2 + (b*c - a*d)^3*FresnelC[a + b*x] - (3*d^2*(b*c - a*d)*FresnelS[a + b* 
x])/Pi + (3*d*(b*c - a*d)^2*Sin[(Pi*(a + b*x)^2)/2])/Pi + (3*d^2*(b*c - a* 
d)*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/Pi + (d^3*(a + b*x)^2*Sin[(Pi*(a + b 
*x)^2)/2])/Pi)/(3*b^3*d)
 

Defintions of rubi rules used

rule 2009

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

rule 3915

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_.)*((g_ 
.) + (h_.)*(x_))^(m_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat 
or[n], 1]}, Simp[k/f^(m + 1)   Subst[Int[ExpandIntegrand[(a + b*Cos[c + d*x 
^(k*n)])^p, x^(k - 1)*(f*g - e*h + h*x^k)^m, x], x], x, (e + f*x)^(1/k)], x 
]] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[p, 0] && IGtQ[m, 0]
 

rule 6983

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

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (a d -c b -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {-\frac {d^{3} \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}+\frac {3 \left (a d -c b \right ) d^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {3 \left (a d -c b \right ) d^{2} \operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {3 \left (a d -c b \right )^{2} d \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -c b \right )^{3} \operatorname {FresnelC}\left (b x +a \right )}{3 b^{2} d}}{b}\)	\(190\)
default	\(\frac {-\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (a d -c b -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {-\frac {d^{3} \left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}+\frac {3 \left (a d -c b \right ) d^{2} \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {3 \left (a d -c b \right ) d^{2} \operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {3 \left (a d -c b \right )^{2} d \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -c b \right )^{3} \operatorname {FresnelC}\left (b x +a \right )}{3 b^{2} d}}{b}\)	\(190\)
parts	\(\frac {\operatorname {FresnelC}\left (b x +a \right ) d^{2} x^{3}}{3}+\operatorname {FresnelC}\left (b x +a \right ) d c \,x^{2}+\operatorname {FresnelC}\left (b x +a \right ) c^{2} x +\frac {\operatorname {FresnelC}\left (b x +a \right ) c^{3}}{3 d}-\frac {b \left (\frac {d^{3} x^{2} \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {d^{3} a \left (\frac {x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {\sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b}-\frac {2 d^{3} \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }+\frac {3 d^{2} c x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {3 d^{2} c a \left (\frac {\sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {3 d^{2} c \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}+\frac {3 c^{2} d \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {3 c^{2} d \sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}+\frac {\sqrt {\pi }\, c^{3} \operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3 d}\)	\(620\)

Input:

int((d*x+c)^2*FresnelC(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/b*(-1/3*FresnelC(b*x+a)*(a*d-c*b-d*(b*x+a))^3/b^2/d+1/3/b^2/d*(-d^3/Pi*( 
b*x+a)^2*sin(1/2*Pi*(b*x+a)^2)-2*d^3/Pi^2*cos(1/2*Pi*(b*x+a)^2)+3*(a*d-b*c 
)*d^2/Pi*(b*x+a)*sin(1/2*Pi*(b*x+a)^2)-3*(a*d-b*c)*d^2/Pi*FresnelS(b*x+a)- 
3*(a*d-b*c)^2*d/Pi*sin(1/2*Pi*(b*x+a)^2)+(a*d-b*c)^3*FresnelC(b*x+a)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.28 \[ \int (c+d x)^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {\pi ^{2} {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b d^{2} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + 3 \, \pi {\left (b c d - a d^{2}\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi ^{2} b^{4} d^{2} x^{3} + 3 \, \pi ^{2} b^{4} c d x^{2} + 3 \, \pi ^{2} b^{4} c^{2} x\right )} \operatorname {C}\left (b x + a\right ) - {\left (\pi b^{3} d^{2} x^{2} + \pi {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x + \pi {\left (3 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )}\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{3 \, \pi ^{2} b^{4}} \] Input:

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

Output:

1/3*(pi^2*(3*a*b^2*c^2 - 3*a^2*b*c*d + a^3*d^2)*sqrt(b^2)*fresnel_cos(sqrt 
(b^2)*(b*x + a)/b) - 2*b*d^2*cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2) + 
 3*pi*(b*c*d - a*d^2)*sqrt(b^2)*fresnel_sin(sqrt(b^2)*(b*x + a)/b) + (pi^2 
*b^4*d^2*x^3 + 3*pi^2*b^4*c*d*x^2 + 3*pi^2*b^4*c^2*x)*fresnel_cos(b*x + a) 
 - (pi*b^3*d^2*x^2 + pi*(3*b^3*c*d - a*b^2*d^2)*x + pi*(3*b^3*c^2 - 3*a*b^ 
2*c*d + a^2*b*d^2))*sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2))/(pi^2*b^4 
)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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