Integrand size = 14, antiderivative size = 193 \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\frac {(b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d (b c-a d) (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {d^2 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi }-\frac {d (b c-a d) \operatorname {FresnelC}(a+b x)}{b^3 \pi }-\frac {(b c-a d)^3 \operatorname {FresnelS}(a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \operatorname {FresnelS}(a+b x)}{3 d}-\frac {2 d^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]
(-a*d+b*c)^2*cos(1/2*Pi*(b*x+a)^2)/b^3/Pi+d*(-a*d+b*c)*(b*x+a)*cos(1/2*Pi* (b*x+a)^2)/b^3/Pi+1/3*d^2*(b*x+a)^2*cos(1/2*Pi*(b*x+a)^2)/b^3/Pi-d*(-a*d+b *c)*FresnelC(b*x+a)/b^3/Pi-1/3*(-a*d+b*c)^3*FresnelS(b*x+a)/b^3/d+1/3*(d*x +c)^3*FresnelS(b*x+a)/d-2/3*d^2*sin(1/2*Pi*(b*x+a)^2)/b^3/Pi^2
Time = 0.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.22 \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\frac {3 b^2 c^2 \pi \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-3 a b c d \pi \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+a^2 d^2 \pi \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+3 b^2 c d \pi x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-a b d^2 \pi x \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+b^2 d^2 \pi x^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )+3 d (-b c+a d) \pi \operatorname {FresnelC}(a+b x)+\pi ^2 \left (3 a b^2 c^2-3 a^2 b c d+a^3 d^2+b^3 x \left (3 c^2+3 c d x+d^2 x^2\right )\right ) \operatorname {FresnelS}(a+b x)-2 d^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]
(3*b^2*c^2*Pi*Cos[(Pi*(a + b*x)^2)/2] - 3*a*b*c*d*Pi*Cos[(Pi*(a + b*x)^2)/ 2] + a^2*d^2*Pi*Cos[(Pi*(a + b*x)^2)/2] + 3*b^2*c*d*Pi*x*Cos[(Pi*(a + b*x) ^2)/2] - a*b*d^2*Pi*x*Cos[(Pi*(a + b*x)^2)/2] + b^2*d^2*Pi*x^2*Cos[(Pi*(a + b*x)^2)/2] + 3*d*(-(b*c) + a*d)*Pi*FresnelC[a + b*x] + 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))*FresnelS[a + b*x] - 2*d^2*Sin[(Pi*(a + b*x)^2)/2])/(3*b^3*Pi^2)
Time = 0.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6982, 3914, 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.
| \(\displaystyle \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6982 |
| \(\displaystyle \frac {(c+d x)^3 \operatorname {FresnelS}(a+b x)}{3 d}-\frac {b \int (c+d x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )dx}{3 d}\) |
| \(\Big \downarrow \) 3914 |
| \(\displaystyle \frac {(c+d x)^3 \operatorname {FresnelS}(a+b x)}{3 d}-\frac {\int \left (\sin \left (\frac {1}{2} \pi (a+b x)^2\right ) (b c-a d)^3+3 d (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) (b c-a d)^2+3 d^2 (a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) (b c-a d)+d^3 (a+b x)^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )\right )d(a+b x)}{3 b^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(c+d x)^3 \operatorname {FresnelS}(a+b x)}{3 d}-\frac {\frac {3 d^2 (b c-a d) \operatorname {FresnelC}(a+b x)}{\pi }-\frac {3 d^2 (a+b x) (b c-a d) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+(b c-a d)^3 \operatorname {FresnelS}(a+b x)-\frac {3 d (b c-a d)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {2 d^3 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi ^2}-\frac {d^3 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }}{3 b^3 d}\) |
((c + d*x)^3*FresnelS[a + b*x])/(3*d) - ((-3*d*(b*c - a*d)^2*Cos[(Pi*(a + b*x)^2)/2])/Pi - (3*d^2*(b*c - a*d)*(a + b*x)*Cos[(Pi*(a + b*x)^2)/2])/Pi - (d^3*(a + b*x)^2*Cos[(Pi*(a + b*x)^2)/2])/Pi + (3*d^2*(b*c - a*d)*Fresne lC[a + b*x])/Pi + (b*c - a*d)^3*FresnelS[a + b*x] + (2*d^3*Sin[(Pi*(a + b* x)^2)/2])/Pi^2)/(3*b^3*d)
3.1.20.3.1 Defintions of rubi rules used
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Module[{k = If[FractionQ[n], Denominat or[n], 1]}, Simp[k/f^(m + 1) Subst[Int[ExpandIntegrand[(a + b*Sin[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]
Int[FresnelS[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> S imp[(c + d*x)^(m + 1)*(FresnelS[a + b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(c + d*x)^(m + 1)*Sin[(Pi/2)*(a + b*x)^2], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
Time = 0.91 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(\frac {-\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {\frac {d^{3} \left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}-\frac {3 \left (a d -b c \right ) d^{2} \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {3 \left (a d -b c \right ) d^{2} \operatorname {FresnelC}\left (b x +a \right )}{\pi }+\frac {3 \left (a d -b c \right )^{2} d \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -b c \right )^{3} \operatorname {FresnelS}\left (b x +a \right )}{3 b^{2} d}}{b}\) | \(189\) |
| default | \(\frac {-\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (a d -b c -d \left (b x +a \right )\right )^{3}}{3 b^{2} d}+\frac {\frac {d^{3} \left (b x +a \right )^{2} \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {2 d^{3} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi ^{2}}-\frac {3 \left (a d -b c \right ) d^{2} \left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {3 \left (a d -b c \right ) d^{2} \operatorname {FresnelC}\left (b x +a \right )}{\pi }+\frac {3 \left (a d -b c \right )^{2} d \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\left (a d -b c \right )^{3} \operatorname {FresnelS}\left (b x +a \right )}{3 b^{2} d}}{b}\) | \(189\) |
| parts | \(\frac {\operatorname {FresnelS}\left (b x +a \right ) d^{2} x^{3}}{3}+\operatorname {FresnelS}\left (b x +a \right ) d c \,x^{2}+\operatorname {FresnelS}\left (b x +a \right ) c^{2} x +\frac {\operatorname {FresnelS}\left (b x +a \right ) c^{3}}{3 d}-\frac {b \left (-\frac {d^{3} x^{2} \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 {d^{3} a \left (-\frac {x \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 {a \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}+\frac {\operatorname {FresnelC}\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 {\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^{2} \pi }-\frac {3 c \,d^{2} x \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 {3 c \,d^{2} a \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}+\frac {3 c \,d^{2} \operatorname {FresnelC}\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 \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 {3 c^{2} d \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 }}+\frac {\sqrt {\pi }\, c^{3} \operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{\sqrt {b^{2} \pi }}\right )}{3 d}\) | \(622\) |
1/b*(-1/3*FresnelS(b*x+a)*(a*d-b*c-d*(b*x+a))^3/b^2/d+1/3/b^2/d*(d^3/Pi*(b *x+a)^2*cos(1/2*Pi*(b*x+a)^2)-2*d^3/Pi^2*sin(1/2*Pi*(b*x+a)^2)-3*(a*d-b*c) *d^2/Pi*(b*x+a)*cos(1/2*Pi*(b*x+a)^2)+3*(a*d-b*c)*d^2/Pi*FresnelC(b*x+a)+3 *(a*d-b*c)^2*d/Pi*cos(1/2*Pi*(b*x+a)^2)+(a*d-b*c)^3*FresnelS(b*x+a)))
Time = 0.27 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.28 \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\frac {\pi ^{2} {\left (3 \, a b^{2} c^{2} - 3 \, a^{2} b c d + a^{3} d^{2}\right )} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b d^{2} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - 3 \, \pi {\left (b c d - a d^{2}\right )} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (\pi b^{3} d^{2} x^{2} + \pi {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x + \pi {\left (3 \, b^{3} c^{2} - 3 \, a b^{2} c d + a^{2} b d^{2}\right )}\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) + {\left (\pi ^{2} b^{4} d^{2} x^{3} + 3 \, \pi ^{2} b^{4} c d x^{2} + 3 \, \pi ^{2} b^{4} c^{2} x\right )} \operatorname {S}\left (b x + a\right )}{3 \, \pi ^{2} b^{4}} \]
1/3*(pi^2*(3*a*b^2*c^2 - 3*a^2*b*c*d + a^3*d^2)*sqrt(b^2)*fresnel_sin(sqrt (b^2)*(b*x + a)/b) - 2*b*d^2*sin(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_cos(sqrt(b^2)*(b*x + a)/b) + (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))*cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2) + (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_sin(b*x + a))/(pi^2*b^4 )
\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\int \left (c + d x\right )^{2} S\left (a + b x\right )\, dx \]
\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {S}\left (b x + a\right ) \,d x } \]
\[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \operatorname {S}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int (c+d x)^2 \operatorname {FresnelS}(a+b x) \, dx=\int \mathrm {FresnelS}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2 \,d x \]