Integrand size = 10, antiderivative size = 148 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2}+\frac {a^3 \operatorname {FresnelC}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {a \operatorname {FresnelS}(a+b x)}{b^3 \pi }-\frac {a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }+\frac {a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^3 \pi }-\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi } \]
-2/3*cos(1/2*Pi*(b*x+a)^2)/b^3/Pi^2+1/3*a^3*FresnelC(b*x+a)/b^3+1/3*x^3*Fr esnelC(b*x+a)-a*FresnelS(b*x+a)/b^3/Pi-a^2*sin(1/2*Pi*(b*x+a)^2)/b^3/Pi+a* (b*x+a)*sin(1/2*Pi*(b*x+a)^2)/b^3/Pi-1/3*(b*x+a)^2*sin(1/2*Pi*(b*x+a)^2)/b ^3/Pi
Time = 0.21 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=-\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )-\pi ^2 \left (a^3+b^3 x^3\right ) \operatorname {FresnelC}(a+b x)+3 a \pi \operatorname {FresnelS}(a+b x)+a^2 \pi \sin \left (\frac {1}{2} \pi (a+b x)^2\right )-a b \pi x \sin \left (\frac {1}{2} \pi (a+b x)^2\right )+b^2 \pi x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{3 b^3 \pi ^2} \]
-1/3*(2*Cos[(Pi*(a + b*x)^2)/2] - Pi^2*(a^3 + b^3*x^3)*FresnelC[a + b*x] + 3*a*Pi*FresnelS[a + b*x] + a^2*Pi*Sin[(Pi*(a + b*x)^2)/2] - a*b*Pi*x*Sin[ (Pi*(a + b*x)^2)/2] + b^2*Pi*x^2*Sin[(Pi*(a + b*x)^2)/2])/(b^3*Pi^2)
Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
| \(\displaystyle \int x^2 \operatorname {FresnelC}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6983 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {1}{3} b \int x^3 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )dx\) |
| \(\Big \downarrow \) 3915 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {\int \left (-\cos \left (\frac {1}{2} \pi (a+b x)^2\right ) a^3+3 (a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) a^2-3 (a+b x)^2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right ) a+(a+b x)^3 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )\right )d(a+b x)}{3 b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 \operatorname {FresnelC}(a+b x)-\frac {a^3 (-\operatorname {FresnelC}(a+b x))+\frac {3 a^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {3 a \operatorname {FresnelS}(a+b x)}{\pi }-\frac {3 a (a+b x) \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {(a+b x)^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi }+\frac {2 \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi ^2}}{3 b^3}\) |
(x^3*FresnelC[a + b*x])/3 - ((2*Cos[(Pi*(a + b*x)^2)/2])/Pi^2 - a^3*Fresne lC[a + b*x] + (3*a*FresnelS[a + b*x])/Pi + (3*a^2*Sin[(Pi*(a + b*x)^2)/2]) /Pi - (3*a*(a + b*x)*Sin[(Pi*(a + b*x)^2)/2])/Pi + ((a + b*x)^2*Sin[(Pi*(a + b*x)^2)/2])/Pi)/(3*b^3)
3.2.35.3.1 Defintions of rubi rules used
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]
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]
Time = 0.53 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {a^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {a \,\operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {\left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(122\) |
| default | \(\frac {\frac {\operatorname {FresnelC}\left (b x +a \right ) b^{3} x^{3}}{3}+\frac {a^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {a^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {a \left (b x +a \right ) \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }-\frac {a \,\operatorname {FresnelS}\left (b x +a \right )}{\pi }-\frac {\left (b x +a \right )^{2} \sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi }-\frac {2 \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{3 \pi ^{2}}}{b^{3}}\) | \(122\) |
| parts | \(\frac {x^{3} \operatorname {FresnelC}\left (b x +a \right )}{3}-\frac {b \left (\frac {x^{2} \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {x \sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (\frac {\sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}-\frac {\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b}-\frac {2 \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }\right )}{3}\) | \(286\) |
1/b^3*(1/3*FresnelC(b*x+a)*b^3*x^3+1/3*a^3*FresnelC(b*x+a)-a^2/Pi*sin(1/2* Pi*(b*x+a)^2)+a/Pi*(b*x+a)*sin(1/2*Pi*(b*x+a)^2)-a/Pi*FresnelS(b*x+a)-1/3/ Pi*(b*x+a)^2*sin(1/2*Pi*(b*x+a)^2)-2/3/Pi^2*cos(1/2*Pi*(b*x+a)^2))
Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {\pi ^{2} b^{4} x^{3} \operatorname {C}\left (b x + a\right ) + \pi ^{2} a^{3} \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 3 \, \pi a \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - {\left (\pi b^{3} x^{2} - \pi a b^{2} x + \pi a^{2} b\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{3 \, \pi ^{2} b^{4}} \]
1/3*(pi^2*b^4*x^3*fresnel_cos(b*x + a) + pi^2*a^3*sqrt(b^2)*fresnel_cos(sq rt(b^2)*(b*x + a)/b) - 3*pi*a*sqrt(b^2)*fresnel_sin(sqrt(b^2)*(b*x + a)/b) - 2*b*cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2) - (pi*b^3*x^2 - pi*a*b^ 2*x + pi*a^2*b)*sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2))/(pi^2*b^4)
\[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int x^{2} C\left (a + b x\right )\, dx \]
Result contains complex when optimal does not.
Time = 0.81 (sec) , antiderivative size = 423, normalized size of antiderivative = 2.86 \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\frac {1}{3} \, x^{3} \operatorname {C}\left (b x + a\right ) - \frac {{\left (12 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{3} + 4 \, {\left (3 \, {\left (-i \, \pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + i \, \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} + 2 \, \Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + 2 \, \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} b x + 8 \, a {\left (\Gamma \left (2, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \Gamma \left (2, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} + \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (\left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} - \left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{3} + 6 \, {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )} a\right )}\right )} b}{24 \, {\left (\pi ^{2} b^{5} x + \pi ^{2} a b^{4}\right )}} \]
1/3*x^3*fresnel_cos(b*x + a) - 1/24*(12*(-I*pi*e^(1/2*I*pi*b^2*x^2 + I*pi* a*b*x + 1/2*I*pi*a^2) + I*pi*e^(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi* a^2))*a^3 + 4*(3*(-I*pi*e^(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + I*pi*e^(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*a^2 + 2*gamma(2, 1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + 2*gamma(2, -1/2*I*pi*b^2*x ^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*b*x + 8*a*(gamma(2, 1/2*I*pi*b^2*x^2 + I* pi*a*b*x + 1/2*I*pi*a^2) + gamma(2, -1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I *pi*a^2)) + sqrt(2*pi*b^2*x^2 + 4*pi*a*b*x + 2*pi*a^2)*(((I - 1)*sqrt(2)*p i^(3/2)*(erf(sqrt(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2)) - 1) - (I + 1)*sqrt(2)*pi^(3/2)*(erf(sqrt(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi *a^2)) - 1))*a^3 + 6*(-(I + 1)*sqrt(2)*gamma(3/2, 1/2*I*pi*b^2*x^2 + I*pi* a*b*x + 1/2*I*pi*a^2) + (I - 1)*sqrt(2)*gamma(3/2, -1/2*I*pi*b^2*x^2 - I*p i*a*b*x - 1/2*I*pi*a^2))*a))*b/(pi^2*b^5*x + pi^2*a*b^4)
\[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int { x^{2} \operatorname {C}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x^2 \operatorname {FresnelC}(a+b x) \, dx=\int x^2\,\mathrm {FresnelC}\left (a+b\,x\right ) \,d x \]