Integrand size = 8, antiderivative size = 74 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi } \] Output:
-3/4*x*cos(1/2*b^2*Pi*x^2)/b^3/Pi^2+3/4*FresnelC(b*x)/b^4/Pi^2+1/4*x^4*Fre snelC(b*x)-1/4*x^3*sin(1/2*b^2*Pi*x^2)/b/Pi
Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {3 x \cos \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b^3 \pi ^2}+\frac {3 \operatorname {FresnelC}(b x)}{4 b^4 \pi ^2}+\frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {x^3 \sin \left (\frac {1}{2} b^2 \pi x^2\right )}{4 b \pi } \] Input:
Integrate[x^3*FresnelC[b*x],x]
Output:
(-3*x*Cos[(b^2*Pi*x^2)/2])/(4*b^3*Pi^2) + (3*FresnelC[b*x])/(4*b^4*Pi^2) + (x^4*FresnelC[b*x])/4 - (x^3*Sin[(b^2*Pi*x^2)/2])/(4*b*Pi)
Time = 0.33 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6981, 3867, 3866, 3833}
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.
\(\displaystyle \int x^3 \operatorname {FresnelC}(b x) \, dx\) |
\(\Big \downarrow \) 6981 |
\(\displaystyle \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {1}{4} b \int x^4 \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx\) |
\(\Big \downarrow \) 3867 |
\(\displaystyle \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {1}{4} b \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \int x^2 \sin \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3866 |
\(\displaystyle \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {1}{4} b \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \left (\frac {\int \cos \left (\frac {1}{2} b^2 \pi x^2\right )dx}{\pi b^2}-\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {1}{4} x^4 \operatorname {FresnelC}(b x)-\frac {1}{4} b \left (\frac {x^3 \sin \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}-\frac {3 \left (\frac {\operatorname {FresnelC}(b x)}{\pi b^3}-\frac {x \cos \left (\frac {1}{2} \pi b^2 x^2\right )}{\pi b^2}\right )}{\pi b^2}\right )\) |
Input:
Int[x^3*FresnelC[b*x],x]
Output:
(x^4*FresnelC[b*x])/4 - (b*((-3*(-((x*Cos[(b^2*Pi*x^2)/2])/(b^2*Pi)) + Fre snelC[b*x]/(b^3*Pi)))/(b^2*Pi) + (x^3*Sin[(b^2*Pi*x^2)/2])/(b^2*Pi)))/4
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Simp[(-e^ (n - 1))*(e*x)^(m - n + 1)*(Cos[c + d*x^n]/(d*n)), x] + Simp[e^n*((m - n + 1)/(d*n)) Int[(e*x)^(m - n)*Cos[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(Sin[c + d*x^n]/(d*n)), x] - Simp[e^n*((m - n + 1)/ (d*n)) Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n, 0] && LtQ[n, m + 1]
Int[FresnelC[(b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1 )*(FresnelC[b*x]/(d*(m + 1))), x] - Simp[b/(d*(m + 1)) Int[(d*x)^(m + 1)* Cos[(Pi/2)*b^2*x^2], x], x] /; FreeQ[{b, d, m}, x] && NeQ[m, -1]
Time = 0.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(\frac {-\frac {3 \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right ) b x}{4}-\frac {\pi \,b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4}+\frac {\left (5 x^{4} \pi ^{2} b^{4}+15\right ) \operatorname {FresnelC}\left (b x \right )}{20}}{b^{4} \pi ^{2}}\) | \(62\) |
derivativedivides | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{4} x^{4}}{4}-\frac {b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {3 \,\operatorname {FresnelC}\left (b x \right )}{4 \pi }}{\pi }}{b^{4}}\) | \(70\) |
default | \(\frac {\frac {\operatorname {FresnelC}\left (b x \right ) b^{4} x^{4}}{4}-\frac {b^{3} x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {-\frac {3 b x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{4 \pi }+\frac {3 \,\operatorname {FresnelC}\left (b x \right )}{4 \pi }}{\pi }}{b^{4}}\) | \(70\) |
parts | \(\frac {x^{4} \operatorname {FresnelC}\left (b x \right )}{4}-\frac {b \left (\frac {x^{3} \sin \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }-\frac {3 \left (-\frac {x \cos \left (\frac {b^{2} \pi \,x^{2}}{2}\right )}{b^{2} \pi }+\frac {\operatorname {FresnelC}\left (\frac {\sqrt {\pi }\, b^{2} x}{\sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \pi }\right )}{4}\) | \(93\) |
Input:
int(x^3*FresnelC(b*x),x,method=_RETURNVERBOSE)
Output:
2/Pi^2/b^4*(-3/8*cos(1/2*b^2*Pi*x^2)*b*x-1/8*Pi*b^3*x^3*sin(1/2*b^2*Pi*x^2 )+1/40*(5*Pi^2*b^4*x^4+15)*FresnelC(b*x))
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.80 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=-\frac {\pi b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 3 \, b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) - {\left (\pi ^{2} b^{4} x^{4} + 3\right )} \operatorname {C}\left (b x\right )}{4 \, \pi ^{2} b^{4}} \] Input:
integrate(x^3*fresnel_cos(b*x),x, algorithm="fricas")
Output:
-1/4*(pi*b^3*x^3*sin(1/2*pi*b^2*x^2) + 3*b*x*cos(1/2*pi*b^2*x^2) - (pi^2*b ^4*x^4 + 3)*fresnel_cos(b*x))/(pi^2*b^4)
Time = 0.65 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\frac {5 x^{4} C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \Gamma \left (\frac {9}{4}\right )} - \frac {5 x^{3} \sin {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi b \Gamma \left (\frac {9}{4}\right )} - \frac {15 x \cos {\left (\frac {\pi b^{2} x^{2}}{2} \right )} \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{3} \Gamma \left (\frac {9}{4}\right )} + \frac {15 C\left (b x\right ) \Gamma \left (\frac {1}{4}\right )}{64 \pi ^{2} b^{4} \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate(x**3*fresnelc(b*x),x)
Output:
5*x**4*fresnelc(b*x)*gamma(1/4)/(64*gamma(9/4)) - 5*x**3*sin(pi*b**2*x**2/ 2)*gamma(1/4)/(64*pi*b*gamma(9/4)) - 15*x*cos(pi*b**2*x**2/2)*gamma(1/4)/( 64*pi**2*b**3*gamma(9/4)) + 15*fresnelc(b*x)*gamma(1/4)/(64*pi**2*b**4*gam ma(9/4))
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.27 \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\frac {1}{4} \, x^{4} \operatorname {C}\left (b x\right ) - \frac {\sqrt {\frac {1}{2}} {\left (4 \, \sqrt {\frac {1}{2}} \pi ^{2} b^{3} x^{3} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + 12 \, \sqrt {\frac {1}{2}} \pi b x \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \left (3 i - 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi } b x\right ) - \left (3 i + 3\right ) \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \pi \operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi } b x\right )\right )}}{8 \, \pi ^{3} b^{4}} \] Input:
integrate(x^3*fresnel_cos(b*x),x, algorithm="maxima")
Output:
1/4*x^4*fresnel_cos(b*x) - 1/8*sqrt(1/2)*(4*sqrt(1/2)*pi^2*b^3*x^3*sin(1/2 *pi*b^2*x^2) + 12*sqrt(1/2)*pi*b*x*cos(1/2*pi*b^2*x^2) + (3*I - 3)*(1/4)^( 1/4)*pi*erf(sqrt(1/2*I*pi)*b*x) - (3*I + 3)*(1/4)^(1/4)*pi*erf(sqrt(-1/2*I *pi)*b*x))/(pi^3*b^4)
\[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\int { x^{3} \operatorname {C}\left (b x\right ) \,d x } \] Input:
integrate(x^3*fresnel_cos(b*x),x, algorithm="giac")
Output:
integrate(x^3*fresnel_cos(b*x), x)
Timed out. \[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\int x^3\,\mathrm {FresnelC}\left (b\,x\right ) \,d x \] Input:
int(x^3*FresnelC(b*x),x)
Output:
int(x^3*FresnelC(b*x), x)
\[ \int x^3 \operatorname {FresnelC}(b x) \, dx=\int x^{3} \mathrm {FresnelC}\left (b x \right )d x \] Input:
int(x^3*FresnelC(b*x),x)
Output:
int(x^3*FresnelC(b*x),x)