\(\int x \operatorname {FresnelS}(a+b x) \, dx\) [27]

Optimal result

Integrand size = 8, antiderivative size = 96 \[ \int x \operatorname {FresnelS}(a+b x) \, dx=-\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {\operatorname {FresnelC}(a+b x)}{2 b^2 \pi }-\frac {a^2 \operatorname {FresnelS}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \operatorname {FresnelS}(a+b x) \]

[Out]

-a*cos(1/2*Pi*(b*x+a)^2)/b^2/Pi+1/2*(b*x+a)*cos(1/2*Pi*(b*x+a)^2)/b^2/Pi-1/2*FresnelC(b*x+a)/b^2/Pi-1/2*a^2*Fr
esnelS(b*x+a)/b^2+1/2*x^2*FresnelS(b*x+a)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6563, 3514, 3432, 3460, 2718, 3466, 3433} \[ \int x \operatorname {FresnelS}(a+b x) \, dx=-\frac {a^2 \operatorname {FresnelS}(a+b x)}{2 b^2}-\frac {\operatorname {FresnelC}(a+b x)}{2 \pi b^2}-\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b^2}+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 \pi b^2}+\frac {1}{2} x^2 \operatorname {FresnelS}(a+b x) \]

[In]

Int[x*FresnelS[a + b*x],x]

[Out]

-((a*Cos[(Pi*(a + b*x)^2)/2])/(b^2*Pi)) + ((a + b*x)*Cos[(Pi*(a + b*x)^2)/2])/(2*b^2*Pi) - FresnelC[a + b*x]/(
2*b^2*Pi) - (a^2*FresnelS[a + b*x])/(2*b^2) + (x^2*FresnelS[a + b*x])/2

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

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]

Rule 3460

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3466

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] + Dist[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]

Rule 3514

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Module[{k = If[FractionQ[n], Denominator[n], 1]}, Dist[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]

Rule 6563

Int[FresnelS[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(FresnelS[a +
b*x]/(d*(m + 1))), x] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \operatorname {FresnelS}(a+b x)-\frac {1}{2} b \int x^2 \sin \left (\frac {1}{2} \pi (a+b x)^2\right ) \, dx \\ & = \frac {1}{2} x^2 \operatorname {FresnelS}(a+b x)-\frac {\text {Subst}\left (\int \left (a^2 \sin \left (\frac {\pi x^2}{2}\right )-2 a x \sin \left (\frac {\pi x^2}{2}\right )+x^2 \sin \left (\frac {\pi x^2}{2}\right )\right ) \, dx,x,a+b x\right )}{2 b^2} \\ & = \frac {1}{2} x^2 \operatorname {FresnelS}(a+b x)-\frac {\text {Subst}\left (\int x^2 \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2}+\frac {a \text {Subst}\left (\int x \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{b^2}-\frac {a^2 \text {Subst}\left (\int \sin \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2} \\ & = \frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {a^2 \operatorname {FresnelS}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \operatorname {FresnelS}(a+b x)+\frac {a \text {Subst}\left (\int \sin \left (\frac {\pi x}{2}\right ) \, dx,x,(a+b x)^2\right )}{2 b^2}-\frac {\text {Subst}\left (\int \cos \left (\frac {\pi x^2}{2}\right ) \, dx,x,a+b x\right )}{2 b^2 \pi } \\ & = -\frac {a \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b^2 \pi }+\frac {(a+b x) \cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{2 b^2 \pi }-\frac {\operatorname {FresnelC}(a+b x)}{2 b^2 \pi }-\frac {a^2 \operatorname {FresnelS}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \operatorname {FresnelS}(a+b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.53 \[ \int x \operatorname {FresnelS}(a+b x) \, dx=-\frac {\operatorname {FresnelC}(a+b x)+(a-b x) \left (\cos \left (\frac {1}{2} \pi (a+b x)^2\right )+\pi (a+b x) \operatorname {FresnelS}(a+b x)\right )}{2 b^2 \pi } \]

[In]

Integrate[x*FresnelS[a + b*x],x]

[Out]

-1/2*(FresnelC[a + b*x] + (a - b*x)*(Cos[(Pi*(a + b*x)^2)/2] + Pi*(a + b*x)*FresnelS[a + b*x]))/(b^2*Pi)

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-\frac {a \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {\left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 \pi }-\frac {\operatorname {FresnelC}\left (b x +a \right )}{2 \pi }}{b^{2}}\)	\(80\)
default	\(\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-\frac {a \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }+\frac {\left (b x +a \right ) \cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{2 \pi }-\frac {\operatorname {FresnelC}\left (b x +a \right )}{2 \pi }}{b^{2}}\)	\(80\)
parts	\(\frac {x^{2} \operatorname {FresnelS}\left (b x +a \right )}{2}-\frac {b \left (-\frac {x \cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {a \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )}{b}+\frac {\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b^{2} \sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{2}\)	\(167\)

[In]

int(x*FresnelS(b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int x \operatorname {FresnelS}(a+b x) \, dx=\frac {\pi b^{3} x^{2} \operatorname {S}\left (b x + a\right ) - \pi a^{2} \sqrt {b^{2}} \operatorname {S}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (b^{2} x - a b\right )} \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right ) - \sqrt {b^{2}} \operatorname {C}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right )}{2 \, \pi b^{3}} \]

[In]

integrate(x*fresnel_sin(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate(x*fresnels(b*x+a),x)

[Out]

Integral(x*fresnels(a + b*x), x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.20 \[ \int x \operatorname {FresnelS}(a+b x) \, dx=\frac {1}{2} \, x^{2} \operatorname {S}\left (b x + a\right ) - \frac {{\left (8 \, {\left (\pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a b x + 8 \, {\left (\pi e^{\left (\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right )} + \pi e^{\left (-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )}\right )} a^{2} - \sqrt {2 \, \pi b^{2} x^{2} + 4 \, \pi a b x + 2 \, \pi a^{2}} {\left ({\left (-\left (i + 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )} + \left (i - 1\right ) \, \sqrt {2} \pi ^{\frac {3}{2}} {\left (\operatorname {erf}\left (\sqrt {-\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}}\right ) - 1\right )}\right )} a^{2} - \left (2 i - 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, \frac {1}{2} i \, \pi b^{2} x^{2} + i \, \pi a b x + \frac {1}{2} i \, \pi a^{2}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \Gamma \left (\frac {3}{2}, -\frac {1}{2} i \, \pi b^{2} x^{2} - i \, \pi a b x - \frac {1}{2} i \, \pi a^{2}\right )\right )}\right )} b}{16 \, {\left (\pi ^{2} b^{4} x + \pi ^{2} a b^{3}\right )}} \]

[In]

integrate(x*fresnel_sin(b*x+a),x, algorithm="maxima")

[Out]

1/2*x^2*fresnel_sin(b*x + a) - 1/16*(8*(pi*e^(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + pi*e^(-1/2*I*pi*
b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*a*b*x + 8*(pi*e^(1/2*I*pi*b^2*x^2 + I*pi*a*b*x + 1/2*I*pi*a^2) + pi*e^(-
1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2))*a^2 - sqrt(2*pi*b^2*x^2 + 4*pi*a*b*x + 2*pi*a^2)*((-(I + 1)*sqr
t(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) + (I - 1)*sqrt(2)*pi^(3/2)*(erf(sq
rt(-1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2)) - 1))*a^2 - (2*I - 2)*sqrt(2)*gamma(3/2, 1/2*I*pi*b^2*x^2 +
I*pi*a*b*x + 1/2*I*pi*a^2) + (2*I + 2)*sqrt(2)*gamma(3/2, -1/2*I*pi*b^2*x^2 - I*pi*a*b*x - 1/2*I*pi*a^2)))*b/
(pi^2*b^4*x + pi^2*a*b^3)

Giac [F]

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

[In]

integrate(x*fresnel_sin(b*x+a),x, algorithm="giac")

[Out]

integrate(x*fresnel_sin(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(x*FresnelS(a + b*x),x)

[Out]

int(x*FresnelS(a + b*x), x)