\(\int \operatorname {FresnelS}(a+b x) \, dx\) [28]

Optimal result

Integrand size = 6, antiderivative size = 36 \[ \int \operatorname {FresnelS}(a+b x) \, dx=\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi }+\frac {(a+b x) \operatorname {FresnelS}(a+b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6553} \[ \int \operatorname {FresnelS}(a+b x) \, dx=\frac {(a+b x) \operatorname {FresnelS}(a+b x)}{b}+\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \]

[In]

Int[FresnelS[a + b*x],x]

[Out]

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

Rule 6553

Int[FresnelS[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(FresnelS[a + b*x]/b), x] + Simp[Cos[(Pi/2)*(a +
b*x)^2]/(b*Pi), x] /; FreeQ[{a, b}, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(36)=72\).

Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.47 \[ \int \operatorname {FresnelS}(a+b x) \, dx=\frac {\cos \left (\frac {a^2 \pi }{2}\right ) \cos \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right )}{b \pi }+\frac {a \operatorname {FresnelS}(a+b x)}{b}+x \operatorname {FresnelS}(a+b x)-\frac {\sin \left (\frac {a^2 \pi }{2}\right ) \sin \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right )}{b \pi } \]

[In]

Integrate[FresnelS[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (b x +a \right )+\frac {\cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(33\)
default	\(\frac {\operatorname {FresnelS}\left (b x +a \right ) \left (b x +a \right )+\frac {\cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(33\)
parts	\(x \,\operatorname {FresnelS}\left (b x +a \right )-b \left (-\frac {\cos \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelS}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )\)	\(86\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.25 \[ \int \operatorname {FresnelS}(a+b x) \, dx=\frac {{\left (\pi b x + \pi a\right )} \operatorname {S}\left (b x + a\right ) + \cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi b} \]

[In]

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

[Out]

((pi*b*x + pi*a)*fresnel_sin(b*x + a) + cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2))/(pi*b)

Sympy [F]

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

[In]

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

[Out]

Integral(fresnels(a + b*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19 \[ \int \operatorname {FresnelS}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {S}\left (b x + a\right ) + \frac {\cos \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi }}{b} \]

[In]

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

[Out]

((b*x + a)*fresnel_sin(b*x + a) + cos(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2)/pi)/b

Giac [F]

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

[In]

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

[Out]

integrate(fresnel_sin(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(FresnelS(a + b*x),x)

[Out]

int(FresnelS(a + b*x), x)