3.22 \(\int S(a+b x) \, dx\)

Optimal. Leaf size=36 \[ \frac{(a+b x) S(a+b x)}{b}+\frac{\cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0063385, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6418} \[ \frac{(a+b x) S(a+b x)}{b}+\frac{\cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]

Antiderivative was successfully verified.

[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 6418

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

Rubi steps

\begin{align*} \int S(a+b x) \, dx &=\frac{\cos \left (\frac{1}{2} \pi (a+b x)^2\right )}{b \pi }+\frac{(a+b x) S(a+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 0.0286256, size = 89, normalized size = 2.47 \[ -\frac{\sin \left (\frac{\pi a^2}{2}\right ) \sin \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+\frac{\cos \left (\frac{\pi a^2}{2}\right ) \cos \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x S(a+b x)+\frac{a S(a+b x)}{b} \]

Antiderivative was successfully verified.

[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]  time = 0.046, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ( \left ( bx+a \right ){\it FresnelS} \left ( bx+a \right ) +{\frac{1}{\pi }\cos \left ({\frac{\pi \, \left ( bx+a \right ) ^{2}}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(b*x+a),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x + a), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\rm fresnels}\left (b x + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnels(b*x + a), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int S\left (a + b x\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(fresnels(a + b*x), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm fresnels}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnels(b*x + a), x)