3.1.22 \(\int S(a+b x) \, dx\) [22]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = {6553} \begin {gather*} \frac {(a+b x) S(a+b x)}{b}+\frac {\cos \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \end {gather*}

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 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*} \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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(36)=72\).
time = 0.02, size = 89, normalized size = 2.47 \begin {gather*} \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 S(a+b x)}{b}+x S(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 } \end {gather*}

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)

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Maple [A]
time = 0.38, size = 33, normalized size = 0.92

method	result	size
derivativedivides	\(\frac {\mathrm {S}\left (b x +a \right ) \left (b x +a \right )+\frac {\cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(33\)
default	\(\frac {\mathrm {S}\left (b x +a \right ) \left (b x +a \right )+\frac {\cos \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Maxima [A]
time = 0.27, size = 43, normalized size = 1.19 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]
time = 0.35, size = 45, normalized size = 1.25 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int S\left (a + b x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(fresnels(a + b*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnel_sin(b*x + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \mathrm {FresnelS}\left (a+b\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelS(a + b*x),x)

[Out]

int(FresnelS(a + b*x), x)

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