3.137 \(\int \text{FresnelC}(a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac{(a+b x) \text{FresnelC}(a+b x)}{b}-\frac{\sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]

[Out]

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

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Rubi [A]  time = 0.0067759, antiderivative size = 37, 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 = {6419} \[ \frac{(a+b x) \text{FresnelC}(a+b x)}{b}-\frac{\sin \left (\frac{1}{2} \pi (a+b x)^2\right )}{\pi b} \]

Antiderivative was successfully verified.

[In]

Int[FresnelC[a + b*x],x]

[Out]

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

Rule 6419

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

Rubi steps

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

Mathematica [B]  time = 0.0290167, size = 90, normalized size = 2.43 \[ -\frac{\sin \left (\frac{\pi a^2}{2}\right ) \cos \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}-\frac{\cos \left (\frac{\pi a^2}{2}\right ) \sin \left (\pi a b x+\frac{1}{2} \pi b^2 x^2\right )}{\pi b}+x \text{FresnelC}(a+b x)+\frac{a \text{FresnelC}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[a + b*x],x]

[Out]

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

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Maple [A]  time = 0.049, size = 34, normalized size = 0.9 \begin{align*}{\frac{1}{b} \left ({\it FresnelC} \left ( bx+a \right ) \left ( bx+a \right ) -{\frac{1}{\pi }\sin \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(FresnelC(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnelc(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(fresnelc(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(fresnelc(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(fresnelc(b*x + a), x)