\(\int \operatorname {FresnelC}(a+b x) \, dx\) [137]

Optimal result

Integrand size = 6, antiderivative size = 37 \[ \int \operatorname {FresnelC}(a+b x) \, dx=\frac {(a+b x) \operatorname {FresnelC}(a+b x)}{b}-\frac {\sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{b \pi } \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, 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 = {6554} \[ \int \operatorname {FresnelC}(a+b x) \, dx=\frac {(a+b x) \operatorname {FresnelC}(a+b x)}{b}-\frac {\sin \left (\frac {1}{2} \pi (a+b x)^2\right )}{\pi b} \]

[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 6554

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(90\) vs. \(2(37)=74\).

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.43 \[ \int \operatorname {FresnelC}(a+b x) \, dx=\frac {a \operatorname {FresnelC}(a+b x)}{b}+x \operatorname {FresnelC}(a+b x)-\frac {\cos \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right ) \sin \left (\frac {a^2 \pi }{2}\right )}{b \pi }-\frac {\cos \left (\frac {a^2 \pi }{2}\right ) \sin \left (a b \pi x+\frac {1}{2} b^2 \pi x^2\right )}{b \pi } \]

[In]

Integrate[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)

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (b x +a \right )-\frac {\sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(34\)
default	\(\frac {\operatorname {FresnelC}\left (b x +a \right ) \left (b x +a \right )-\frac {\sin \left (\frac {\pi \left (b x +a \right )^{2}}{2}\right )}{\pi }}{b}\)	\(34\)
parts	\(x \,\operatorname {FresnelC}\left (b x +a \right )-b \left (\frac {\sin \left (\frac {1}{2} b^{2} \pi \,x^{2}+\pi a b x +\frac {1}{2} \pi \,a^{2}\right )}{b^{2} \pi }-\frac {\sqrt {\pi }\, a \,\operatorname {FresnelC}\left (\frac {b^{2} \pi x +\pi b a}{\sqrt {\pi }\, \sqrt {b^{2} \pi }}\right )}{b \sqrt {b^{2} \pi }}\right )\)	\(85\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int \operatorname {FresnelC}(a+b x) \, dx=\frac {{\left (\pi b x + \pi a\right )} \operatorname {C}\left (b x + a\right ) - \sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi b} \]

[In]

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

[Out]

((pi*b*x + pi*a)*fresnel_cos(b*x + a) - sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2))/(pi*b)

Sympy [F]

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

[In]

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

[Out]

Integral(fresnelc(a + b*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.19 \[ \int \operatorname {FresnelC}(a+b x) \, dx=\frac {{\left (b x + a\right )} \operatorname {C}\left (b x + a\right ) - \frac {\sin \left (\frac {1}{2} \, \pi b^{2} x^{2} + \pi a b x + \frac {1}{2} \, \pi a^{2}\right )}{\pi }}{b} \]

[In]

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

[Out]

((b*x + a)*fresnel_cos(b*x + a) - sin(1/2*pi*b^2*x^2 + pi*a*b*x + 1/2*pi*a^2)/pi)/b

Giac [F]

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

[In]

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

[Out]

integrate(fresnel_cos(b*x + a), x)

Mupad [F(-1)]

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

[In]

int(FresnelC(a + b*x),x)

[Out]

int(FresnelC(a + b*x), x)