3.87 \(\int \cos ((a+b x)^2) \, dx\)

Optimal. Leaf size=29 \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]

[Out]

(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b

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Rubi [A]  time = 0.0058369, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3352} \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[(a + b*x)^2],x]

[Out]

(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [A]  time = 0.0080303, size = 29, normalized size = 1. \[ \frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} (a+b x)\right )}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[(a + b*x)^2],x]

[Out]

(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*(a + b*x)])/b

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Maple [A]  time = 0.028, size = 36, normalized size = 1.2 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{2}{\it FresnelC} \left ({\frac{\sqrt{2} \left ({b}^{2}x+ab \right ) }{\sqrt{\pi }}{\frac{1}{\sqrt{{b}^{2}}}}} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos((b*x+a)^2),x)

[Out]

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

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Maxima [C]  time = 1.76279, size = 113, normalized size = 3.9 \begin{align*} -\frac{\sqrt{\pi }{\left (\left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (-1\right )^{\frac{3}{4}}{\left (i \, b x + i \, a\right )}\right ) + \left (i - 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, b x + 2 i \, a\right )}\right ) - \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (-\left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2}{\left (2 i \, b x + 2 i \, a\right )}\right ) + \left (i + 1\right ) \, \sqrt{2} \operatorname{erf}\left (\frac{i \, b x + i \, a}{\sqrt{-i}}\right )\right )}}{16 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos((b*x+a)^2),x, algorithm="maxima")

[Out]

-1/16*sqrt(pi)*((I - 1)*sqrt(2)*erf(-(-1)^(3/4)*(I*b*x + I*a)) + (I - 1)*sqrt(2)*erf(-(1/4*I - 1/4)*sqrt(2)*(2
*I*b*x + 2*I*a)) - (I + 1)*sqrt(2)*erf(-(1/4*I + 1/4)*sqrt(2)*(2*I*b*x + 2*I*a)) + (I + 1)*sqrt(2)*erf((I*b*x
+ I*a)/sqrt(-I)))/b

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Fricas [A]  time = 1.54214, size = 107, normalized size = 3.69 \begin{align*} \frac{\sqrt{2} \pi \sqrt{\frac{b^{2}}{\pi }} \operatorname{C}\left (\frac{\sqrt{2}{\left (b x + a\right )} \sqrt{\frac{b^{2}}{\pi }}}{b}\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos((b*x+a)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos{\left (\left (a + b x\right )^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos((b*x+a)**2),x)

[Out]

Integral(cos((a + b*x)**2), x)

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Giac [C]  time = 1.09037, size = 74, normalized size = 2.55 \begin{align*} -\frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2}{\left (x + \frac{a}{b}\right )}{\left | b \right |}\right )}{8 \,{\left | b \right |}} + \frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2}{\left (x + \frac{a}{b}\right )}{\left | b \right |}\right )}{8 \,{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos((b*x+a)^2),x, algorithm="giac")

[Out]

-(1/8*I + 1/8)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*(x + a/b)*abs(b))/abs(b) + (1/8*I - 1/8)*sqrt(2)*sqr
t(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*(x + a/b)*abs(b))/abs(b)