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

Optimal. Leaf size=70 \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3354, 3352, 3351} \[ \frac {\sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right )}{\sqrt {b}}-\frac {\sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^2],x]

[Out]

(Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/Sqrt[b] - (Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]
)/Sqrt[b]

Rule 3351

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

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]

Rule 3354

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 57, normalized size = 0.81 \[ \frac {\sqrt {\frac {\pi }{2}} \left (\cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )\right )}{\sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^2],x]

[Out]

(Sqrt[Pi/2]*(Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x] - FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a]))/Sqrt[b]

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fricas [A]  time = 0.82, size = 61, normalized size = 0.87 \[ \frac {\sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \relax (a) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) - \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \relax (a)}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x*sqrt(b/pi)) - sqrt(2)*pi*sqrt(b/pi)*fresnel_sin(sqrt(2
)*x*sqrt(b/pi))*sin(a))/b

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giac [C]  time = 0.39, size = 95, normalized size = 1.36 \[ -\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{4 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{4 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/((-I*b/abs(b) + 1)*sqrt(abs(b
))) - 1/4*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/((I*b/abs(b) + 1)*sqrt(a
bs(b)))

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maple [A]  time = 0.02, size = 44, normalized size = 0.63 \[ \frac {\sqrt {2}\, \sqrt {\pi }\, \left (\cos \relax (a ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \relax (a ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{2 \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.81, size = 48, normalized size = 0.69 \[ -\frac {\sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \relax (a) + \left (i + 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \relax (a) - \left (i - 1\right ) \, \sin \relax (a)\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )}}{8 \, \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt(2)*sqrt(pi)*(((I - 1)*cos(a) + (I + 1)*sin(a))*erf(sqrt(I*b)*x) + (-(I + 1)*cos(a) - (I - 1)*sin(a))
*erf(sqrt(-I*b)*x))/sqrt(b)

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mupad [B]  time = 0.36, size = 51, normalized size = 0.73 \[ \frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {C}\left (\frac {\sqrt {2}\,\sqrt {b}\,x}{\sqrt {\pi }}\right )\,\cos \relax (a)}{2\,\sqrt {b}}-\frac {\sqrt {2}\,\sqrt {\pi }\,\mathrm {S}\left (\frac {\sqrt {2}\,\sqrt {b}\,x}{\sqrt {\pi }}\right )\,\sin \relax (a)}{2\,\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^2),x)

[Out]

(2^(1/2)*pi^(1/2)*fresnelc((2^(1/2)*b^(1/2)*x)/pi^(1/2))*cos(a))/(2*b^(1/2)) - (2^(1/2)*pi^(1/2)*fresnels((2^(
1/2)*b^(1/2)*x)/pi^(1/2))*sin(a))/(2*b^(1/2))

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sympy [A]  time = 0.44, size = 61, normalized size = 0.87 \[ \frac {\sqrt {2} \sqrt {\pi } \left (- \sin {\relax (a )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\relax (a )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(pi)*(-sin(a)*fresnels(sqrt(2)*sqrt(b)*x/sqrt(pi)) + cos(a)*fresnelc(sqrt(2)*sqrt(b)*x/sqrt(pi)))*
sqrt(1/b)/2

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