3.1.18 \(\int \cos ^3(a+b x^2) \, dx\) [18]

Optimal. Leaf size=153 \[ \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}} \]

[Out]

1/24*cos(3*a)*FresnelC(x*b^(1/2)*6^(1/2)/Pi^(1/2))*6^(1/2)*Pi^(1/2)/b^(1/2)-1/24*FresnelS(x*b^(1/2)*6^(1/2)/Pi
^(1/2))*sin(3*a)*6^(1/2)*Pi^(1/2)/b^(1/2)+3/8*cos(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(
1/2)-3/8*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/2)*Pi^(1/2)/b^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3439, 3435, 3433, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {b} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^2]^3,x]

[Out]

(3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x])/(4*Sqrt[b]) + (Sqrt[Pi/6]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[
6/Pi]*x])/(4*Sqrt[b]) - (3*Sqrt[Pi/2]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(4*Sqrt[b]) - (Sqrt[Pi/6]*Fresnel
S[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(4*Sqrt[b])

Rule 3432

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

Rule 3433

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

Rule 3435

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]

Rule 3439

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 1] && IGtQ[n, 1]

Rubi steps

\begin {align*} \int \cos ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} \cos \left (a+b x^2\right )+\frac {1}{4} \cos \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \cos \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int \cos \left (a+b x^2\right ) \, dx\\ &=\frac {1}{4} (3 \cos (a)) \int \cos \left (b x^2\right ) \, dx+\frac {1}{4} \cos (3 a) \int \cos \left (3 b x^2\right ) \, dx-\frac {1}{4} (3 \sin (a)) \int \sin \left (b x^2\right ) \, dx-\frac {1}{4} \sin (3 a) \int \sin \left (3 b x^2\right ) \, dx\\ &=\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{4 \sqrt {b}}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{4 \sqrt {b}}-\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{4 \sqrt {b}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.15, size = 116, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\frac {\pi }{6}} \left (3 \sqrt {3} \cos (a) \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )+\cos (3 a) \text {FresnelC}\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-3 \sqrt {3} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)-S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)\right )}{4 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^2]^3,x]

[Out]

(Sqrt[Pi/6]*(3*Sqrt[3]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x] + Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x] - 3*Sqr
t[3]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a] - FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a]))/(4*Sqrt[b])

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 101, normalized size = 0.66

method	result	size
default	\(\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{8 \sqrt {b}}+\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )-\sin \left (3 a \right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{24 \sqrt {b}}\)	\(101\)
risch	\(\frac {{\mathrm e}^{-3 i a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (\sqrt {3}\, \sqrt {i b}\, x \right )}{48 \sqrt {i b}}+\frac {3 \,{\mathrm e}^{-i a} \sqrt {\pi }\, \erf \left (\sqrt {i b}\, x \right )}{16 \sqrt {i b}}+\frac {{\mathrm e}^{3 i a} \sqrt {\pi }\, \erf \left (\sqrt {-3 i b}\, x \right )}{16 \sqrt {-3 i b}}+\frac {3 \,{\mathrm e}^{i a} \sqrt {\pi }\, \erf \left (\sqrt {-i b}\, x \right )}{16 \sqrt {-i b}}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

3/8*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)))+1/24*2^(1/2)*Pi^(1/2)*3^(1/2)/b^(1/2)*(cos(3*a)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x)-sin(3*a)
*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x))

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 112, normalized size = 0.73 \begin {gather*} -\frac {9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) + \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) - \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} - 9 \, \sqrt {2} \sqrt {\pi } {\left ({\left (-\left (i - 1\right ) \, \cos \left (a\right ) - \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (\left (i + 1\right ) \, \cos \left (a\right ) + \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{96 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/96*(9^(1/4)*sqrt(2)*sqrt(pi)*(((I - 1)*cos(3*a) + (I + 1)*sin(3*a))*erf(sqrt(3*I*b)*x) + (-(I + 1)*cos(3*a)
- (I - 1)*sin(3*a))*erf(sqrt(-3*I*b)*x))*b^(3/2) - 9*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))*b^(3/2))/b^2

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 121, normalized size = 0.79 \begin {gather*} \frac {\sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) - \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) - 9 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right )}{24 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/24*(sqrt(6)*pi*sqrt(b/pi)*cos(3*a)*fresnel_cos(sqrt(6)*x*sqrt(b/pi)) + 9*sqrt(2)*pi*sqrt(b/pi)*cos(a)*fresne
l_cos(sqrt(2)*x*sqrt(b/pi)) - sqrt(6)*pi*sqrt(b/pi)*fresnel_sin(sqrt(6)*x*sqrt(b/pi))*sin(3*a) - 9*sqrt(2)*pi*
sqrt(b/pi)*fresnel_sin(sqrt(2)*x*sqrt(b/pi))*sin(a))/b

________________________________________________________________________________________

Sympy [A]
time = 0.71, size = 129, normalized size = 0.84 \begin {gather*} \frac {3 \sqrt {2} \sqrt {\pi } \left (- \sin {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{8} + \frac {\sqrt {6} \sqrt {\pi } \left (- \sin {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right ) + \cos {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )\right ) \sqrt {\frac {1}{b}}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x**2+a)**3,x)

[Out]

3*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)/8 + sqrt(6)*sqrt(pi)*(-sin(3*a)*fresnels(sqrt(6)*sqrt(b)*x/sqrt(pi)) + cos(3*a)*fresnelc(sqrt(6)*s
qrt(b)*x/sqrt(pi)))*sqrt(1/b)/24

________________________________________________________________________________________

Giac [C] Result contains complex when optimal does not.
time = 0.45, size = 185, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{48 \, \sqrt {b} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {3 \, \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 )}}{16 \, {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 \, \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 )}}{16 \, {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{48 \, \sqrt {b} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/48*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(-I*b/abs(b) + 1))*e^(3*I*a)/(sqrt(b)*(-I*b/abs(b) + 1)) - 3
/16*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)
)) - 3/16*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))) - 1/48*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(I*b/abs(b) + 1))*e^(-3*I*a)/(sqrt(b)*(I*b/abs(b) +
1))

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (b\,x^2+a\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^2)^3,x)

[Out]

int(cos(a + b*x^2)^3, x)

________________________________________________________________________________________