3.16 \(\int x^2 \cos ^3(a+b x^2) \, dx\)

Optimal. Leaf size=188 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sin (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}+\frac{3 x \sin \left (a+b x^2\right )}{8 b}+\frac{x \sin \left (3 a+3 b x^2\right )}{24 b} \]

[Out]

(-3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/(8*b^(3/2)) - (Sqrt[Pi/6]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt
[6/Pi]*x])/(24*b^(3/2)) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(8*b^(3/2)) - (Sqrt[Pi/6]*Fresn
elC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(24*b^(3/2)) + (3*x*Sin[a + b*x^2])/(8*b) + (x*Sin[3*a + 3*b*x^2])/(24*b)

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Rubi [A]  time = 0.182, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {3404, 3386, 3353, 3352, 3351} \[ -\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \sin (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}+\frac{3 x \sin \left (a+b x^2\right )}{8 b}+\frac{x \sin \left (3 a+3 b x^2\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[a + b*x^2]^3,x]

[Out]

(-3*Sqrt[Pi/2]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x])/(8*b^(3/2)) - (Sqrt[Pi/6]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt
[6/Pi]*x])/(24*b^(3/2)) - (3*Sqrt[Pi/2]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(8*b^(3/2)) - (Sqrt[Pi/6]*Fresn
elC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(24*b^(3/2)) + (3*x*Sin[a + b*x^2])/(8*b) + (x*Sin[3*a + 3*b*x^2])/(24*b)

Rule 3404

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rule 3386

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e^(n - 1)*(e*x)^(m - n + 1)*Sin[c + d*
x^n])/(d*n), x] - Dist[(e^n*(m - n + 1))/(d*n), Int[(e*x)^(m - n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x
] && IGtQ[n, 0] && LtQ[n, m + 1]

Rule 3353

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

Rubi steps

\begin{align*} \int x^2 \cos ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac{3}{4} x^2 \cos \left (a+b x^2\right )+\frac{1}{4} x^2 \cos \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \cos \left (3 a+3 b x^2\right ) \, dx+\frac{3}{4} \int x^2 \cos \left (a+b x^2\right ) \, dx\\ &=\frac{3 x \sin \left (a+b x^2\right )}{8 b}+\frac{x \sin \left (3 a+3 b x^2\right )}{24 b}-\frac{\int \sin \left (3 a+3 b x^2\right ) \, dx}{24 b}-\frac{3 \int \sin \left (a+b x^2\right ) \, dx}{8 b}\\ &=\frac{3 x \sin \left (a+b x^2\right )}{8 b}+\frac{x \sin \left (3 a+3 b x^2\right )}{24 b}-\frac{(3 \cos (a)) \int \sin \left (b x^2\right ) \, dx}{8 b}-\frac{\cos (3 a) \int \sin \left (3 b x^2\right ) \, dx}{24 b}-\frac{(3 \sin (a)) \int \cos \left (b x^2\right ) \, dx}{8 b}-\frac{\sin (3 a) \int \cos \left (3 b x^2\right ) \, dx}{24 b}\\ &=-\frac{3 \sqrt{\frac{\pi }{2}} \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}}+\frac{3 x \sin \left (a+b x^2\right )}{8 b}+\frac{x \sin \left (3 a+3 b x^2\right )}{24 b}\\ \end{align*}

Mathematica [A]  time = 0.431522, size = 160, normalized size = 0.85 \[ \frac{-27 \sqrt{2 \pi } \sin (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sqrt{6 \pi } \sin (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )-27 \sqrt{2 \pi } \cos (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )-\sqrt{6 \pi } \cos (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )+54 \sqrt{b} x \sin \left (a+b x^2\right )+6 \sqrt{b} x \sin \left (3 \left (a+b x^2\right )\right )}{144 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[a + b*x^2]^3,x]

[Out]

(-27*Sqrt[2*Pi]*Cos[a]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x] - Sqrt[6*Pi]*Cos[3*a]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x] - 2
7*Sqrt[2*Pi]*FresnelC[Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a] - Sqrt[6*Pi]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a] + 54*S
qrt[b]*x*Sin[a + b*x^2] + 6*Sqrt[b]*x*Sin[3*(a + b*x^2)])/(144*b^(3/2))

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Maple [A]  time = 0.033, size = 130, normalized size = 0.7 \begin{align*}{\frac{3\,x\sin \left ( b{x}^{2}+a \right ) }{8\,b}}-{\frac{3\,\sqrt{2}\sqrt{\pi }}{16} \left ( \cos \left ( a \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) +\sin \left ( a \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}}+{\frac{x\sin \left ( 3\,b{x}^{2}+3\,a \right ) }{24\,b}}-{\frac{\sqrt{2}\sqrt{\pi }\sqrt{3}}{144} \left ( \cos \left ( 3\,a \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}\sqrt{b}} \right ) +\sin \left ( 3\,a \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*x*sin(b*x^2+a)/b-3/16/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2))+sin(a)*Fresnel
C(x*b^(1/2)*2^(1/2)/Pi^(1/2)))+1/24*x*sin(3*b*x^2+3*a)/b-1/144/b^(3/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*(cos(3*a)*Fres
nelS(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x)+sin(3*a)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x))

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Maxima [C]  time = 2.1503, size = 697, normalized size = 3.71 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(24*x*abs(b)*sin(3*b*x^2 + 3*a) + 216*x*abs(b)*sin(b*x^2 + a) + sqrt(3)*sqrt(pi)*(((-I*cos(1/4*pi + 1/2*
arctan2(0, b)) - I*cos(-1/4*pi + 1/2*arctan2(0, b)) - sin(1/4*pi + 1/2*arctan2(0, b)) + sin(-1/4*pi + 1/2*arct
an2(0, b)))*cos(3*a) - (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) - I*sin(1/4*pi + 1/
2*arctan2(0, b)) + I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(3*a))*erf(sqrt(3*I*b)*x) + ((I*cos(1/4*pi + 1/2*arc
tan2(0, b)) + I*cos(-1/4*pi + 1/2*arctan2(0, b)) - sin(1/4*pi + 1/2*arctan2(0, b)) + sin(-1/4*pi + 1/2*arctan2
(0, b)))*cos(3*a) - (cos(1/4*pi + 1/2*arctan2(0, b)) + cos(-1/4*pi + 1/2*arctan2(0, b)) + I*sin(1/4*pi + 1/2*a
rctan2(0, b)) - I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(3*a))*erf(sqrt(-3*I*b)*x))*sqrt(abs(b)) + sqrt(pi)*(((
-27*I*cos(1/4*pi + 1/2*arctan2(0, b)) - 27*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 27*sin(1/4*pi + 1/2*arctan2(0,
b)) + 27*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (27*cos(1/4*pi + 1/2*arctan2(0, b)) + 27*cos(-1/4*pi + 1/
2*arctan2(0, b)) - 27*I*sin(1/4*pi + 1/2*arctan2(0, b)) + 27*I*sin(-1/4*pi + 1/2*arctan2(0, b)))*sin(a))*erf(s
qrt(I*b)*x) + ((27*I*cos(1/4*pi + 1/2*arctan2(0, b)) + 27*I*cos(-1/4*pi + 1/2*arctan2(0, b)) - 27*sin(1/4*pi +
1/2*arctan2(0, b)) + 27*sin(-1/4*pi + 1/2*arctan2(0, b)))*cos(a) - (27*cos(1/4*pi + 1/2*arctan2(0, b)) + 27*c
os(-1/4*pi + 1/2*arctan2(0, b)) + 27*I*sin(1/4*pi + 1/2*arctan2(0, b)) - 27*I*sin(-1/4*pi + 1/2*arctan2(0, b))
)*sin(a))*erf(sqrt(-I*b)*x))*sqrt(abs(b)))/(b*abs(b))

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Fricas [A]  time = 1.78099, size = 450, normalized size = 2.39 \begin{align*} -\frac{\sqrt{6} \pi \sqrt{\frac{b}{\pi }} \cos \left (3 \, a\right ) \operatorname{S}\left (\sqrt{6} x \sqrt{\frac{b}{\pi }}\right ) + 27 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \cos \left (a\right ) \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) + \sqrt{6} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{6} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (3 \, a\right ) + 27 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{C}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) - 24 \,{\left (b x \cos \left (b x^{2} + a\right )^{2} + 2 \, b x\right )} \sin \left (b x^{2} + a\right )}{144 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(sqrt(6)*pi*sqrt(b/pi)*cos(3*a)*fresnel_sin(sqrt(6)*x*sqrt(b/pi)) + 27*sqrt(2)*pi*sqrt(b/pi)*cos(a)*fre
snel_sin(sqrt(2)*x*sqrt(b/pi)) + sqrt(6)*pi*sqrt(b/pi)*fresnel_cos(sqrt(6)*x*sqrt(b/pi))*sin(3*a) + 27*sqrt(2)
*pi*sqrt(b/pi)*fresnel_cos(sqrt(2)*x*sqrt(b/pi))*sin(a) - 24*(b*x*cos(b*x^2 + a)^2 + 2*b*x)*sin(b*x^2 + a))/b^
2

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Sympy [B]  time = 5.78754, size = 439, normalized size = 2.34 \begin{align*} \frac{3 b^{\frac{3}{2}} x^{5} \sqrt{\frac{1}{b}} \sin{\left (a \right )} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle |{- \frac{b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{9}{4}\right )} + \frac{3 b^{\frac{3}{2}} x^{5} \sqrt{\frac{1}{b}} \sin{\left (3 a \right )} \Gamma \left (\frac{3}{4}\right ) \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4} \\ \frac{3}{2}, \frac{7}{4}, \frac{9}{4} \end{matrix}\middle |{- \frac{9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{7}{4}\right ) \Gamma \left (\frac{9}{4}\right )} - \frac{3 \sqrt{b} x^{3} \sqrt{\frac{1}{b}} \cos{\left (a \right )} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} - \frac{\sqrt{b} x^{3} \sqrt{\frac{1}{b}} \cos{\left (3 a \right )} \Gamma \left (\frac{1}{4}\right ) \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{1}{2}, \frac{5}{4}, \frac{7}{4} \end{matrix}\middle |{- \frac{9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac{5}{4}\right ) \Gamma \left (\frac{7}{4}\right )} - \frac{3 \sqrt{2} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \sin{\left (a \right )} S\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right )}{8} - \frac{\sqrt{6} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \sin{\left (3 a \right )} S\left (\frac{\sqrt{6} \sqrt{b} x}{\sqrt{\pi }}\right )}{24} + \frac{3 \sqrt{2} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \cos{\left (a \right )} C\left (\frac{\sqrt{2} \sqrt{b} x}{\sqrt{\pi }}\right )}{8} + \frac{\sqrt{6} \sqrt{\pi } x^{2} \sqrt{\frac{1}{b}} \cos{\left (3 a \right )} C\left (\frac{\sqrt{6} \sqrt{b} x}{\sqrt{\pi }}\right )}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*b**(3/2)*x**5*sqrt(1/b)*sin(a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -b**2*x**4/4)/(32*ga
mma(7/4)*gamma(9/4)) + 3*b**(3/2)*x**5*sqrt(1/b)*sin(3*a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9
/4), -9*b**2*x**4/4)/(32*gamma(7/4)*gamma(9/4)) - 3*sqrt(b)*x**3*sqrt(1/b)*cos(a)*gamma(1/4)*gamma(3/4)*hyper(
(1/4, 3/4), (1/2, 5/4, 7/4), -b**2*x**4/4)/(32*gamma(5/4)*gamma(7/4)) - sqrt(b)*x**3*sqrt(1/b)*cos(3*a)*gamma(
1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -9*b**2*x**4/4)/(32*gamma(5/4)*gamma(7/4)) - 3*sqrt(2)*sqrt
(pi)*x**2*sqrt(1/b)*sin(a)*fresnels(sqrt(2)*sqrt(b)*x/sqrt(pi))/8 - sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*sin(3*a)*f
resnels(sqrt(6)*sqrt(b)*x/sqrt(pi))/24 + 3*sqrt(2)*sqrt(pi)*x**2*sqrt(1/b)*cos(a)*fresnelc(sqrt(2)*sqrt(b)*x/s
qrt(pi))/8 + sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*cos(3*a)*fresnelc(sqrt(6)*sqrt(b)*x/sqrt(pi))/24

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Giac [C]  time = 1.19145, size = 350, normalized size = 1.86 \begin{align*} -\frac{i \, x e^{\left (3 i \, b x^{2} + 3 i \, a\right )}}{48 \, b} - \frac{3 i \, x e^{\left (i \, b x^{2} + i \, a\right )}}{16 \, b} + \frac{3 i \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{16 \, b} + \frac{i \, x e^{\left (-3 i \, b x^{2} - 3 i \, a\right )}}{48 \, b} - \frac{i \, \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 )}}{288 \, b^{\frac{3}{2}}{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )}} - \frac{3 i \, \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 )}}{32 \, b{\left (-\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} + \frac{3 i \, \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 )}}{32 \, b{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )} \sqrt{{\left | b \right |}}} + \frac{i \, \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 )}}{288 \, b^{\frac{3}{2}}{\left (\frac{i \, b}{{\left | b \right |}} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*I*x*e^(3*I*b*x^2 + 3*I*a)/b - 3/16*I*x*e^(I*b*x^2 + I*a)/b + 3/16*I*x*e^(-I*b*x^2 - I*a)/b + 1/48*I*x*e^
(-3*I*b*x^2 - 3*I*a)/b - 1/288*I*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(-I*b/abs(b) + 1))*e^(3*I*a)/(b^(
3/2)*(-I*b/abs(b) + 1)) - 3/32*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(
b*(-I*b/abs(b) + 1)*sqrt(abs(b))) + 3/32*I*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(I*b/abs(b) + 1)*sqrt(abs(b)))*
e^(-I*a)/(b*(I*b/abs(b) + 1)*sqrt(abs(b))) + 1/288*I*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(I*b/abs(b) +
1))*e^(-3*I*a)/(b^(3/2)*(I*b/abs(b) + 1))