3.28 \(\int x^2 \sin ^3(a+b x^2) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.224809, 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 = {3403, 3385, 3354, 3352, 3351} \[ \frac{3 \sqrt{\frac{\pi }{2}} \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )}{8 b^{3/2}}-\frac{\sqrt{\frac{\pi }{6}} \cos (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )}{24 b^{3/2}}-\frac{3 \sqrt{\frac{\pi }{2}} \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )}{8 b^{3/2}}+\frac{\sqrt{\frac{\pi }{6}} \sin (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )}{24 b^{3/2}}-\frac{3 x \cos \left (a+b x^2\right )}{8 b}+\frac{x \cos \left (3 a+3 b x^2\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[a + b*x^2]^3,x]

[Out]

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

Rule 3403

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

Rule 3385

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

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]

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

Mathematica [A]  time = 0.433139, size = 159, normalized size = 0.85 \[ \frac{27 \sqrt{2 \pi } \cos (a) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{b} x\right )-\sqrt{6 \pi } \cos (3 a) \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{b} x\right )-27 \sqrt{2 \pi } \sin (a) S\left (\sqrt{b} \sqrt{\frac{2}{\pi }} x\right )+\sqrt{6 \pi } \sin (3 a) S\left (\sqrt{b} \sqrt{\frac{6}{\pi }} x\right )-54 \sqrt{b} x \cos \left (a+b x^2\right )+6 \sqrt{b} x \cos \left (3 \left (a+b x^2\right )\right )}{144 b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[a + b*x^2]^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 132, normalized size = 0.7 \begin{align*} -{\frac{3\,x\cos \left ( b{x}^{2}+a \right ) }{8\,b}}+{\frac{3\,\sqrt{2}\sqrt{\pi }}{16} \left ( \cos \left ( a \right ){\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) -\sin \left ( a \right ){\it FresnelS} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}\sqrt{b}} \right ) \right ){b}^{-{\frac{3}{2}}}}+{\frac{x\cos \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 FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}\sqrt{b}} \right ) -\sin \left ( 3\,a \right ){\it FresnelS} \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*sin(b*x^2+a)^3,x)

[Out]

-3/8*x*cos(b*x^2+a)/b+3/16/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*Fresne
lS(x*b^(1/2)*2^(1/2)/Pi^(1/2)))+1/24*x*cos(3*b*x^2+3*a)/b-1/144/b^(3/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*(cos(3*a)*Fre
snelC(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))

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Maxima [C]  time = 2.24511, size = 695, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(24*x*abs(b)*cos(3*b*x^2 + 3*a) - 216*x*abs(b)*cos(b*x^2 + a) - sqrt(3)*sqrt(pi)*(((cos(1/4*pi + 1/2*arc
tan2(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*arcta
n2(0, b)))*cos(3*a) - (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*arctan2(0, b)))*sin(3*a))*erf(sqrt(3*I*b)*x) + ((cos(1/4*pi + 1/2*arctan
2(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)))*cos(3*a) - (-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*arctan2(0, b)))*sin(3*a))*erf(sqrt(-3*I*b)*x))*sqrt(abs(b)) + sqrt(pi)*(((
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)))*cos(a) + (-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)))*sin(a))*erf(s
qrt(I*b)*x) + ((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)))*cos(a) + (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))
)*sin(a))*erf(sqrt(-I*b)*x))*sqrt(abs(b)))/(b*abs(b))

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Fricas [A]  time = 2.46357, size = 447, normalized size = 2.38 \begin{align*} \frac{24 \, b x \cos \left (b x^{2} + a\right )^{3} - \sqrt{6} \pi \sqrt{\frac{b}{\pi }} \cos \left (3 \, a\right ) \operatorname{C}\left (\sqrt{6} x \sqrt{\frac{b}{\pi }}\right ) + 27 \, \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 ) - 27 \, \sqrt{2} \pi \sqrt{\frac{b}{\pi }} \operatorname{S}\left (\sqrt{2} x \sqrt{\frac{b}{\pi }}\right ) \sin \left (a\right ) - 72 \, b x \cos \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*sin(b*x^2+a)^3,x, algorithm="fricas")

[Out]

1/144*(24*b*x*cos(b*x^2 + a)^3 - sqrt(6)*pi*sqrt(b/pi)*cos(3*a)*fresnel_cos(sqrt(6)*x*sqrt(b/pi)) + 27*sqrt(2)
*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x*sqrt(b/pi)) + sqrt(6)*pi*sqrt(b/pi)*fresnel_sin(sqrt(6)*x*sqrt(b/p
i))*sin(3*a) - 27*sqrt(2)*pi*sqrt(b/pi)*fresnel_sin(sqrt(2)*x*sqrt(b/pi))*sin(a) - 72*b*x*cos(b*x^2 + a))/b^2

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Sympy [B]  time = 6.3889, size = 439, normalized size = 2.34 \begin{align*} - \frac{3 b^{\frac{3}{2}} x^{5} \sqrt{\frac{1}{b}} \cos{\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}} \cos{\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}} \sin{\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}} \sin{\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 )} C\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 )} C\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 )} S\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 )} S\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*sin(b*x**2+a)**3,x)

[Out]

-3*b**(3/2)*x**5*sqrt(1/b)*cos(a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -b**2*x**4/4)/(32*g
amma(7/4)*gamma(9/4)) + 3*b**(3/2)*x**5*sqrt(1/b)*cos(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)*sin(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)*sin(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)*sqr
t(pi)*x**2*sqrt(1/b)*sin(a)*fresnelc(sqrt(2)*sqrt(b)*x/sqrt(pi))/8 - sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*sin(3*a)*
fresnelc(sqrt(6)*sqrt(b)*x/sqrt(pi))/24 + 3*sqrt(2)*sqrt(pi)*x**2*sqrt(1/b)*cos(a)*fresnels(sqrt(2)*sqrt(b)*x/
sqrt(pi))/8 - sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*cos(3*a)*fresnels(sqrt(6)*sqrt(b)*x/sqrt(pi))/24

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Giac [C]  time = 1.14382, size = 350, normalized size = 1.86 \begin{align*} \frac{x e^{\left (3 i \, b x^{2} + 3 i \, a\right )}}{48 \, b} - \frac{3 \, x e^{\left (i \, b x^{2} + i \, a\right )}}{16 \, b} - \frac{3 \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{16 \, b} + \frac{x e^{\left (-3 i \, b x^{2} - 3 i \, a\right )}}{48 \, b} + \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 )}}{288 \, b^{\frac{3}{2}}{\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 )}}{32 \, b{\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 )}}{32 \, b{\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 )}}{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*sin(b*x^2+a)^3,x, algorithm="giac")

[Out]

1/48*x*e^(3*I*b*x^2 + 3*I*a)/b - 3/16*x*e^(I*b*x^2 + I*a)/b - 3/16*x*e^(-I*b*x^2 - I*a)/b + 1/48*x*e^(-3*I*b*x
^2 - 3*I*a)/b + 1/288*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*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*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*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))