3.31 \(\int x^2 \sin ^3(x^2) \, dx\)

Optimal. Leaf size=71 \[ \frac{3}{8} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )-\frac{1}{24} \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )+\frac{1}{6} x \cos ^3\left (x^2\right )-\frac{1}{2} x \cos \left (x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0532944, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3403, 3385, 3352} \[ \frac{3}{8} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )-\frac{1}{24} \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )-\frac{3}{8} x \cos \left (x^2\right )+\frac{1}{24} x \cos \left (3 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[x^2]^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0672248, size = 63, normalized size = 0.89 \[ \frac{1}{144} \left (27 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )-\sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )+6 x \left (\cos \left (3 x^2\right )-9 \cos \left (x^2\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[x^2]^3,x]

[Out]

(6*x*(-9*Cos[x^2] + Cos[3*x^2]) + 27*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*x] - Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*x])/14
4

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Maple [A]  time = 0.011, size = 58, normalized size = 0.8 \begin{align*} -{\frac{3\,x\cos \left ({x}^{2} \right ) }{8}}+{\frac{3\,\sqrt{2}\sqrt{\pi }}{16}{\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}} \right ) }+{\frac{x\cos \left ( 3\,{x}^{2} \right ) }{24}}-{\frac{\sqrt{2}\sqrt{\pi }\sqrt{3}}{144}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}x}{\sqrt{\pi }}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(x^2)^3,x)

[Out]

-3/8*x*cos(x^2)+3/16*FresnelC(x*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)+1/24*x*cos(3*x^2)-1/144*2^(1/2)*Pi^(1/2)*3^
(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*3^(1/2)*x)

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Maxima [C]  time = 1.52743, size = 131, normalized size = 1.85 \begin{align*} \frac{1}{24} \, x \cos \left (3 \, x^{2}\right ) - \frac{3}{8} \, x \cos \left (x^{2}\right ) + \frac{1}{1152} \, \sqrt{\pi }{\left (\left (2 i - 2\right ) \, \sqrt{3} \sqrt{2} \operatorname{erf}\left (\sqrt{3 i} x\right ) - \left (2 i + 2\right ) \, \sqrt{3} \sqrt{2} \operatorname{erf}\left (\sqrt{-3 i} x\right ) - \left (27 i - 27\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} x\right ) - \left (27 i + 27\right ) \, \sqrt{2} \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} x\right ) + \left (27 i + 27\right ) \, \sqrt{2} \operatorname{erf}\left (\sqrt{-i} x\right ) - \left (27 i - 27\right ) \, \sqrt{2} \operatorname{erf}\left (\left (-1\right )^{\frac{1}{4}} x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*x*cos(3*x^2) - 3/8*x*cos(x^2) + 1/1152*sqrt(pi)*((2*I - 2)*sqrt(3)*sqrt(2)*erf(sqrt(3*I)*x) - (2*I + 2)*s
qrt(3)*sqrt(2)*erf(sqrt(-3*I)*x) - (27*I - 27)*sqrt(2)*erf((1/2*I + 1/2)*sqrt(2)*x) - (27*I + 27)*sqrt(2)*erf(
(1/2*I - 1/2)*sqrt(2)*x) + (27*I + 27)*sqrt(2)*erf(sqrt(-I)*x) - (27*I - 27)*sqrt(2)*erf((-1)^(1/4)*x))

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Fricas [A]  time = 2.18939, size = 200, normalized size = 2.82 \begin{align*} \frac{1}{6} \, x \cos \left (x^{2}\right )^{3} - \frac{1}{2} \, x \cos \left (x^{2}\right ) - \frac{1}{144} \, \sqrt{6} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{6} x}{\sqrt{\pi }}\right ) + \frac{3}{16} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} x}{\sqrt{\pi }}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(x^2)^3,x, algorithm="fricas")

[Out]

1/6*x*cos(x^2)^3 - 1/2*x*cos(x^2) - 1/144*sqrt(6)*sqrt(pi)*fresnel_cos(sqrt(6)*x/sqrt(pi)) + 3/16*sqrt(2)*sqrt
(pi)*fresnel_cos(sqrt(2)*x/sqrt(pi))

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Sympy [A]  time = 4.98922, size = 116, normalized size = 1.63 \begin{align*} - \frac{15 x \cos{\left (x^{2} \right )} \Gamma \left (\frac{5}{4}\right )}{32 \Gamma \left (\frac{9}{4}\right )} + \frac{5 x \cos{\left (3 x^{2} \right )} \Gamma \left (\frac{5}{4}\right )}{96 \Gamma \left (\frac{9}{4}\right )} + \frac{15 \sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} x}{\sqrt{\pi }}\right ) \Gamma \left (\frac{5}{4}\right )}{64 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \sqrt{6} \sqrt{\pi } C\left (\frac{\sqrt{6} x}{\sqrt{\pi }}\right ) \Gamma \left (\frac{5}{4}\right )}{576 \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sin(x**2)**3,x)

[Out]

-15*x*cos(x**2)*gamma(5/4)/(32*gamma(9/4)) + 5*x*cos(3*x**2)*gamma(5/4)/(96*gamma(9/4)) + 15*sqrt(2)*sqrt(pi)*
fresnelc(sqrt(2)*x/sqrt(pi))*gamma(5/4)/(64*gamma(9/4)) - 5*sqrt(6)*sqrt(pi)*fresnelc(sqrt(6)*x/sqrt(pi))*gamm
a(5/4)/(576*gamma(9/4))

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Giac [C]  time = 1.10356, size = 131, normalized size = 1.85 \begin{align*} \left (\frac{1}{576} i + \frac{1}{576}\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{6} x\right ) - \left (\frac{1}{576} i - \frac{1}{576}\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{6} x\right ) - \left (\frac{3}{64} i + \frac{3}{64}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} x\right ) + \left (\frac{3}{64} i - \frac{3}{64}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} x\right ) + \frac{1}{48} \, x e^{\left (3 i \, x^{2}\right )} - \frac{3}{16} \, x e^{\left (i \, x^{2}\right )} - \frac{3}{16} \, x e^{\left (-i \, x^{2}\right )} + \frac{1}{48} \, x e^{\left (-3 i \, x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(x^2)^3,x, algorithm="giac")

[Out]

(1/576*I + 1/576)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*x) - (1/576*I - 1/576)*sqrt(6)*sqrt(pi)*erf(-(1/2
*I + 1/2)*sqrt(6)*x) - (3/64*I + 3/64)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*x) + (3/64*I - 3/64)*sqrt(2)
*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*x) + 1/48*x*e^(3*I*x^2) - 3/16*x*e^(I*x^2) - 3/16*x*e^(-I*x^2) + 1/48*x*e
^(-3*I*x^2)