3.32 \(\int x^4 \cos (x^2) \sin ^2(x^2) \, dx\)

Optimal. Leaf size=84 \[ -\frac{3}{16} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )+\frac{1}{48} \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )+\frac{1}{6} x^3 \sin ^3\left (x^2\right )-\frac{1}{12} x \cos ^3\left (x^2\right )+\frac{1}{4} x \cos \left (x^2\right ) \]

[Out]

(x*Cos[x^2])/4 - (x*Cos[x^2]^3)/12 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x])/16 + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi
]*x])/48 + (x^3*Sin[x^2]^3)/6

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Rubi [A]  time = 0.0776766, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3443, 3403, 3385, 3352} \[ -\frac{3}{16} \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )+\frac{1}{48} \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )+\frac{1}{6} x^3 \sin ^3\left (x^2\right )+\frac{3}{16} x \cos \left (x^2\right )-\frac{1}{48} x \cos \left (3 x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^4*Cos[x^2]*Sin[x^2]^2,x]

[Out]

(3*x*Cos[x^2])/16 - (x*Cos[3*x^2])/48 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*x])/16 + (Sqrt[Pi/6]*FresnelC[Sqrt[6
/Pi]*x])/48 + (x^3*Sin[x^2]^3)/6

Rule 3443

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m - n
+ 1)*Sin[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sin[a + b*x^n]^
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

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^4 \cos \left (x^2\right ) \sin ^2\left (x^2\right ) \, dx &=\frac{1}{6} x^3 \sin ^3\left (x^2\right )-\frac{1}{2} \int x^2 \sin ^3\left (x^2\right ) \, dx\\ &=\frac{1}{6} x^3 \sin ^3\left (x^2\right )-\frac{1}{2} \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\\ &=\frac{1}{6} x^3 \sin ^3\left (x^2\right )+\frac{1}{8} \int x^2 \sin \left (3 x^2\right ) \, dx-\frac{3}{8} \int x^2 \sin \left (x^2\right ) \, dx\\ &=\frac{3}{16} x \cos \left (x^2\right )-\frac{1}{48} x \cos \left (3 x^2\right )+\frac{1}{6} x^3 \sin ^3\left (x^2\right )+\frac{1}{48} \int \cos \left (3 x^2\right ) \, dx-\frac{3}{16} \int \cos \left (x^2\right ) \, dx\\ &=\frac{3}{16} x \cos \left (x^2\right )-\frac{1}{48} x \cos \left (3 x^2\right )-\frac{3}{16} \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} x\right )+\frac{1}{48} \sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} x\right )+\frac{1}{6} x^3 \sin ^3\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.148881, size = 75, normalized size = 0.89 \[ \frac{1}{288} \left (-27 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} x\right )+\sqrt{6 \pi } \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} x\right )+6 x \left (8 x^2 \sin ^3\left (x^2\right )+9 \cos \left (x^2\right )-\cos \left (3 x^2\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Cos[x^2]*Sin[x^2]^2,x]

[Out]

(-27*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*x] + Sqrt[6*Pi]*FresnelC[Sqrt[6/Pi]*x] + 6*x*(9*Cos[x^2] - Cos[3*x^2] + 8*
x^2*Sin[x^2]^3))/288

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Maple [A]  time = 0.019, size = 78, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\sin \left ({x}^{2} \right ) }{8}}+{\frac{3\,x\cos \left ({x}^{2} \right ) }{16}}-{\frac{3\,\sqrt{2}\sqrt{\pi }}{32}{\it FresnelC} \left ({\frac{x\sqrt{2}}{\sqrt{\pi }}} \right ) }-{\frac{{x}^{3}\sin \left ( 3\,{x}^{2} \right ) }{24}}-{\frac{x\cos \left ( 3\,{x}^{2} \right ) }{48}}+{\frac{\sqrt{2}\sqrt{\pi }\sqrt{3}}{288}{\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^4*cos(x^2)*sin(x^2)^2,x)

[Out]

1/8*x^3*sin(x^2)+3/16*x*cos(x^2)-3/32*FresnelC(x*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)-1/24*x^3*sin(3*x^2)-1/48*x
*cos(3*x^2)+1/288*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 = 1.53842, size = 158, normalized size = 1.88 \begin{align*} -\frac{1}{24} \, x^{3} \sin \left (3 \, x^{2}\right ) + \frac{1}{8} \, x^{3} \sin \left (x^{2}\right ) - \frac{1}{48} \, x \cos \left (3 \, x^{2}\right ) + \frac{3}{16} \, x \cos \left (x^{2}\right ) - \frac{1}{2304} \, \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^4*cos(x^2)*sin(x^2)^2,x, algorithm="maxima")

[Out]

-1/24*x^3*sin(3*x^2) + 1/8*x^3*sin(x^2) - 1/48*x*cos(3*x^2) + 3/16*x*cos(x^2) - 1/2304*sqrt(pi)*((2*I - 2)*sqr
t(3)*sqrt(2)*erf(sqrt(3*I)*x) - (2*I + 2)*sqrt(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) - (2
7*I - 27)*sqrt(2)*erf((-1)^(1/4)*x))

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Fricas [A]  time = 2.19345, size = 254, normalized size = 3.02 \begin{align*} -\frac{1}{12} \, x \cos \left (x^{2}\right )^{3} + \frac{1}{4} \, x \cos \left (x^{2}\right ) + \frac{1}{288} \, \sqrt{6} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{6} x}{\sqrt{\pi }}\right ) - \frac{3}{32} \, \sqrt{2} \sqrt{\pi } \operatorname{C}\left (\frac{\sqrt{2} x}{\sqrt{\pi }}\right ) - \frac{1}{6} \,{\left (x^{3} \cos \left (x^{2}\right )^{2} - x^{3}\right )} \sin \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cos(x^2)*sin(x^2)^2,x, algorithm="fricas")

[Out]

-1/12*x*cos(x^2)^3 + 1/4*x*cos(x^2) + 1/288*sqrt(6)*sqrt(pi)*fresnel_cos(sqrt(6)*x/sqrt(pi)) - 3/32*sqrt(2)*sq
rt(pi)*fresnel_cos(sqrt(2)*x/sqrt(pi)) - 1/6*(x^3*cos(x^2)^2 - x^3)*sin(x^2)

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Sympy [B]  time = 6.09948, size = 291, normalized size = 3.46 \begin{align*} - \frac{9 x^{5} \Gamma \left (- \frac{9}{4}\right )}{40 \Gamma \left (- \frac{5}{4}\right )} + \frac{9 x^{3} \sin{\left (x^{2} \right )} \Gamma \left (- \frac{9}{4}\right )}{32 \Gamma \left (- \frac{5}{4}\right )} - \frac{5 x^{3} \sin{\left (x^{2} \right )} \Gamma \left (- \frac{5}{4}\right )}{16 \Gamma \left (- \frac{1}{4}\right )} + \frac{3 x^{3} \sin{\left (3 x^{2} \right )} \Gamma \left (- \frac{9}{4}\right )}{32 \Gamma \left (- \frac{5}{4}\right )} + \frac{27 x \cos{\left (x^{2} \right )} \Gamma \left (- \frac{9}{4}\right )}{64 \Gamma \left (- \frac{5}{4}\right )} - \frac{15 x \cos{\left (x^{2} \right )} \Gamma \left (- \frac{5}{4}\right )}{32 \Gamma \left (- \frac{1}{4}\right )} + \frac{3 x \cos{\left (3 x^{2} \right )} \Gamma \left (- \frac{9}{4}\right )}{64 \Gamma \left (- \frac{5}{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{1}{4}\right )} - \frac{27 \sqrt{2} \sqrt{\pi } C\left (\frac{\sqrt{2} x}{\sqrt{\pi }}\right ) \Gamma \left (- \frac{9}{4}\right )}{128 \Gamma \left (- \frac{5}{4}\right )} - \frac{\sqrt{6} \sqrt{\pi } C\left (\frac{\sqrt{6} x}{\sqrt{\pi }}\right ) \Gamma \left (- \frac{9}{4}\right )}{128 \Gamma \left (- \frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*cos(x**2)*sin(x**2)**2,x)

[Out]

-9*x**5*gamma(-9/4)/(40*gamma(-5/4)) + 9*x**3*sin(x**2)*gamma(-9/4)/(32*gamma(-5/4)) - 5*x**3*sin(x**2)*gamma(
-5/4)/(16*gamma(-1/4)) + 3*x**3*sin(3*x**2)*gamma(-9/4)/(32*gamma(-5/4)) + 27*x*cos(x**2)*gamma(-9/4)/(64*gamm
a(-5/4)) - 15*x*cos(x**2)*gamma(-5/4)/(32*gamma(-1/4)) + 3*x*cos(3*x**2)*gamma(-9/4)/(64*gamma(-5/4)) + 15*sqr
t(2)*sqrt(pi)*fresnelc(sqrt(2)*x/sqrt(pi))*gamma(-5/4)/(64*gamma(-1/4)) - 27*sqrt(2)*sqrt(pi)*fresnelc(sqrt(2)
*x/sqrt(pi))*gamma(-9/4)/(128*gamma(-5/4)) - sqrt(6)*sqrt(pi)*fresnelc(sqrt(6)*x/sqrt(pi))*gamma(-9/4)/(128*ga
mma(-5/4))

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Giac [C]  time = 1.13131, size = 169, normalized size = 2.01 \begin{align*} -\left (\frac{1}{1152} i + \frac{1}{1152}\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{6} x\right ) + \left (\frac{1}{1152} i - \frac{1}{1152}\right ) \, \sqrt{6} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{6} x\right ) + \left (\frac{3}{128} i + \frac{3}{128}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} x\right ) - \left (\frac{3}{128} i - \frac{3}{128}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} x\right ) - \frac{1}{96} \,{\left (-2 i \, x^{3} + x\right )} e^{\left (3 i \, x^{2}\right )} - \frac{1}{32} \,{\left (2 i \, x^{3} - 3 \, x\right )} e^{\left (i \, x^{2}\right )} - \frac{1}{32} \,{\left (-2 i \, x^{3} - 3 \, x\right )} e^{\left (-i \, x^{2}\right )} - \frac{1}{96} \,{\left (2 i \, x^{3} + x\right )} e^{\left (-3 i \, x^{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*cos(x^2)*sin(x^2)^2,x, algorithm="giac")

[Out]

-(1/1152*I + 1/1152)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(6)*x) + (1/1152*I - 1/1152)*sqrt(6)*sqrt(pi)*erf(
-(1/2*I + 1/2)*sqrt(6)*x) + (3/128*I + 3/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*x) - (3/128*I - 3/128
)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*x) - 1/96*(-2*I*x^3 + x)*e^(3*I*x^2) - 1/32*(2*I*x^3 - 3*x)*e^(I
*x^2) - 1/32*(-2*I*x^3 - 3*x)*e^(-I*x^2) - 1/96*(2*I*x^3 + x)*e^(-3*I*x^2)