3.485 \(\int x^2 \sin ^6(x) \, dx\)

Optimal. Leaf size=105 \[ \frac{5 x^3}{48}-\frac{1}{6} x^2 \sin ^5(x) \cos (x)-\frac{5}{24} x^2 \sin ^3(x) \cos (x)-\frac{5}{16} x^2 \sin (x) \cos (x)-\frac{245 x}{1152}+\frac{1}{18} x \sin ^6(x)+\frac{5}{48} x \sin ^4(x)+\frac{5}{16} x \sin ^2(x)+\frac{1}{108} \sin ^5(x) \cos (x)+\frac{65 \sin ^3(x) \cos (x)}{1728}+\frac{245 \sin (x) \cos (x)}{1152} \]

[Out]

(-245*x)/1152 + (5*x^3)/48 + (245*Cos[x]*Sin[x])/1152 - (5*x^2*Cos[x]*Sin[x])/16 + (5*x*Sin[x]^2)/16 + (65*Cos
[x]*Sin[x]^3)/1728 - (5*x^2*Cos[x]*Sin[x]^3)/24 + (5*x*Sin[x]^4)/48 + (Cos[x]*Sin[x]^5)/108 - (x^2*Cos[x]*Sin[
x]^5)/6 + (x*Sin[x]^6)/18

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Rubi [A]  time = 0.109744, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3311, 30, 2635, 8} \[ \frac{5 x^3}{48}-\frac{1}{6} x^2 \sin ^5(x) \cos (x)-\frac{5}{24} x^2 \sin ^3(x) \cos (x)-\frac{5}{16} x^2 \sin (x) \cos (x)-\frac{245 x}{1152}+\frac{1}{18} x \sin ^6(x)+\frac{5}{48} x \sin ^4(x)+\frac{5}{16} x \sin ^2(x)+\frac{1}{108} \sin ^5(x) \cos (x)+\frac{65 \sin ^3(x) \cos (x)}{1728}+\frac{245 \sin (x) \cos (x)}{1152} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[x]^6,x]

[Out]

(-245*x)/1152 + (5*x^3)/48 + (245*Cos[x]*Sin[x])/1152 - (5*x^2*Cos[x]*Sin[x])/16 + (5*x*Sin[x]^2)/16 + (65*Cos
[x]*Sin[x]^3)/1728 - (5*x^2*Cos[x]*Sin[x]^3)/24 + (5*x*Sin[x]^4)/48 + (Cos[x]*Sin[x]^5)/108 - (x^2*Cos[x]*Sin[
x]^5)/6 + (x*Sin[x]^6)/18

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x^2 \sin ^6(x) \, dx &=-\frac{1}{6} x^2 \cos (x) \sin ^5(x)+\frac{1}{18} x \sin ^6(x)-\frac{1}{18} \int \sin ^6(x) \, dx+\frac{5}{6} \int x^2 \sin ^4(x) \, dx\\ &=-\frac{5}{24} x^2 \cos (x) \sin ^3(x)+\frac{5}{48} x \sin ^4(x)+\frac{1}{108} \cos (x) \sin ^5(x)-\frac{1}{6} x^2 \cos (x) \sin ^5(x)+\frac{1}{18} x \sin ^6(x)-\frac{5}{108} \int \sin ^4(x) \, dx-\frac{5}{48} \int \sin ^4(x) \, dx+\frac{5}{8} \int x^2 \sin ^2(x) \, dx\\ &=-\frac{5}{16} x^2 \cos (x) \sin (x)+\frac{5}{16} x \sin ^2(x)+\frac{65 \cos (x) \sin ^3(x)}{1728}-\frac{5}{24} x^2 \cos (x) \sin ^3(x)+\frac{5}{48} x \sin ^4(x)+\frac{1}{108} \cos (x) \sin ^5(x)-\frac{1}{6} x^2 \cos (x) \sin ^5(x)+\frac{1}{18} x \sin ^6(x)-\frac{5}{144} \int \sin ^2(x) \, dx-\frac{5}{64} \int \sin ^2(x) \, dx+\frac{5 \int x^2 \, dx}{16}-\frac{5}{16} \int \sin ^2(x) \, dx\\ &=\frac{5 x^3}{48}+\frac{245 \cos (x) \sin (x)}{1152}-\frac{5}{16} x^2 \cos (x) \sin (x)+\frac{5}{16} x \sin ^2(x)+\frac{65 \cos (x) \sin ^3(x)}{1728}-\frac{5}{24} x^2 \cos (x) \sin ^3(x)+\frac{5}{48} x \sin ^4(x)+\frac{1}{108} \cos (x) \sin ^5(x)-\frac{1}{6} x^2 \cos (x) \sin ^5(x)+\frac{1}{18} x \sin ^6(x)-\frac{5 \int 1 \, dx}{288}-\frac{5 \int 1 \, dx}{128}-\frac{5 \int 1 \, dx}{32}\\ &=-\frac{245 x}{1152}+\frac{5 x^3}{48}+\frac{245 \cos (x) \sin (x)}{1152}-\frac{5}{16} x^2 \cos (x) \sin (x)+\frac{5}{16} x \sin ^2(x)+\frac{65 \cos (x) \sin ^3(x)}{1728}-\frac{5}{24} x^2 \cos (x) \sin ^3(x)+\frac{5}{48} x \sin ^4(x)+\frac{1}{108} \cos (x) \sin ^5(x)-\frac{1}{6} x^2 \cos (x) \sin ^5(x)+\frac{1}{18} x \sin ^6(x)\\ \end{align*}

Mathematica [A]  time = 0.086196, size = 70, normalized size = 0.67 \[ \frac{1440 x^3-1620 \left (2 x^2-1\right ) \sin (2 x)+81 \left (8 x^2-1\right ) \sin (4 x)-4 \left (18 x^2-1\right ) \sin (6 x)-3240 x \cos (2 x)+324 x \cos (4 x)-24 x \cos (6 x)}{13824} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[x]^6,x]

[Out]

(1440*x^3 - 3240*x*Cos[2*x] + 324*x*Cos[4*x] - 24*x*Cos[6*x] - 1620*(-1 + 2*x^2)*Sin[2*x] + 81*(-1 + 8*x^2)*Si
n[4*x] - 4*(-1 + 18*x^2)*Sin[6*x])/13824

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Maple [A]  time = 0.03, size = 96, normalized size = 0.9 \begin{align*}{x}^{2} \left ( -{\frac{\cos \left ( x \right ) }{6} \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) }+{\frac{5\,x}{16}} \right ) +{\frac{x \left ( \sin \left ( x \right ) \right ) ^{6}}{18}}+{\frac{\cos \left ( x \right ) }{108} \left ( \left ( \sin \left ( x \right ) \right ) ^{5}+{\frac{5\, \left ( \sin \left ( x \right ) \right ) ^{3}}{4}}+{\frac{15\,\sin \left ( x \right ) }{8}} \right ) }+{\frac{115\,x}{1152}}+{\frac{5\,x \left ( \sin \left ( x \right ) \right ) ^{4}}{48}}+{\frac{5\,\cos \left ( x \right ) }{192} \left ( \left ( \sin \left ( x \right ) \right ) ^{3}+{\frac{3\,\sin \left ( x \right ) }{2}} \right ) }-{\frac{5\,x \left ( \cos \left ( x \right ) \right ) ^{2}}{16}}+{\frac{5\,\cos \left ( x \right ) \sin \left ( x \right ) }{32}}-{\frac{5\,{x}^{3}}{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(x)^6,x)

[Out]

x^2*(-1/6*(sin(x)^5+5/4*sin(x)^3+15/8*sin(x))*cos(x)+5/16*x)+1/18*x*sin(x)^6+1/108*(sin(x)^5+5/4*sin(x)^3+15/8
*sin(x))*cos(x)+115/1152*x+5/48*x*sin(x)^4+5/192*(sin(x)^3+3/2*sin(x))*cos(x)-5/16*x*cos(x)^2+5/32*cos(x)*sin(
x)-5/24*x^3

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Maxima [A]  time = 0.942559, size = 89, normalized size = 0.85 \begin{align*} \frac{5}{48} \, x^{3} - \frac{1}{576} \, x \cos \left (6 \, x\right ) + \frac{3}{128} \, x \cos \left (4 \, x\right ) - \frac{15}{64} \, x \cos \left (2 \, x\right ) - \frac{1}{3456} \,{\left (18 \, x^{2} - 1\right )} \sin \left (6 \, x\right ) + \frac{3}{512} \,{\left (8 \, x^{2} - 1\right )} \sin \left (4 \, x\right ) - \frac{15}{128} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(x)^6,x, algorithm="maxima")

[Out]

5/48*x^3 - 1/576*x*cos(6*x) + 3/128*x*cos(4*x) - 15/64*x*cos(2*x) - 1/3456*(18*x^2 - 1)*sin(6*x) + 3/512*(8*x^
2 - 1)*sin(4*x) - 15/128*(2*x^2 - 1)*sin(2*x)

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Fricas [A]  time = 2.31521, size = 240, normalized size = 2.29 \begin{align*} -\frac{1}{18} \, x \cos \left (x\right )^{6} + \frac{13}{48} \, x \cos \left (x\right )^{4} + \frac{5}{48} \, x^{3} - \frac{11}{16} \, x \cos \left (x\right )^{2} - \frac{1}{3456} \,{\left (32 \,{\left (18 \, x^{2} - 1\right )} \cos \left (x\right )^{5} - 2 \,{\left (936 \, x^{2} - 97\right )} \cos \left (x\right )^{3} + 3 \,{\left (792 \, x^{2} - 299\right )} \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{299}{1152} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(x)^6,x, algorithm="fricas")

[Out]

-1/18*x*cos(x)^6 + 13/48*x*cos(x)^4 + 5/48*x^3 - 11/16*x*cos(x)^2 - 1/3456*(32*(18*x^2 - 1)*cos(x)^5 - 2*(936*
x^2 - 97)*cos(x)^3 + 3*(792*x^2 - 299)*cos(x))*sin(x) + 299/1152*x

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Sympy [A]  time = 5.74307, size = 192, normalized size = 1.83 \begin{align*} \frac{5 x^{3} \sin ^{6}{\left (x \right )}}{48} + \frac{5 x^{3} \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac{5 x^{3} \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{16} + \frac{5 x^{3} \cos ^{6}{\left (x \right )}}{48} - \frac{11 x^{2} \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{16} - \frac{5 x^{2} \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac{5 x^{2} \sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{16} + \frac{299 x \sin ^{6}{\left (x \right )}}{1152} + \frac{35 x \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{384} - \frac{125 x \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{384} - \frac{245 x \cos ^{6}{\left (x \right )}}{1152} + \frac{299 \sin ^{5}{\left (x \right )} \cos{\left (x \right )}}{1152} + \frac{25 \sin ^{3}{\left (x \right )} \cos ^{3}{\left (x \right )}}{54} + \frac{245 \sin{\left (x \right )} \cos ^{5}{\left (x \right )}}{1152} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sin(x)**6,x)

[Out]

5*x**3*sin(x)**6/48 + 5*x**3*sin(x)**4*cos(x)**2/16 + 5*x**3*sin(x)**2*cos(x)**4/16 + 5*x**3*cos(x)**6/48 - 11
*x**2*sin(x)**5*cos(x)/16 - 5*x**2*sin(x)**3*cos(x)**3/6 - 5*x**2*sin(x)*cos(x)**5/16 + 299*x*sin(x)**6/1152 +
35*x*sin(x)**4*cos(x)**2/384 - 125*x*sin(x)**2*cos(x)**4/384 - 245*x*cos(x)**6/1152 + 299*sin(x)**5*cos(x)/11
52 + 25*sin(x)**3*cos(x)**3/54 + 245*sin(x)*cos(x)**5/1152

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Giac [A]  time = 1.06228, size = 89, normalized size = 0.85 \begin{align*} \frac{5}{48} \, x^{3} - \frac{1}{576} \, x \cos \left (6 \, x\right ) + \frac{3}{128} \, x \cos \left (4 \, x\right ) - \frac{15}{64} \, x \cos \left (2 \, x\right ) - \frac{1}{3456} \,{\left (18 \, x^{2} - 1\right )} \sin \left (6 \, x\right ) + \frac{3}{512} \,{\left (8 \, x^{2} - 1\right )} \sin \left (4 \, x\right ) - \frac{15}{128} \,{\left (2 \, x^{2} - 1\right )} \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sin(x)^6,x, algorithm="giac")

[Out]

5/48*x^3 - 1/576*x*cos(6*x) + 3/128*x*cos(4*x) - 15/64*x*cos(2*x) - 1/3456*(18*x^2 - 1)*sin(6*x) + 3/512*(8*x^
2 - 1)*sin(4*x) - 15/128*(2*x^2 - 1)*sin(2*x)