\(\int x \sin ^4(x) \, dx\) [300]

Optimal result

Integrand size = 6, antiderivative size = 44 \[ \int x \sin ^4(x) \, dx=\frac {3 x^2}{16}-\frac {3}{8} x \cos (x) \sin (x)+\frac {3 \sin ^2(x)}{16}-\frac {1}{4} x \cos (x) \sin ^3(x)+\frac {\sin ^4(x)}{16} \]

[Out]

3/16*x^2-3/8*x*cos(x)*sin(x)+3/16*sin(x)^2-1/4*x*cos(x)*sin(x)^3+1/16*sin(x)^4

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3391, 30} \[ \int x \sin ^4(x) \, dx=\frac {3 x^2}{16}+\frac {\sin ^4(x)}{16}+\frac {3 \sin ^2(x)}{16}-\frac {1}{4} x \sin ^3(x) \cos (x)-\frac {3}{8} x \sin (x) \cos (x) \]

[In]

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

[Out]

(3*x^2)/16 - (3*x*Cos[x]*Sin[x])/8 + (3*Sin[x]^2)/16 - (x*Cos[x]*Sin[x]^3)/4 + Sin[x]^4/16

Rule 30

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

Rule 3391

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{4} x \cos (x) \sin ^3(x)+\frac {\sin ^4(x)}{16}+\frac {3}{4} \int x \sin ^2(x) \, dx \\ & = -\frac {3}{8} x \cos (x) \sin (x)+\frac {3 \sin ^2(x)}{16}-\frac {1}{4} x \cos (x) \sin ^3(x)+\frac {\sin ^4(x)}{16}+\frac {3 \int x \, dx}{8} \\ & = \frac {3 x^2}{16}-\frac {3}{8} x \cos (x) \sin (x)+\frac {3 \sin ^2(x)}{16}-\frac {1}{4} x \cos (x) \sin ^3(x)+\frac {\sin ^4(x)}{16} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int x \sin ^4(x) \, dx=\frac {3 x^2}{16}-\frac {1}{8} \cos (2 x)+\frac {1}{128} \cos (4 x)-\frac {1}{4} x \sin (2 x)+\frac {1}{32} x \sin (4 x) \]

[In]

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

[Out]

(3*x^2)/16 - Cos[2*x]/8 + Cos[4*x]/128 - (x*Sin[2*x])/4 + (x*Sin[4*x])/32

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.75

method	result	size
risch	\(\frac {3 x^{2}}{16}+\frac {\cos \left (4 x \right )}{128}+\frac {x \sin \left (4 x \right )}{32}-\frac {\cos \left (2 x \right )}{8}-\frac {x \sin \left (2 x \right )}{4}\)	\(33\)
default	\(x \left (-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{4}+\frac {3 x}{8}\right )-\frac {3 x^{2}}{16}+\frac {\left (2 \cos \left (x \right )^{2}-5\right )^{2}}{64}\)	\(38\)
norman	\(\frac {\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{4}+\frac {3 \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {5 \tan \left (\frac {x}{2}\right )^{4}}{2}+\frac {3 x^{2}}{16}-\frac {3 x \tan \left (\frac {x}{2}\right )}{4}-\frac {11 x \tan \left (\frac {x}{2}\right )^{3}}{4}+\frac {11 x \tan \left (\frac {x}{2}\right )^{5}}{4}+\frac {3 x \tan \left (\frac {x}{2}\right )^{7}}{4}+\frac {3 x^{2} \tan \left (\frac {x}{2}\right )^{2}}{4}+\frac {9 x^{2} \tan \left (\frac {x}{2}\right )^{4}}{8}+\frac {3 x^{2} \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {3 x^{2} \tan \left (\frac {x}{2}\right )^{8}}{16}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{4}}\)	\(120\)

[In]

int(x*sin(x)^4,x,method=_RETURNVERBOSE)

[Out]

3/16*x^2+1/128*cos(4*x)+1/32*x*sin(4*x)-1/8*cos(2*x)-1/4*x*sin(2*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.80 \[ \int x \sin ^4(x) \, dx=\frac {1}{16} \, \cos \left (x\right )^{4} + \frac {3}{16} \, x^{2} - \frac {5}{16} \, \cos \left (x\right )^{2} + \frac {1}{8} \, {\left (2 \, x \cos \left (x\right )^{3} - 5 \, x \cos \left (x\right )\right )} \sin \left (x\right ) \]

[In]

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

[Out]

1/16*cos(x)^4 + 3/16*x^2 - 5/16*cos(x)^2 + 1/8*(2*x*cos(x)^3 - 5*x*cos(x))*sin(x)

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.89 \[ \int x \sin ^4(x) \, dx=\frac {3 x^{2} \sin ^{4}{\left (x \right )}}{16} + \frac {3 x^{2} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{8} + \frac {3 x^{2} \cos ^{4}{\left (x \right )}}{16} - \frac {5 x \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{8} - \frac {3 x \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{8} + \frac {5 \sin ^{4}{\left (x \right )}}{32} - \frac {3 \cos ^{4}{\left (x \right )}}{32} \]

[In]

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

[Out]

3*x**2*sin(x)**4/16 + 3*x**2*sin(x)**2*cos(x)**2/8 + 3*x**2*cos(x)**4/16 - 5*x*sin(x)**3*cos(x)/8 - 3*x*sin(x)
*cos(x)**3/8 + 5*sin(x)**4/32 - 3*cos(x)**4/32

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int x \sin ^4(x) \, dx=\frac {3}{16} \, x^{2} + \frac {1}{32} \, x \sin \left (4 \, x\right ) - \frac {1}{4} \, x \sin \left (2 \, x\right ) + \frac {1}{128} \, \cos \left (4 \, x\right ) - \frac {1}{8} \, \cos \left (2 \, x\right ) \]

[In]

integrate(x*sin(x)^4,x, algorithm="maxima")

[Out]

3/16*x^2 + 1/32*x*sin(4*x) - 1/4*x*sin(2*x) + 1/128*cos(4*x) - 1/8*cos(2*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.73 \[ \int x \sin ^4(x) \, dx=\frac {3}{16} \, x^{2} + \frac {1}{32} \, x \sin \left (4 \, x\right ) - \frac {1}{4} \, x \sin \left (2 \, x\right ) + \frac {1}{128} \, \cos \left (4 \, x\right ) - \frac {1}{8} \, \cos \left (2 \, x\right ) \]

[In]

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

[Out]

3/16*x^2 + 1/32*x*sin(4*x) - 1/4*x*sin(2*x) + 1/128*cos(4*x) - 1/8*cos(2*x)

Mupad [B] (verification not implemented)

Time = 16.93 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.86 \[ \int x \sin ^4(x) \, dx=\frac {{\cos \left (2\,x\right )}^2}{64}-\frac {x\,\sin \left (2\,x\right )}{4}-\frac {\cos \left (2\,x\right )}{8}+\frac {3\,x^2}{16}+\frac {x\,\cos \left (2\,x\right )\,\sin \left (2\,x\right )}{16} \]

[In]

int(x*sin(x)^4,x)

[Out]

cos(2*x)^2/64 - (x*sin(2*x))/4 - cos(2*x)/8 + (3*x^2)/16 + (x*cos(2*x)*sin(2*x))/16