3.2.1 \(\int x \cos ^3(x) \, dx\) [101]

Optimal. Leaf size=33 \[ \frac {2 \cos (x)}{3}+\frac {\cos ^3(x)}{9}+\frac {2}{3} x \sin (x)+\frac {1}{3} x \cos ^2(x) \sin (x) \]

[Out]

2/3*cos(x)+1/9*cos(x)^3+2/3*x*sin(x)+1/3*x*cos(x)^2*sin(x)

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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3391, 3377, 2718} \begin {gather*} \frac {2}{3} x \sin (x)+\frac {\cos ^3(x)}{9}+\frac {2 \cos (x)}{3}+\frac {1}{3} x \sin (x) \cos ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*Cos[x]^3,x]

[Out]

(2*Cos[x])/3 + Cos[x]^3/9 + (2*x*Sin[x])/3 + (x*Cos[x]^2*Sin[x])/3

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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*} \int x \cos ^3(x) \, dx &=\frac {\cos ^3(x)}{9}+\frac {1}{3} x \cos ^2(x) \sin (x)+\frac {2}{3} \int x \cos (x) \, dx\\ &=\frac {\cos ^3(x)}{9}+\frac {2}{3} x \sin (x)+\frac {1}{3} x \cos ^2(x) \sin (x)-\frac {2}{3} \int \sin (x) \, dx\\ &=\frac {2 \cos (x)}{3}+\frac {\cos ^3(x)}{9}+\frac {2}{3} x \sin (x)+\frac {1}{3} x \cos ^2(x) \sin (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.94 \begin {gather*} \frac {3 \cos (x)}{4}+\frac {1}{36} \cos (3 x)+\frac {3}{4} x \sin (x)+\frac {1}{12} x \sin (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[x]^3,x]

[Out]

(3*Cos[x])/4 + Cos[3*x]/36 + (3*x*Sin[x])/4 + (x*Sin[3*x])/12

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Maple [A]
time = 0.03, size = 23, normalized size = 0.70

method	result	size
default	\(\frac {x \left (2+\cos ^{2}\left (x \right )\right ) \sin \left (x \right )}{3}+\frac {\left (\cos ^{3}\left (x \right )\right )}{9}+\frac {2 \cos \left (x \right )}{3}\)	\(23\)
risch	\(\frac {3 \cos \left (x \right )}{4}+\frac {3 x \sin \left (x \right )}{4}+\frac {\cos \left (3 x \right )}{36}+\frac {x \sin \left (3 x \right )}{12}\)	\(24\)
norman	\(\frac {-\frac {2 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3}-\frac {8 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{9}+2 x \tan \left (\frac {x}{2}\right )+\frac {4 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3}+2 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+\frac {2}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{3}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)^3,x,method=_RETURNVERBOSE)

[Out]

1/3*x*(2+cos(x)^2)*sin(x)+1/9*cos(x)^3+2/3*cos(x)

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Maxima [A]
time = 1.38, size = 23, normalized size = 0.70 \begin {gather*} \frac {1}{12} \, x \sin \left (3 \, x\right ) + \frac {3}{4} \, x \sin \left (x\right ) + \frac {1}{36} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x*sin(3*x) + 3/4*x*sin(x) + 1/36*cos(3*x) + 3/4*cos(x)

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Fricas [A]
time = 0.47, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{9} \, \cos \left (x\right )^{3} + \frac {1}{3} \, {\left (x \cos \left (x\right )^{2} + 2 \, x\right )} \sin \left (x\right ) + \frac {2}{3} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*cos(x)^3 + 1/3*(x*cos(x)^2 + 2*x)*sin(x) + 2/3*cos(x)

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Sympy [A]
time = 0.13, size = 39, normalized size = 1.18 \begin {gather*} \frac {2 x \sin ^{3}{\left (x \right )}}{3} + x \sin {\left (x \right )} \cos ^{2}{\left (x \right )} + \frac {2 \sin ^{2}{\left (x \right )} \cos {\left (x \right )}}{3} + \frac {7 \cos ^{3}{\left (x \right )}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(x)**3,x)

[Out]

2*x*sin(x)**3/3 + x*sin(x)*cos(x)**2 + 2*sin(x)**2*cos(x)/3 + 7*cos(x)**3/9

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Giac [A]
time = 0.63, size = 23, normalized size = 0.70 \begin {gather*} \frac {1}{12} \, x \sin \left (3 \, x\right ) + \frac {3}{4} \, x \sin \left (x\right ) + \frac {1}{36} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x*sin(3*x) + 3/4*x*sin(x) + 1/36*cos(3*x) + 3/4*cos(x)

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Mupad [B]
time = 0.17, size = 25, normalized size = 0.76 \begin {gather*} \frac {{\cos \left (x\right )}^3}{9}+\frac {x\,\sin \left (x\right )\,{\cos \left (x\right )}^2}{3}+\frac {2\,\cos \left (x\right )}{3}+\frac {2\,x\,\sin \left (x\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cos(x)^3,x)

[Out]

(2*cos(x))/3 + cos(x)^3/9 + (2*x*sin(x))/3 + (x*cos(x)^2*sin(x))/3

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