\(\int x^2 \sin (\log (x)) \, dx\) [263]

Optimal result

Integrand size = 7, antiderivative size = 21 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {1}{10} x^3 \cos (\log (x))+\frac {3}{10} x^3 \sin (\log (x)) \]

[Out]

-1/10*x^3*cos(ln(x))+3/10*x^3*sin(ln(x))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4573} \[ \int x^2 \sin (\log (x)) \, dx=\frac {3}{10} x^3 \sin (\log (x))-\frac {1}{10} x^3 \cos (\log (x)) \]

[In]

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

[Out]

-1/10*(x^3*Cos[Log[x]]) + (3*x^3*Sin[Log[x]])/10

Rule 4573

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[(m + 1)*(e*x)^(m +
1)*(Sin[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)), x] - Simp[b*d*n*(e*x)^(m + 1)*(Cos[d*(a + b*Log[
c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{10} x^3 \cos (\log (x))+\frac {3}{10} x^3 \sin (\log (x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {1}{10} x^3 \cos (\log (x))+\frac {3}{10} x^3 \sin (\log (x)) \]

[In]

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

[Out]

-1/10*(x^3*Cos[Log[x]]) + (3*x^3*Sin[Log[x]])/10

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24

method	result	size
risch	\(\left (-\frac {1}{20}-\frac {3 i}{20}\right ) x^{3} x^{i}+\left (-\frac {1}{20}+\frac {3 i}{20}\right ) x^{3} x^{-i}\)	\(26\)
norman	\(\frac {-\frac {x^{3}}{10}+\frac {3 x^{3} \tan \left (\frac {\ln \left (x \right )}{2}\right )}{5}+\frac {x^{3} \tan \left (\frac {\ln \left (x \right )}{2}\right )^{2}}{10}}{1+\tan \left (\frac {\ln \left (x \right )}{2}\right )^{2}}\)	\(41\)

[In]

int(x^2*sin(ln(x)),x,method=_RETURNVERBOSE)

[Out]

(-1/20-3/20*I)*x^3*x^I+(-1/20+3/20*I)*x^3/(x^I)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {1}{10} \, x^{3} \cos \left (\log \left (x\right )\right ) + \frac {3}{10} \, x^{3} \sin \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

-1/10*x^3*cos(log(x)) + 3/10*x^3*sin(log(x))

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int x^2 \sin (\log (x)) \, dx=\frac {3 x^{3} \sin {\left (\log {\left (x \right )} \right )}}{10} - \frac {x^{3} \cos {\left (\log {\left (x \right )} \right )}}{10} \]

[In]

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

[Out]

3*x**3*sin(log(x))/10 - x**3*cos(log(x))/10

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {1}{10} \, x^{3} {\left (\cos \left (\log \left (x\right )\right ) - 3 \, \sin \left (\log \left (x\right )\right )\right )} \]

[In]

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

[Out]

-1/10*x^3*(cos(log(x)) - 3*sin(log(x)))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {1}{10} \, x^{3} \cos \left (\log \left (x\right )\right ) + \frac {3}{10} \, x^{3} \sin \left (\log \left (x\right )\right ) \]

[In]

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

[Out]

-1/10*x^3*cos(log(x)) + 3/10*x^3*sin(log(x))

Mupad [B] (verification not implemented)

Time = 16.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67 \[ \int x^2 \sin (\log (x)) \, dx=-\frac {\sqrt {10}\,x^3\,\cos \left (\mathrm {atan}\left (3\right )+\ln \left (x\right )\right )}{10} \]

[In]

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

[Out]

-(10^(1/2)*x^3*cos(atan(3) + log(x)))/10