\(\int x^3 \log ^3(x) \, dx\) [67]

Optimal result

Integrand size = 8, antiderivative size = 39 \[ \int x^3 \log ^3(x) \, dx=-\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (x)-\frac {3}{16} x^4 \log ^2(x)+\frac {1}{4} x^4 \log ^3(x) \]

[Out]

-3/128*x^4+3/32*x^4*ln(x)-3/16*x^4*ln(x)^2+1/4*x^4*ln(x)^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \[ \int x^3 \log ^3(x) \, dx=-\frac {3 x^4}{128}+\frac {1}{4} x^4 \log ^3(x)-\frac {3}{16} x^4 \log ^2(x)+\frac {3}{32} x^4 \log (x) \]

[In]

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

[Out]

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

Rule 2341

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

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \log ^3(x)-\frac {3}{4} \int x^3 \log ^2(x) \, dx \\ & = -\frac {3}{16} x^4 \log ^2(x)+\frac {1}{4} x^4 \log ^3(x)+\frac {3}{8} \int x^3 \log (x) \, dx \\ & = -\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (x)-\frac {3}{16} x^4 \log ^2(x)+\frac {1}{4} x^4 \log ^3(x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int x^3 \log ^3(x) \, dx=-\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (x)-\frac {3}{16} x^4 \log ^2(x)+\frac {1}{4} x^4 \log ^3(x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x \right )}{32}-\frac {3 x^{4} \ln \left (x \right )^{2}}{16}+\frac {x^{4} \ln \left (x \right )^{3}}{4}\)	\(32\)
norman	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x \right )}{32}-\frac {3 x^{4} \ln \left (x \right )^{2}}{16}+\frac {x^{4} \ln \left (x \right )^{3}}{4}\)	\(32\)
risch	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x \right )}{32}-\frac {3 x^{4} \ln \left (x \right )^{2}}{16}+\frac {x^{4} \ln \left (x \right )^{3}}{4}\)	\(32\)
parallelrisch	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x \right )}{32}-\frac {3 x^{4} \ln \left (x \right )^{2}}{16}+\frac {x^{4} \ln \left (x \right )^{3}}{4}\)	\(32\)
parts	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x \right )}{32}-\frac {3 x^{4} \ln \left (x \right )^{2}}{16}+\frac {x^{4} \ln \left (x \right )^{3}}{4}\)	\(32\)

[In]

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

[Out]

-3/128*x^4+3/32*x^4*ln(x)-3/16*x^4*ln(x)^2+1/4*x^4*ln(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int x^3 \log ^3(x) \, dx=\frac {1}{4} \, x^{4} \log \left (x\right )^{3} - \frac {3}{16} \, x^{4} \log \left (x\right )^{2} + \frac {3}{32} \, x^{4} \log \left (x\right ) - \frac {3}{128} \, x^{4} \]

[In]

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

[Out]

1/4*x^4*log(x)^3 - 3/16*x^4*log(x)^2 + 3/32*x^4*log(x) - 3/128*x^4

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int x^3 \log ^3(x) \, dx=\frac {x^{4} \log {\left (x \right )}^{3}}{4} - \frac {3 x^{4} \log {\left (x \right )}^{2}}{16} + \frac {3 x^{4} \log {\left (x \right )}}{32} - \frac {3 x^{4}}{128} \]

[In]

integrate(x**3*ln(x)**3,x)

[Out]

x**4*log(x)**3/4 - 3*x**4*log(x)**2/16 + 3*x**4*log(x)/32 - 3*x**4/128

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.59 \[ \int x^3 \log ^3(x) \, dx=\frac {1}{128} \, {\left (32 \, \log \left (x\right )^{3} - 24 \, \log \left (x\right )^{2} + 12 \, \log \left (x\right ) - 3\right )} x^{4} \]

[In]

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

[Out]

1/128*(32*log(x)^3 - 24*log(x)^2 + 12*log(x) - 3)*x^4

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.79 \[ \int x^3 \log ^3(x) \, dx=\frac {1}{4} \, x^{4} \log \left (x\right )^{3} - \frac {3}{16} \, x^{4} \log \left (x\right )^{2} + \frac {3}{32} \, x^{4} \log \left (x\right ) - \frac {3}{128} \, x^{4} \]

[In]

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

[Out]

1/4*x^4*log(x)^3 - 3/16*x^4*log(x)^2 + 3/32*x^4*log(x) - 3/128*x^4

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.59 \[ \int x^3 \log ^3(x) \, dx=\frac {3\,x^4\,\left (\frac {32\,{\ln \left (x\right )}^3}{3}-8\,{\ln \left (x\right )}^2+4\,\ln \left (x\right )-1\right )}{128} \]

[In]

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

[Out]

(3*x^4*(4*log(x) - 8*log(x)^2 + (32*log(x)^3)/3 - 1))/128