\(\int x^3 \log ^3(c x) \, dx\) [15]

Optimal result

Integrand size = 10, antiderivative size = 45 \[ \int x^3 \log ^3(c x) \, dx=-\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (c x)-\frac {3}{16} x^4 \log ^2(c x)+\frac {1}{4} x^4 \log ^3(c x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2342, 2341} \[ \int x^3 \log ^3(c x) \, dx=\frac {1}{4} x^4 \log ^3(c x)-\frac {3}{16} x^4 \log ^2(c x)+\frac {3}{32} x^4 \log (c x)-\frac {3 x^4}{128} \]

[In]

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

[Out]

(-3*x^4)/128 + (3*x^4*Log[c*x])/32 - (3*x^4*Log[c*x]^2)/16 + (x^4*Log[c*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(c x)-\frac {3}{4} \int x^3 \log ^2(c x) \, dx \\ & = -\frac {3}{16} x^4 \log ^2(c x)+\frac {1}{4} x^4 \log ^3(c x)+\frac {3}{8} \int x^3 \log (c x) \, dx \\ & = -\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (c x)-\frac {3}{16} x^4 \log ^2(c x)+\frac {1}{4} x^4 \log ^3(c x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x^3 \log ^3(c x) \, dx=-\frac {3 x^4}{128}+\frac {3}{32} x^4 \log (c x)-\frac {3}{16} x^4 \log ^2(c x)+\frac {1}{4} x^4 \log ^3(c x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84

method	result	size
norman	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x c \right )}{32}-\frac {3 x^{4} \ln \left (x c \right )^{2}}{16}+\frac {x^{4} \ln \left (x c \right )^{3}}{4}\)	\(38\)
risch	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x c \right )}{32}-\frac {3 x^{4} \ln \left (x c \right )^{2}}{16}+\frac {x^{4} \ln \left (x c \right )^{3}}{4}\)	\(38\)
parallelrisch	\(-\frac {3 x^{4}}{128}+\frac {3 x^{4} \ln \left (x c \right )}{32}-\frac {3 x^{4} \ln \left (x c \right )^{2}}{16}+\frac {x^{4} \ln \left (x c \right )^{3}}{4}\)	\(38\)
parts	\(\frac {x^{4} \ln \left (x c \right )^{3}}{4}-\frac {3 \left (\frac {x^{4} c^{4} \ln \left (x c \right )^{2}}{4}-\frac {x^{4} c^{4} \ln \left (x c \right )}{8}+\frac {x^{4} c^{4}}{32}\right )}{4 c^{4}}\)	\(53\)
derivativedivides	\(\frac {\frac {x^{4} c^{4} \ln \left (x c \right )^{3}}{4}-\frac {3 x^{4} c^{4} \ln \left (x c \right )^{2}}{16}+\frac {3 x^{4} c^{4} \ln \left (x c \right )}{32}-\frac {3 x^{4} c^{4}}{128}}{c^{4}}\)	\(54\)
default	\(\frac {\frac {x^{4} c^{4} \ln \left (x c \right )^{3}}{4}-\frac {3 x^{4} c^{4} \ln \left (x c \right )^{2}}{16}+\frac {3 x^{4} c^{4} \ln \left (x c \right )}{32}-\frac {3 x^{4} c^{4}}{128}}{c^{4}}\)	\(54\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x^3 \log ^3(c x) \, dx=\frac {1}{4} \, x^{4} \log \left (c x\right )^{3} - \frac {3}{16} \, x^{4} \log \left (c x\right )^{2} + \frac {3}{32} \, x^{4} \log \left (c x\right ) - \frac {3}{128} \, x^{4} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x^3 \log ^3(c x) \, dx=\frac {x^{4} \log {\left (c x \right )}^{3}}{4} - \frac {3 x^{4} \log {\left (c x \right )}^{2}}{16} + \frac {3 x^{4} \log {\left (c x \right )}}{32} - \frac {3 x^{4}}{128} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x^3 \log ^3(c x) \, dx=\frac {1}{128} \, {\left (32 \, \log \left (c x\right )^{3} - 24 \, \log \left (c x\right )^{2} + 12 \, \log \left (c x\right ) - 3\right )} x^{4} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x^3 \log ^3(c x) \, dx=\frac {1}{4} \, x^{4} \log \left (c x\right )^{3} - \frac {3}{16} \, x^{4} \log \left (c x\right )^{2} + \frac {3}{32} \, x^{4} \log \left (c x\right ) - \frac {3}{128} \, x^{4} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x^3 \log ^3(c x) \, dx=\frac {x^4\,\left (32\,{\ln \left (c\,x\right )}^3-24\,{\ln \left (c\,x\right )}^2+12\,\ln \left (c\,x\right )-3\right )}{128} \]

[In]

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

[Out]

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