\(\int x \log ^3(c x) \, dx\) [17]

Optimal result

Integrand size = 8, antiderivative size = 45 \[ \int x \log ^3(c x) \, dx=-\frac {3 x^2}{8}+\frac {3}{4} x^2 \log (c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x) \]

[Out]

-3/8*x^2+3/4*x^2*ln(c*x)-3/4*x^2*ln(c*x)^2+1/2*x^2*ln(c*x)^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \[ \int x \log ^3(c x) \, dx=\frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

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}{2} x^2 \log ^3(c x)-\frac {3}{2} \int x \log ^2(c x) \, dx \\ & = -\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x)+\frac {3}{2} \int x \log (c x) \, dx \\ & = -\frac {3 x^2}{8}+\frac {3}{4} x^2 \log (c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int x \log ^3(c x) \, dx=-\frac {3 x^2}{8}+\frac {3}{4} x^2 \log (c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x) \]

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

-3/8*x^2+3/4*x^2*ln(x*c)-3/4*x^2*ln(x*c)^2+1/2*x^2*ln(x*c)^3

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x \log ^3(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int x \log ^3(c x) \, dx=\frac {x^{2} \log {\left (c x \right )}^{3}}{2} - \frac {3 x^{2} \log {\left (c x \right )}^{2}}{4} + \frac {3 x^{2} \log {\left (c x \right )}}{4} - \frac {3 x^{2}}{8} \]

[In]

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

[Out]

x**2*log(c*x)**3/2 - 3*x**2*log(c*x)**2/4 + 3*x**2*log(c*x)/4 - 3*x**2/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x \log ^3(c x) \, dx=\frac {1}{8} \, {\left (4 \, \log \left (c x\right )^{3} - 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 3\right )} x^{2} \]

[In]

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

[Out]

1/8*(4*log(c*x)^3 - 6*log(c*x)^2 + 6*log(c*x) - 3)*x^2

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int x \log ^3(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.64 \[ \int x \log ^3(c x) \, dx=\frac {x^2\,\left (4\,{\ln \left (c\,x\right )}^3-6\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-3\right )}{8} \]

[In]

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

[Out]

(x^2*(6*log(c*x) - 6*log(c*x)^2 + 4*log(c*x)^3 - 3))/8