\(\int x \log ^2(c x) \, dx\) [10]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \[ \int x \log ^2(c x) \, dx=\frac {1}{2} x^2 \log ^2(c x)-\frac {1}{2} x^2 \log (c x)+\frac {x^2}{4} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

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

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
norman	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\)	\(27\)
risch	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\)	\(27\)
parallelrisch	\(\frac {x^{2}}{4}-\frac {x^{2} \ln \left (x c \right )}{2}+\frac {x^{2} \ln \left (x c \right )^{2}}{2}\)	\(27\)
parts	\(\frac {x^{2} \ln \left (x c \right )^{2}}{2}-\frac {\frac {x^{2} c^{2} \ln \left (x c \right )}{2}-\frac {x^{2} c^{2}}{4}}{c^{2}}\)	\(39\)
derivativedivides	\(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )^{2}}{2}-\frac {x^{2} c^{2} \ln \left (x c \right )}{2}+\frac {x^{2} c^{2}}{4}}{c^{2}}\)	\(40\)
default	\(\frac {\frac {x^{2} c^{2} \ln \left (x c \right )^{2}}{2}-\frac {x^{2} c^{2} \ln \left (x c \right )}{2}+\frac {x^{2} c^{2}}{4}}{c^{2}}\)	\(40\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {x^{2} \log {\left (c x \right )}^{2}}{2} - \frac {x^{2} \log {\left (c x \right )}}{2} + \frac {x^{2}}{4} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x \log ^2(c x) \, dx=\frac {1}{4} \, {\left (2 \, \log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 1\right )} x^{2} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x \log ^2(c x) \, dx=\frac {1}{2} \, x^{2} \log \left (c x\right )^{2} - \frac {1}{2} \, x^{2} \log \left (c x\right ) + \frac {1}{4} \, x^{2} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66 \[ \int x \log ^2(c x) \, dx=\frac {x^2\,\left (2\,{\ln \left (c\,x\right )}^2-2\,\ln \left (c\,x\right )+1\right )}{4} \]

[In]

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

[Out]

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