3.10 \(\int x \log ^2(c x) \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]

[Out]

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

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Rubi [A]  time = 0.0108939, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2305, 2304} \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]

Antiderivative was successfully verified.

[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 2305

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]

Rule 2304

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
]

Rubi steps

\begin{align*} \int x \log ^2(c x) \, dx &=\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]  time = 0.0009494, size = 32, normalized size = 1. \[ \frac{1}{2} x^2 \log ^2(c x)-\frac{1}{2} x^2 \log (c x)+\frac{x^2}{4} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.036, size = 27, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}}-{\frac{{x}^{2}\ln \left ( cx \right ) }{2}}+{\frac{{x}^{2} \left ( \ln \left ( cx \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.977959, size = 28, normalized size = 0.88 \begin{align*} \frac{1}{4} \,{\left (2 \, \log \left (c x\right )^{2} - 2 \, \log \left (c x\right ) + 1\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 0.744731, size = 66, normalized size = 2.06 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.112876, size = 26, normalized size = 0.81 \begin{align*} \frac{x^{2} \log{\left (c x \right )}^{2}}{2} - \frac{x^{2} \log{\left (c x \right )}}{2} + \frac{x^{2}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.10867, size = 35, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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