3.101.38 \(\int (1-4 x^3+4 x^2 \log (x)+6 x^2 \log ^2(x)-4 x \log ^3(x)-2 x \log ^4(x)) \, dx\)

Optimal. Leaf size=18 \[ -8+x-x^2 \left (-x+\log ^2(x)\right )^2 \]

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Rubi [A]  time = 0.09, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 11, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2304, 2305} \begin {gather*} -x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x)+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 - 4*x^3 + 4*x^2*Log[x] + 6*x^2*Log[x]^2 - 4*x*Log[x]^3 - 2*x*Log[x]^4,x]

[Out]

x - x^4 + 2*x^3*Log[x]^2 - x^2*Log[x]^4

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
]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x-x^4-2 \int x \log ^4(x) \, dx+4 \int x^2 \log (x) \, dx-4 \int x \log ^3(x) \, dx+6 \int x^2 \log ^2(x) \, dx\\ &=x-\frac {4 x^3}{9}-x^4+\frac {4}{3} x^3 \log (x)+2 x^3 \log ^2(x)-2 x^2 \log ^3(x)-x^2 \log ^4(x)-4 \int x^2 \log (x) \, dx+4 \int x \log ^3(x) \, dx+6 \int x \log ^2(x) \, dx\\ &=x-x^4+3 x^2 \log ^2(x)+2 x^3 \log ^2(x)-x^2 \log ^4(x)-6 \int x \log (x) \, dx-6 \int x \log ^2(x) \, dx\\ &=x+\frac {3 x^2}{2}-x^4-3 x^2 \log (x)+2 x^3 \log ^2(x)-x^2 \log ^4(x)+6 \int x \log (x) \, dx\\ &=x-x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.39 \begin {gather*} x-x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 - 4*x^3 + 4*x^2*Log[x] + 6*x^2*Log[x]^2 - 4*x*Log[x]^3 - 2*x*Log[x]^4,x]

[Out]

x - x^4 + 2*x^3*Log[x]^2 - x^2*Log[x]^4

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fricas [A]  time = 1.64, size = 25, normalized size = 1.39 \begin {gather*} -x^{2} \log \relax (x)^{4} + 2 \, x^{3} \log \relax (x)^{2} - x^{4} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x*log(x)^4-4*x*log(x)^3+6*x^2*log(x)^2+4*x^2*log(x)-4*x^3+1,x, algorithm="fricas")

[Out]

-x^2*log(x)^4 + 2*x^3*log(x)^2 - x^4 + x

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giac [A]  time = 0.16, size = 25, normalized size = 1.39 \begin {gather*} -x^{2} \log \relax (x)^{4} + 2 \, x^{3} \log \relax (x)^{2} - x^{4} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x*log(x)^4-4*x*log(x)^3+6*x^2*log(x)^2+4*x^2*log(x)-4*x^3+1,x, algorithm="giac")

[Out]

-x^2*log(x)^4 + 2*x^3*log(x)^2 - x^4 + x

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maple [A]  time = 0.03, size = 26, normalized size = 1.44

method	result	size
default	\(-x^{4}+x -x^{2} \ln \relax (x )^{4}+2 x^{3} \ln \relax (x )^{2}\)	\(26\)
norman	\(-x^{4}+x -x^{2} \ln \relax (x )^{4}+2 x^{3} \ln \relax (x )^{2}\)	\(26\)
risch	\(-x^{4}+x -x^{2} \ln \relax (x )^{4}+2 x^{3} \ln \relax (x )^{2}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2*x*ln(x)^4-4*x*ln(x)^3+6*x^2*ln(x)^2+4*x^2*ln(x)-4*x^3+1,x,method=_RETURNVERBOSE)

[Out]

-x^4+x-x^2*ln(x)^4+2*x^3*ln(x)^2

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maxima [B]  time = 0.35, size = 88, normalized size = 4.89 \begin {gather*} \frac {2}{9} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} - x^{4} + \frac {4}{3} \, x^{3} \log \relax (x) - \frac {1}{2} \, {\left (2 \, \log \relax (x)^{4} - 4 \, \log \relax (x)^{3} + 6 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 3\right )} x^{2} - \frac {1}{2} \, {\left (4 \, \log \relax (x)^{3} - 6 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 3\right )} x^{2} - \frac {4}{9} \, x^{3} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x*log(x)^4-4*x*log(x)^3+6*x^2*log(x)^2+4*x^2*log(x)-4*x^3+1,x, algorithm="maxima")

[Out]

2/9*(9*log(x)^2 - 6*log(x) + 2)*x^3 - x^4 + 4/3*x^3*log(x) - 1/2*(2*log(x)^4 - 4*log(x)^3 + 6*log(x)^2 - 6*log
(x) + 3)*x^2 - 1/2*(4*log(x)^3 - 6*log(x)^2 + 6*log(x) - 3)*x^2 - 4/9*x^3 + x

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mupad [B]  time = 7.06, size = 25, normalized size = 1.39 \begin {gather*} -x^4+2\,x^3\,{\ln \relax (x)}^2-x^2\,{\ln \relax (x)}^4+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x^2*log(x) - 4*x*log(x)^3 - 2*x*log(x)^4 + 6*x^2*log(x)^2 - 4*x^3 + 1,x)

[Out]

x + 2*x^3*log(x)^2 - x^2*log(x)^4 - x^4

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sympy [A]  time = 0.11, size = 22, normalized size = 1.22 \begin {gather*} - x^{4} + 2 x^{3} \log {\relax (x )}^{2} - x^{2} \log {\relax (x )}^{4} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2*x*ln(x)**4-4*x*ln(x)**3+6*x**2*ln(x)**2+4*x**2*ln(x)-4*x**3+1,x)

[Out]

-x**4 + 2*x**3*log(x)**2 - x**2*log(x)**4 + x

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