3.613 \(\int (-1-8 \log ^2(x)+3 \log ^3(x)) \, dx\)

Optimal. Leaf size=23 \[ -35 x+3 x \log ^3(x)-17 x \log ^2(x)+34 x \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2296, 2295} \[ -35 x+3 x \log ^3(x)-17 x \log ^2(x)+34 x \log (x) \]

Antiderivative was successfully verified.

[In]

Int[-1 - 8*Log[x]^2 + 3*Log[x]^3,x]

[Out]

-35*x + 34*x*Log[x] - 17*x*Log[x]^2 + 3*x*Log[x]^3

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx &=-x+3 \int \log ^3(x) \, dx-8 \int \log ^2(x) \, dx\\ &=-x-8 x \log ^2(x)+3 x \log ^3(x)-9 \int \log ^2(x) \, dx+16 \int \log (x) \, dx\\ &=-17 x+16 x \log (x)-17 x \log ^2(x)+3 x \log ^3(x)+18 \int \log (x) \, dx\\ &=-35 x+34 x \log (x)-17 x \log ^2(x)+3 x \log ^3(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \[ -35 x+3 x \log ^3(x)-17 x \log ^2(x)+34 x \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[-1 - 8*Log[x]^2 + 3*Log[x]^3,x]

[Out]

-35*x + 34*x*Log[x] - 17*x*Log[x]^2 + 3*x*Log[x]^3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[-1 - 8*Log[x]^2 + 3*Log[x]^3,x]

[Out]

Could not integrate

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fricas [A]  time = 0.80, size = 23, normalized size = 1.00 \[ 3 \, x \log \relax (x)^{3} - 17 \, x \log \relax (x)^{2} + 34 \, x \log \relax (x) - 35 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1-8*log(x)^2+3*log(x)^3,x, algorithm="fricas")

[Out]

3*x*log(x)^3 - 17*x*log(x)^2 + 34*x*log(x) - 35*x

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giac [A]  time = 0.58, size = 23, normalized size = 1.00 \[ 3 \, x \log \relax (x)^{3} - 17 \, x \log \relax (x)^{2} + 34 \, x \log \relax (x) - 35 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1-8*log(x)^2+3*log(x)^3,x, algorithm="giac")

[Out]

3*x*log(x)^3 - 17*x*log(x)^2 + 34*x*log(x) - 35*x

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maple [A]  time = 0.02, size = 24, normalized size = 1.04

method	result	size
default	\(-35 x +34 x \ln \relax (x )-17 x \ln \relax (x )^{2}+3 x \ln \relax (x )^{3}\)	\(24\)
norman	\(-35 x +34 x \ln \relax (x )-17 x \ln \relax (x )^{2}+3 x \ln \relax (x )^{3}\)	\(24\)
risch	\(-35 x +34 x \ln \relax (x )-17 x \ln \relax (x )^{2}+3 x \ln \relax (x )^{3}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1-8*ln(x)^2+3*ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

-35*x+34*x*ln(x)-17*x*ln(x)^2+3*x*ln(x)^3

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maxima [A]  time = 0.47, size = 36, normalized size = 1.57 \[ 3 \, {\left (\log \relax (x)^{3} - 3 \, \log \relax (x)^{2} + 6 \, \log \relax (x) - 6\right )} x - 8 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1-8*log(x)^2+3*log(x)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.29, size = 20, normalized size = 0.87 \[ x\,\left (3\,{\ln \relax (x)}^3-17\,{\ln \relax (x)}^2+34\,\ln \relax (x)-35\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*log(x)^3 - 8*log(x)^2 - 1,x)

[Out]

x*(34*log(x) - 17*log(x)^2 + 3*log(x)^3 - 35)

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sympy [A]  time = 0.11, size = 26, normalized size = 1.13 \[ 3 x \log {\relax (x )}^{3} - 17 x \log {\relax (x )}^{2} + 34 x \log {\relax (x )} - 35 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1-8*ln(x)**2+3*ln(x)**3,x)

[Out]

3*x*log(x)**3 - 17*x*log(x)**2 + 34*x*log(x) - 35*x

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