\(\int (-1-8 \log ^2(x)+3 \log ^3(x)) \, dx\) [613]

Optimal result

Integrand size = 14, antiderivative size = 23 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=-35 x+34 x \log (x)-17 x \log ^2(x)+3 x \log ^3(x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2333, 2332} \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=-35 x+3 x \log ^3(x)-17 x \log ^2(x)+34 x \log (x) \]

[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 2332

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

Rule 2333

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*} \text {integral}& = -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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=-35 x+34 x \log (x)-17 x \log ^2(x)+3 x \log ^3(x) \]

[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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=3 \, x \log \left (x\right )^{3} - 17 \, x \log \left (x\right )^{2} + 34 \, x \log \left (x\right ) - 35 \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=3 x \log {\left (x \right )}^{3} - 17 x \log {\left (x \right )}^{2} + 34 x \log {\left (x \right )} - 35 x \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=3 \, {\left (\log \left (x\right )^{3} - 3 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 6\right )} x - 8 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=3 \, x \log \left (x\right )^{3} - 17 \, x \log \left (x\right )^{2} + 34 \, x \log \left (x\right ) - 35 \, x \]

[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

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \left (-1-8 \log ^2(x)+3 \log ^3(x)\right ) \, dx=x\,\left (3\,{\ln \left (x\right )}^3-17\,{\ln \left (x\right )}^2+34\,\ln \left (x\right )-35\right ) \]

[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)