3.72.55 \(\int \frac {6-3 x-2 x^2+88 x^4+40 x^5-170 x^6-70 x^7-128 x^8+400 x^{10}-300 x^{12}+(-2+2 x-32 x^4+60 x^6) \log (x)}{x} \, dx\)

Optimal. Leaf size=24 \[ x-\left (3+x+x^4 \left (-4+5 x^2\right )-\log (x)\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.10, antiderivative size = 73, normalized size of antiderivative = 3.04, number of steps used = 10, number of rules used = 5, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {14, 2357, 2295, 2301, 2304} \begin {gather*} -25 x^{12}+40 x^{10}-16 x^8-10 x^7-30 x^6+10 x^6 \log (x)+8 x^5+24 x^4-8 x^4 \log (x)-x^2-5 x-\log ^2(x)+2 x \log (x)+6 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - 3*x - 2*x^2 + 88*x^4 + 40*x^5 - 170*x^6 - 70*x^7 - 128*x^8 + 400*x^10 - 300*x^12 + (-2 + 2*x - 32*x^4
+ 60*x^6)*Log[x])/x,x]

[Out]

-5*x - x^2 + 24*x^4 + 8*x^5 - 30*x^6 - 10*x^7 - 16*x^8 + 40*x^10 - 25*x^12 + 6*Log[x] + 2*x*Log[x] - 8*x^4*Log
[x] + 10*x^6*Log[x] - Log[x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {6-3 x-2 x^2+88 x^4+40 x^5-170 x^6-70 x^7-128 x^8+400 x^{10}-300 x^{12}}{x}+\frac {2 \left (-1+x-16 x^4+30 x^6\right ) \log (x)}{x}\right ) \, dx\\ &=2 \int \frac {\left (-1+x-16 x^4+30 x^6\right ) \log (x)}{x} \, dx+\int \frac {6-3 x-2 x^2+88 x^4+40 x^5-170 x^6-70 x^7-128 x^8+400 x^{10}-300 x^{12}}{x} \, dx\\ &=2 \int \left (\log (x)-\frac {\log (x)}{x}-16 x^3 \log (x)+30 x^5 \log (x)\right ) \, dx+\int \left (-3+\frac {6}{x}-2 x+88 x^3+40 x^4-170 x^5-70 x^6-128 x^7+400 x^9-300 x^{11}\right ) \, dx\\ &=-3 x-x^2+22 x^4+8 x^5-\frac {85 x^6}{3}-10 x^7-16 x^8+40 x^{10}-25 x^{12}+6 \log (x)+2 \int \log (x) \, dx-2 \int \frac {\log (x)}{x} \, dx-32 \int x^3 \log (x) \, dx+60 \int x^5 \log (x) \, dx\\ &=-5 x-x^2+24 x^4+8 x^5-30 x^6-10 x^7-16 x^8+40 x^{10}-25 x^{12}+6 \log (x)+2 x \log (x)-8 x^4 \log (x)+10 x^6 \log (x)-\log ^2(x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 73, normalized size = 3.04 \begin {gather*} -5 x-x^2+24 x^4+8 x^5-30 x^6-10 x^7-16 x^8+40 x^{10}-25 x^{12}+6 \log (x)+2 x \log (x)-8 x^4 \log (x)+10 x^6 \log (x)-\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 3*x - 2*x^2 + 88*x^4 + 40*x^5 - 170*x^6 - 70*x^7 - 128*x^8 + 400*x^10 - 300*x^12 + (-2 + 2*x -
32*x^4 + 60*x^6)*Log[x])/x,x]

[Out]

-5*x - x^2 + 24*x^4 + 8*x^5 - 30*x^6 - 10*x^7 - 16*x^8 + 40*x^10 - 25*x^12 + 6*Log[x] + 2*x*Log[x] - 8*x^4*Log
[x] + 10*x^6*Log[x] - Log[x]^2

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 67, normalized size = 2.79 \begin {gather*} -25 \, x^{12} + 40 \, x^{10} - 16 \, x^{8} - 10 \, x^{7} - 30 \, x^{6} + 8 \, x^{5} + 24 \, x^{4} - x^{2} + 2 \, {\left (5 \, x^{6} - 4 \, x^{4} + x + 3\right )} \log \relax (x) - \log \relax (x)^{2} - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^6-32*x^4+2*x-2)*log(x)-300*x^12+400*x^10-128*x^8-70*x^7-170*x^6+40*x^5+88*x^4-2*x^2-3*x+6)/x,
x, algorithm="fricas")

[Out]

-25*x^12 + 40*x^10 - 16*x^8 - 10*x^7 - 30*x^6 + 8*x^5 + 24*x^4 - x^2 + 2*(5*x^6 - 4*x^4 + x + 3)*log(x) - log(
x)^2 - 5*x

________________________________________________________________________________________

giac [B]  time = 0.19, size = 70, normalized size = 2.92 \begin {gather*} -25 \, x^{12} + 40 \, x^{10} - 16 \, x^{8} - 10 \, x^{7} - 30 \, x^{6} + 8 \, x^{5} + 24 \, x^{4} - x^{2} + 2 \, {\left (5 \, x^{6} - 4 \, x^{4} + x\right )} \log \relax (x) - \log \relax (x)^{2} - 5 \, x + 6 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^6-32*x^4+2*x-2)*log(x)-300*x^12+400*x^10-128*x^8-70*x^7-170*x^6+40*x^5+88*x^4-2*x^2-3*x+6)/x,
x, algorithm="giac")

[Out]

-25*x^12 + 40*x^10 - 16*x^8 - 10*x^7 - 30*x^6 + 8*x^5 + 24*x^4 - x^2 + 2*(5*x^6 - 4*x^4 + x)*log(x) - log(x)^2
- 5*x + 6*log(x)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 72, normalized size = 3.00

method	result	size
risch	\(-\ln \relax (x )^{2}+\left (10 x^{6}-8 x^{4}+2 x \right ) \ln \relax (x )-25 x^{12}+40 x^{10}-16 x^{8}-10 x^{7}-30 x^{6}+8 x^{5}+24 x^{4}-x^{2}-5 x +6 \ln \relax (x )\)	\(72\)
default	\(-25 x^{12}+40 x^{10}-16 x^{8}+10 x^{6} \ln \relax (x )-30 x^{6}-10 x^{7}-8 x^{4} \ln \relax (x )+24 x^{4}+8 x^{5}+2 x \ln \relax (x )-5 x -x^{2}-\ln \relax (x )^{2}+6 \ln \relax (x )\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*x^6-32*x^4+2*x-2)*ln(x)-300*x^12+400*x^10-128*x^8-70*x^7-170*x^6+40*x^5+88*x^4-2*x^2-3*x+6)/x,x,metho
d=_RETURNVERBOSE)

[Out]

-ln(x)^2+(10*x^6-8*x^4+2*x)*ln(x)-25*x^12+40*x^10-16*x^8-10*x^7-30*x^6+8*x^5+24*x^4-x^2-5*x+6*ln(x)

________________________________________________________________________________________

maxima [B]  time = 0.50, size = 73, normalized size = 3.04 \begin {gather*} -25 \, x^{12} + 40 \, x^{10} - 16 \, x^{8} - 10 \, x^{7} + 10 \, x^{6} \log \relax (x) - 30 \, x^{6} + 8 \, x^{5} - 8 \, x^{4} \log \relax (x) + 24 \, x^{4} - x^{2} + 2 \, x \log \relax (x) - \log \relax (x)^{2} - 5 \, x + 6 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^6-32*x^4+2*x-2)*log(x)-300*x^12+400*x^10-128*x^8-70*x^7-170*x^6+40*x^5+88*x^4-2*x^2-3*x+6)/x,
x, algorithm="maxima")

[Out]

-25*x^12 + 40*x^10 - 16*x^8 - 10*x^7 + 10*x^6*log(x) - 30*x^6 + 8*x^5 - 8*x^4*log(x) + 24*x^4 - x^2 + 2*x*log(
x) - log(x)^2 - 5*x + 6*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.25, size = 73, normalized size = 3.04 \begin {gather*} -25\,x^{12}+40\,x^{10}-16\,x^8-10\,x^7+10\,x^6\,\ln \relax (x)-30\,x^6+8\,x^5-8\,x^4\,\ln \relax (x)+24\,x^4-x^2+2\,x\,\ln \relax (x)-5\,x-{\ln \relax (x)}^2+6\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 2*x^2 - 88*x^4 - 40*x^5 + 170*x^6 + 70*x^7 + 128*x^8 - 400*x^10 + 300*x^12 - log(x)*(2*x - 32*x^4
+ 60*x^6 - 2) - 6)/x,x)

[Out]

6*log(x) - 5*x - 8*x^4*log(x) + 10*x^6*log(x) - log(x)^2 + 2*x*log(x) - x^2 + 24*x^4 + 8*x^5 - 30*x^6 - 10*x^7
- 16*x^8 + 40*x^10 - 25*x^12

________________________________________________________________________________________

sympy [B]  time = 0.19, size = 68, normalized size = 2.83 \begin {gather*} - 25 x^{12} + 40 x^{10} - 16 x^{8} - 10 x^{7} - 30 x^{6} + 8 x^{5} + 24 x^{4} - x^{2} - 5 x + \left (10 x^{6} - 8 x^{4} + 2 x\right ) \log {\relax (x )} - \log {\relax (x )}^{2} + 6 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x**6-32*x**4+2*x-2)*ln(x)-300*x**12+400*x**10-128*x**8-70*x**7-170*x**6+40*x**5+88*x**4-2*x**2-
3*x+6)/x,x)

[Out]

-25*x**12 + 40*x**10 - 16*x**8 - 10*x**7 - 30*x**6 + 8*x**5 + 24*x**4 - x**2 - 5*x + (10*x**6 - 8*x**4 + 2*x)*
log(x) - log(x)**2 + 6*log(x)

________________________________________________________________________________________