3.1.37 \(\int \frac {-20 x+364 x^2-288 x^4+72 x^6-6 x^8+(-20-20 x-576 x^3+288 x^5-36 x^7) \log (x)+(-288 x^2+432 x^4-90 x^6) \log ^2(x)+(288 x^3-120 x^5) \log ^3(x)+(72 x^2-90 x^4) \log ^4(x)-36 x^3 \log ^5(x)-6 x^2 \log ^6(x)}{-192 x+144 x^3-36 x^5+3 x^7+(288 x^2-144 x^4+18 x^6) \log (x)+(144 x-216 x^3+45 x^5) \log ^2(x)+(-144 x^2+60 x^4) \log ^3(x)+(-36 x+45 x^3) \log ^4(x)+18 x^2 \log ^5(x)+3 x \log ^6(x)} \, dx\)

Optimal. Leaf size=21 \[ 2-x^2+\frac {5}{3 \left (-4+(x+\log (x))^2\right )^2} \]

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Rubi [B]  time = 1.53, antiderivative size = 58, normalized size of antiderivative = 2.76, number of steps used = 8, number of rules used = 4, integrand size = 224, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {6688, 12, 6742, 6686} \begin {gather*} -x^2+\frac {5}{96 (-x-\log (x)+2)}+\frac {5}{96 (x+\log (x)+2)}+\frac {5}{48 (-x-\log (x)+2)^2}+\frac {5}{48 (x+\log (x)+2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-20*x + 364*x^2 - 288*x^4 + 72*x^6 - 6*x^8 + (-20 - 20*x - 576*x^3 + 288*x^5 - 36*x^7)*Log[x] + (-288*x^2
+ 432*x^4 - 90*x^6)*Log[x]^2 + (288*x^3 - 120*x^5)*Log[x]^3 + (72*x^2 - 90*x^4)*Log[x]^4 - 36*x^3*Log[x]^5 -
6*x^2*Log[x]^6)/(-192*x + 144*x^3 - 36*x^5 + 3*x^7 + (288*x^2 - 144*x^4 + 18*x^6)*Log[x] + (144*x - 216*x^3 +
45*x^5)*Log[x]^2 + (-144*x^2 + 60*x^4)*Log[x]^3 + (-36*x + 45*x^3)*Log[x]^4 + 18*x^2*Log[x]^5 + 3*x*Log[x]^6),
x]

[Out]

-x^2 + 5/(48*(2 - x - Log[x])^2) + 5/(96*(2 - x - Log[x])) + 5/(48*(2 + x + Log[x])^2) + 5/(96*(2 + x + Log[x]
))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (x \left (10-182 x+144 x^3-36 x^5+3 x^7\right )+2 \left (5+5 x+144 x^3-72 x^5+9 x^7\right ) \log (x)+9 x^2 \left (16-24 x^2+5 x^4\right ) \log ^2(x)+12 x^3 \left (-12+5 x^2\right ) \log ^3(x)+9 x^2 \left (-4+5 x^2\right ) \log ^4(x)+18 x^3 \log ^5(x)+3 x^2 \log ^6(x)\right )}{3 x \left (4-x^2-2 x \log (x)-\log ^2(x)\right )^3} \, dx\\ &=\frac {2}{3} \int \frac {x \left (10-182 x+144 x^3-36 x^5+3 x^7\right )+2 \left (5+5 x+144 x^3-72 x^5+9 x^7\right ) \log (x)+9 x^2 \left (16-24 x^2+5 x^4\right ) \log ^2(x)+12 x^3 \left (-12+5 x^2\right ) \log ^3(x)+9 x^2 \left (-4+5 x^2\right ) \log ^4(x)+18 x^3 \log ^5(x)+3 x^2 \log ^6(x)}{x \left (4-x^2-2 x \log (x)-\log ^2(x)\right )^3} \, dx\\ &=\frac {2}{3} \int \left (-3 x-\frac {5 (1+x)}{16 x (-2+x+\log (x))^3}+\frac {5 (1+x)}{64 x (-2+x+\log (x))^2}-\frac {5 (1+x)}{16 x (2+x+\log (x))^3}-\frac {5 (1+x)}{64 x (2+x+\log (x))^2}\right ) \, dx\\ &=-x^2+\frac {5}{96} \int \frac {1+x}{x (-2+x+\log (x))^2} \, dx-\frac {5}{96} \int \frac {1+x}{x (2+x+\log (x))^2} \, dx-\frac {5}{24} \int \frac {1+x}{x (-2+x+\log (x))^3} \, dx-\frac {5}{24} \int \frac {1+x}{x (2+x+\log (x))^3} \, dx\\ &=-x^2+\frac {5}{48 (2-x-\log (x))^2}+\frac {5}{96 (2-x-\log (x))}+\frac {5}{48 (2+x+\log (x))^2}+\frac {5}{96 (2+x+\log (x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 32, normalized size = 1.52 \begin {gather*} -\frac {2}{3} \left (\frac {3 x^2}{2}-\frac {5}{2 \left (-4+x^2+2 x \log (x)+\log ^2(x)\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20*x + 364*x^2 - 288*x^4 + 72*x^6 - 6*x^8 + (-20 - 20*x - 576*x^3 + 288*x^5 - 36*x^7)*Log[x] + (-2
88*x^2 + 432*x^4 - 90*x^6)*Log[x]^2 + (288*x^3 - 120*x^5)*Log[x]^3 + (72*x^2 - 90*x^4)*Log[x]^4 - 36*x^3*Log[x
]^5 - 6*x^2*Log[x]^6)/(-192*x + 144*x^3 - 36*x^5 + 3*x^7 + (288*x^2 - 144*x^4 + 18*x^6)*Log[x] + (144*x - 216*
x^3 + 45*x^5)*Log[x]^2 + (-144*x^2 + 60*x^4)*Log[x]^3 + (-36*x + 45*x^3)*Log[x]^4 + 18*x^2*Log[x]^5 + 3*x*Log[
x]^6),x]

[Out]

(-2*((3*x^2)/2 - 5/(2*(-4 + x^2 + 2*x*Log[x] + Log[x]^2)^2)))/3

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fricas [B]  time = 0.84, size = 114, normalized size = 5.43 \begin {gather*} -\frac {3 \, x^{6} + 12 \, x^{3} \log \relax (x)^{3} + 3 \, x^{2} \log \relax (x)^{4} - 24 \, x^{4} + 6 \, {\left (3 \, x^{4} - 4 \, x^{2}\right )} \log \relax (x)^{2} + 48 \, x^{2} + 12 \, {\left (x^{5} - 4 \, x^{3}\right )} \log \relax (x) - 5}{3 \, {\left (x^{4} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} + 2 \, {\left (3 \, x^{2} - 4\right )} \log \relax (x)^{2} - 8 \, x^{2} + 4 \, {\left (x^{3} - 4 \, x\right )} \log \relax (x) + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*log(x)^6-36*x^3*log(x)^5+(-90*x^4+72*x^2)*log(x)^4+(-120*x^5+288*x^3)*log(x)^3+(-90*x^6+432*
x^4-288*x^2)*log(x)^2+(-36*x^7+288*x^5-576*x^3-20*x-20)*log(x)-6*x^8+72*x^6-288*x^4+364*x^2-20*x)/(3*x*log(x)^
6+18*x^2*log(x)^5+(45*x^3-36*x)*log(x)^4+(60*x^4-144*x^2)*log(x)^3+(45*x^5-216*x^3+144*x)*log(x)^2+(18*x^6-144
*x^4+288*x^2)*log(x)+3*x^7-36*x^5+144*x^3-192*x),x, algorithm="fricas")

[Out]

-1/3*(3*x^6 + 12*x^3*log(x)^3 + 3*x^2*log(x)^4 - 24*x^4 + 6*(3*x^4 - 4*x^2)*log(x)^2 + 48*x^2 + 12*(x^5 - 4*x^
3)*log(x) - 5)/(x^4 + 4*x*log(x)^3 + log(x)^4 + 2*(3*x^2 - 4)*log(x)^2 - 8*x^2 + 4*(x^3 - 4*x)*log(x) + 16)

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giac [B]  time = 0.87, size = 117, normalized size = 5.57 \begin {gather*} -x^{2} + \frac {5 \, {\left (x + 1\right )}}{3 \, {\left (x^{5} + 4 \, x^{4} \log \relax (x) + 6 \, x^{3} \log \relax (x)^{2} + 4 \, x^{2} \log \relax (x)^{3} + x \log \relax (x)^{4} + x^{4} + 4 \, x^{3} \log \relax (x) + 6 \, x^{2} \log \relax (x)^{2} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} - 8 \, x^{3} - 16 \, x^{2} \log \relax (x) - 8 \, x \log \relax (x)^{2} - 8 \, x^{2} - 16 \, x \log \relax (x) - 8 \, \log \relax (x)^{2} + 16 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*log(x)^6-36*x^3*log(x)^5+(-90*x^4+72*x^2)*log(x)^4+(-120*x^5+288*x^3)*log(x)^3+(-90*x^6+432*
x^4-288*x^2)*log(x)^2+(-36*x^7+288*x^5-576*x^3-20*x-20)*log(x)-6*x^8+72*x^6-288*x^4+364*x^2-20*x)/(3*x*log(x)^
6+18*x^2*log(x)^5+(45*x^3-36*x)*log(x)^4+(60*x^4-144*x^2)*log(x)^3+(45*x^5-216*x^3+144*x)*log(x)^2+(18*x^6-144
*x^4+288*x^2)*log(x)+3*x^7-36*x^5+144*x^3-192*x),x, algorithm="giac")

[Out]

-x^2 + 5/3*(x + 1)/(x^5 + 4*x^4*log(x) + 6*x^3*log(x)^2 + 4*x^2*log(x)^3 + x*log(x)^4 + x^4 + 4*x^3*log(x) + 6
*x^2*log(x)^2 + 4*x*log(x)^3 + log(x)^4 - 8*x^3 - 16*x^2*log(x) - 8*x*log(x)^2 - 8*x^2 - 16*x*log(x) - 8*log(x
)^2 + 16*x + 16)

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maple [A]  time = 0.02, size = 25, normalized size = 1.19

method	result	size
risch	\(-x^{2}+\frac {5}{3 \left (\ln \relax (x )^{2}+2 x \ln \relax (x )+x^{2}-4\right )^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x^2*ln(x)^6-36*x^3*ln(x)^5+(-90*x^4+72*x^2)*ln(x)^4+(-120*x^5+288*x^3)*ln(x)^3+(-90*x^6+432*x^4-288*x^
2)*ln(x)^2+(-36*x^7+288*x^5-576*x^3-20*x-20)*ln(x)-6*x^8+72*x^6-288*x^4+364*x^2-20*x)/(3*x*ln(x)^6+18*x^2*ln(x
)^5+(45*x^3-36*x)*ln(x)^4+(60*x^4-144*x^2)*ln(x)^3+(45*x^5-216*x^3+144*x)*ln(x)^2+(18*x^6-144*x^4+288*x^2)*ln(
x)+3*x^7-36*x^5+144*x^3-192*x),x,method=_RETURNVERBOSE)

[Out]

-x^2+5/3/(ln(x)^2+2*x*ln(x)+x^2-4)^2

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maxima [B]  time = 0.53, size = 114, normalized size = 5.43 \begin {gather*} -\frac {3 \, x^{6} + 12 \, x^{3} \log \relax (x)^{3} + 3 \, x^{2} \log \relax (x)^{4} - 24 \, x^{4} + 6 \, {\left (3 \, x^{4} - 4 \, x^{2}\right )} \log \relax (x)^{2} + 48 \, x^{2} + 12 \, {\left (x^{5} - 4 \, x^{3}\right )} \log \relax (x) - 5}{3 \, {\left (x^{4} + 4 \, x \log \relax (x)^{3} + \log \relax (x)^{4} + 2 \, {\left (3 \, x^{2} - 4\right )} \log \relax (x)^{2} - 8 \, x^{2} + 4 \, {\left (x^{3} - 4 \, x\right )} \log \relax (x) + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x^2*log(x)^6-36*x^3*log(x)^5+(-90*x^4+72*x^2)*log(x)^4+(-120*x^5+288*x^3)*log(x)^3+(-90*x^6+432*
x^4-288*x^2)*log(x)^2+(-36*x^7+288*x^5-576*x^3-20*x-20)*log(x)-6*x^8+72*x^6-288*x^4+364*x^2-20*x)/(3*x*log(x)^
6+18*x^2*log(x)^5+(45*x^3-36*x)*log(x)^4+(60*x^4-144*x^2)*log(x)^3+(45*x^5-216*x^3+144*x)*log(x)^2+(18*x^6-144
*x^4+288*x^2)*log(x)+3*x^7-36*x^5+144*x^3-192*x),x, algorithm="maxima")

[Out]

-1/3*(3*x^6 + 12*x^3*log(x)^3 + 3*x^2*log(x)^4 - 24*x^4 + 6*(3*x^4 - 4*x^2)*log(x)^2 + 48*x^2 + 12*(x^5 - 4*x^
3)*log(x) - 5)/(x^4 + 4*x*log(x)^3 + log(x)^4 + 2*(3*x^2 - 4)*log(x)^2 - 8*x^2 + 4*(x^3 - 4*x)*log(x) + 16)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {20\,x+\ln \relax (x)\,\left (36\,x^7-288\,x^5+576\,x^3+20\,x+20\right )-{\ln \relax (x)}^4\,\left (72\,x^2-90\,x^4\right )-{\ln \relax (x)}^3\,\left (288\,x^3-120\,x^5\right )+6\,x^2\,{\ln \relax (x)}^6+36\,x^3\,{\ln \relax (x)}^5+{\ln \relax (x)}^2\,\left (90\,x^6-432\,x^4+288\,x^2\right )-364\,x^2+288\,x^4-72\,x^6+6\,x^8}{3\,x\,{\ln \relax (x)}^6-{\ln \relax (x)}^4\,\left (36\,x-45\,x^3\right )-192\,x+{\ln \relax (x)}^2\,\left (45\,x^5-216\,x^3+144\,x\right )+\ln \relax (x)\,\left (18\,x^6-144\,x^4+288\,x^2\right )-{\ln \relax (x)}^3\,\left (144\,x^2-60\,x^4\right )+18\,x^2\,{\ln \relax (x)}^5+144\,x^3-36\,x^5+3\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x + log(x)*(20*x + 576*x^3 - 288*x^5 + 36*x^7 + 20) - log(x)^4*(72*x^2 - 90*x^4) - log(x)^3*(288*x^3
- 120*x^5) + 6*x^2*log(x)^6 + 36*x^3*log(x)^5 + log(x)^2*(288*x^2 - 432*x^4 + 90*x^6) - 364*x^2 + 288*x^4 - 72
*x^6 + 6*x^8)/(3*x*log(x)^6 - log(x)^4*(36*x - 45*x^3) - 192*x + log(x)^2*(144*x - 216*x^3 + 45*x^5) + log(x)*
(288*x^2 - 144*x^4 + 18*x^6) - log(x)^3*(144*x^2 - 60*x^4) + 18*x^2*log(x)^5 + 144*x^3 - 36*x^5 + 3*x^7),x)

[Out]

int(-(20*x + log(x)*(20*x + 576*x^3 - 288*x^5 + 36*x^7 + 20) - log(x)^4*(72*x^2 - 90*x^4) - log(x)^3*(288*x^3
- 120*x^5) + 6*x^2*log(x)^6 + 36*x^3*log(x)^5 + log(x)^2*(288*x^2 - 432*x^4 + 90*x^6) - 364*x^2 + 288*x^4 - 72
*x^6 + 6*x^8)/(3*x*log(x)^6 - log(x)^4*(36*x - 45*x^3) - 192*x + log(x)^2*(144*x - 216*x^3 + 45*x^5) + log(x)*
(288*x^2 - 144*x^4 + 18*x^6) - log(x)^3*(144*x^2 - 60*x^4) + 18*x^2*log(x)^5 + 144*x^3 - 36*x^5 + 3*x^7), x)

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sympy [B]  time = 0.23, size = 54, normalized size = 2.57 \begin {gather*} - x^{2} + \frac {5}{3 x^{4} - 24 x^{2} + 12 x \log {\relax (x )}^{3} + \left (18 x^{2} - 24\right ) \log {\relax (x )}^{2} + \left (12 x^{3} - 48 x\right ) \log {\relax (x )} + 3 \log {\relax (x )}^{4} + 48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x**2*ln(x)**6-36*x**3*ln(x)**5+(-90*x**4+72*x**2)*ln(x)**4+(-120*x**5+288*x**3)*ln(x)**3+(-90*x*
*6+432*x**4-288*x**2)*ln(x)**2+(-36*x**7+288*x**5-576*x**3-20*x-20)*ln(x)-6*x**8+72*x**6-288*x**4+364*x**2-20*
x)/(3*x*ln(x)**6+18*x**2*ln(x)**5+(45*x**3-36*x)*ln(x)**4+(60*x**4-144*x**2)*ln(x)**3+(45*x**5-216*x**3+144*x)
*ln(x)**2+(18*x**6-144*x**4+288*x**2)*ln(x)+3*x**7-36*x**5+144*x**3-192*x),x)

[Out]

-x**2 + 5/(3*x**4 - 24*x**2 + 12*x*log(x)**3 + (18*x**2 - 24)*log(x)**2 + (12*x**3 - 48*x)*log(x) + 3*log(x)**
4 + 48)

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