3.88.36 \(\int \frac {-2 x-4 x^2-2 x^3+(4 x+4 x^2) \log (3)-2 x \log ^2(3)+(-8 x-4 x^2+4 x^3+(6 x-6 x^2) \log (3)+2 x \log ^2(3)) \log (x)+(20 x+30 x^2-20 x \log (3)) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx\)

Optimal. Leaf size=21 \[ x^2 \left (-5-\frac {1+x-\log (3)}{\log (x)}\right )^2 \]

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Rubi [C]  time = 0.80, antiderivative size = 193, normalized size of antiderivative = 9.19, number of steps used = 38, number of rules used = 10, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 6688, 12, 6742, 27, 2353, 2306, 2309, 2178, 2356} \begin {gather*} -4 \left (4-\log ^2(3)-\log (27)\right ) \text {Ei}(2 \log (x))+30 \text {Ei}(3 \log (x))-6 (2+\log (27)) \text {Ei}(3 \log (x))-(2-\log (9))^2 \text {Ei}(2 \log (x))+10 (2-\log (9)) \text {Ei}(2 \log (x))-9 (2-\log (9)) \text {Ei}(3 \log (x))+\frac {x^4}{\log ^2(x)}+\frac {x^3 (2-\log (9))}{\log ^2(x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}+\frac {3 x^3 (2-\log (9))}{\log (x)}+25 x^2+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {x^2 (2-\log (9))^2}{4 \log ^2(x)}+\frac {x^2 (2-\log (9))^2}{2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x - 4*x^2 - 2*x^3 + (4*x + 4*x^2)*Log[3] - 2*x*Log[3]^2 + (-8*x - 4*x^2 + 4*x^3 + (6*x - 6*x^2)*Log[3]
+ 2*x*Log[3]^2)*Log[x] + (20*x + 30*x^2 - 20*x*Log[3])*Log[x]^2 + 50*x*Log[x]^3)/Log[x]^3,x]

[Out]

25*x^2 + 30*ExpIntegralEi[3*Log[x]] + 10*ExpIntegralEi[2*Log[x]]*(2 - Log[9]) - 9*ExpIntegralEi[3*Log[x]]*(2 -
Log[9]) - ExpIntegralEi[2*Log[x]]*(2 - Log[9])^2 - 4*ExpIntegralEi[2*Log[x]]*(4 - Log[3]^2 - Log[27]) - 6*Exp
IntegralEi[3*Log[x]]*(2 + Log[27]) + x^4/Log[x]^2 + (x^3*(2 - Log[9]))/Log[x]^2 + (x^2*(2 - Log[9])^2)/(4*Log[
x]^2) + (3*x^3*(2 - Log[9]))/Log[x] + (x^2*(2 - Log[9])^2)/(2*Log[x]) + (2*x^2*(4 - Log[3]^2 - Log[27]))/Log[x
] + (2*x^3*(2 + Log[27]))/Log[x]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 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] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

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 {-4 x^2-2 x^3+\left (4 x+4 x^2\right ) \log (3)+x \left (-2-2 \log ^2(3)\right )+\left (-8 x-4 x^2+4 x^3+\left (6 x-6 x^2\right ) \log (3)+2 x \log ^2(3)\right ) \log (x)+\left (20 x+30 x^2-20 x \log (3)\right ) \log ^2(x)+50 x \log ^3(x)}{\log ^3(x)} \, dx\\ &=\int \frac {2 x \left (-1-x^2-\log ^2(3)+x (-2+\log (9))+\log (9)+\left (-4+2 x^2+\log ^2(3)+\log (27)-x (2+\log (27))\right ) \log (x)+5 (2+3 x-\log (9)) \log ^2(x)+25 \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \frac {x \left (-1-x^2-\log ^2(3)+x (-2+\log (9))+\log (9)+\left (-4+2 x^2+\log ^2(3)+\log (27)-x (2+\log (27))\right ) \log (x)+5 (2+3 x-\log (9)) \log ^2(x)+25 \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \left (25 x+\frac {x \left (-1-x^2-\log ^2(3)-x (2-\log (9))+\log (9)\right )}{\log ^3(x)}+\frac {x \left (-4+2 x^2+\log ^2(3)+\log (27)-x (2+\log (27))\right )}{\log ^2(x)}+\frac {5 x (2+3 x-\log (9))}{\log (x)}\right ) \, dx\\ &=25 x^2+2 \int \frac {x \left (-1-x^2-\log ^2(3)-x (2-\log (9))+\log (9)\right )}{\log ^3(x)} \, dx+2 \int \frac {x \left (-4+2 x^2+\log ^2(3)+\log (27)-x (2+\log (27))\right )}{\log ^2(x)} \, dx+10 \int \frac {x (2+3 x-\log (9))}{\log (x)} \, dx\\ &=25 x^2+2 \int \left (\frac {2 x^3}{\log ^2(x)}+\frac {x^2 (-2-\log (27))}{\log ^2(x)}+\frac {x \left (-4+\log ^2(3)+\log (27)\right )}{\log ^2(x)}\right ) \, dx+2 \int -\frac {x (2+2 x-\log (9))^2}{4 \log ^3(x)} \, dx+10 \int \left (\frac {3 x^2}{\log (x)}-\frac {x (-2+\log (9))}{\log (x)}\right ) \, dx\\ &=25 x^2-\frac {1}{2} \int \frac {x (2+2 x-\log (9))^2}{\log ^3(x)} \, dx+4 \int \frac {x^3}{\log ^2(x)} \, dx+30 \int \frac {x^2}{\log (x)} \, dx+(10 (2-\log (9))) \int \frac {x}{\log (x)} \, dx-\left (2 \left (4-\log ^2(3)-\log (27)\right )\right ) \int \frac {x}{\log ^2(x)} \, dx-(2 (2+\log (27))) \int \frac {x^2}{\log ^2(x)} \, dx\\ &=25 x^2-\frac {4 x^4}{\log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}-\frac {1}{2} \int \left (\frac {4 x^3}{\log ^3(x)}-\frac {4 x^2 (-2+\log (9))}{\log ^3(x)}+\frac {x (-2+\log (9))^2}{\log ^3(x)}\right ) \, dx+16 \int \frac {x^3}{\log (x)} \, dx+30 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+(10 (2-\log (9))) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-\left (4 \left (4-\log ^2(3)-\log (27)\right )\right ) \int \frac {x}{\log (x)} \, dx-(6 (2+\log (27))) \int \frac {x^2}{\log (x)} \, dx\\ &=25 x^2+30 \text {Ei}(3 \log (x))+10 \text {Ei}(2 \log (x)) (2-\log (9))-\frac {4 x^4}{\log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}-2 \int \frac {x^3}{\log ^3(x)} \, dx+16 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-(2 (2-\log (9))) \int \frac {x^2}{\log ^3(x)} \, dx-\frac {1}{2} (2-\log (9))^2 \int \frac {x}{\log ^3(x)} \, dx-\left (4 \left (4-\log ^2(3)-\log (27)\right )\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-(6 (2+\log (27))) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )\\ &=25 x^2+30 \text {Ei}(3 \log (x))+16 \text {Ei}(4 \log (x))+10 \text {Ei}(2 \log (x)) (2-\log (9))-4 \text {Ei}(2 \log (x)) \left (4-\log ^2(3)-\log (27)\right )-6 \text {Ei}(3 \log (x)) (2+\log (27))+\frac {x^4}{\log ^2(x)}+\frac {x^3 (2-\log (9))}{\log ^2(x)}+\frac {x^2 (2-\log (9))^2}{4 \log ^2(x)}-\frac {4 x^4}{\log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}-4 \int \frac {x^3}{\log ^2(x)} \, dx-(3 (2-\log (9))) \int \frac {x^2}{\log ^2(x)} \, dx-\frac {1}{2} (2-\log (9))^2 \int \frac {x}{\log ^2(x)} \, dx\\ &=25 x^2+30 \text {Ei}(3 \log (x))+16 \text {Ei}(4 \log (x))+10 \text {Ei}(2 \log (x)) (2-\log (9))-4 \text {Ei}(2 \log (x)) \left (4-\log ^2(3)-\log (27)\right )-6 \text {Ei}(3 \log (x)) (2+\log (27))+\frac {x^4}{\log ^2(x)}+\frac {x^3 (2-\log (9))}{\log ^2(x)}+\frac {x^2 (2-\log (9))^2}{4 \log ^2(x)}+\frac {3 x^3 (2-\log (9))}{\log (x)}+\frac {x^2 (2-\log (9))^2}{2 \log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}-16 \int \frac {x^3}{\log (x)} \, dx-(9 (2-\log (9))) \int \frac {x^2}{\log (x)} \, dx-(2-\log (9))^2 \int \frac {x}{\log (x)} \, dx\\ &=25 x^2+30 \text {Ei}(3 \log (x))+16 \text {Ei}(4 \log (x))+10 \text {Ei}(2 \log (x)) (2-\log (9))-4 \text {Ei}(2 \log (x)) \left (4-\log ^2(3)-\log (27)\right )-6 \text {Ei}(3 \log (x)) (2+\log (27))+\frac {x^4}{\log ^2(x)}+\frac {x^3 (2-\log (9))}{\log ^2(x)}+\frac {x^2 (2-\log (9))^2}{4 \log ^2(x)}+\frac {3 x^3 (2-\log (9))}{\log (x)}+\frac {x^2 (2-\log (9))^2}{2 \log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}-16 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-(9 (2-\log (9))) \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-(2-\log (9))^2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=25 x^2+30 \text {Ei}(3 \log (x))+10 \text {Ei}(2 \log (x)) (2-\log (9))-9 \text {Ei}(3 \log (x)) (2-\log (9))-\text {Ei}(2 \log (x)) (2-\log (9))^2-4 \text {Ei}(2 \log (x)) \left (4-\log ^2(3)-\log (27)\right )-6 \text {Ei}(3 \log (x)) (2+\log (27))+\frac {x^4}{\log ^2(x)}+\frac {x^3 (2-\log (9))}{\log ^2(x)}+\frac {x^2 (2-\log (9))^2}{4 \log ^2(x)}+\frac {3 x^3 (2-\log (9))}{\log (x)}+\frac {x^2 (2-\log (9))^2}{2 \log (x)}+\frac {2 x^2 \left (4-\log ^2(3)-\log (27)\right )}{\log (x)}+\frac {2 x^3 (2+\log (27))}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.28, size = 46, normalized size = 2.19 \begin {gather*} \frac {x^2 \left (1+x^2+\log ^2(3)-x (-2+\log (9))-\log (9)+(10+10 x-\log (59049)) \log (x)+25 \log ^2(x)\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - 4*x^2 - 2*x^3 + (4*x + 4*x^2)*Log[3] - 2*x*Log[3]^2 + (-8*x - 4*x^2 + 4*x^3 + (6*x - 6*x^2)*
Log[3] + 2*x*Log[3]^2)*Log[x] + (20*x + 30*x^2 - 20*x*Log[3])*Log[x]^2 + 50*x*Log[x]^3)/Log[x]^3,x]

[Out]

(x^2*(1 + x^2 + Log[3]^2 - x*(-2 + Log[9]) - Log[9] + (10 + 10*x - Log[59049])*Log[x] + 25*Log[x]^2))/Log[x]^2

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fricas [B]  time = 0.69, size = 63, normalized size = 3.00 \begin {gather*} \frac {x^{4} + x^{2} \log \relax (3)^{2} + 25 \, x^{2} \log \relax (x)^{2} + 2 \, x^{3} + x^{2} - 2 \, {\left (x^{3} + x^{2}\right )} \log \relax (3) + 10 \, {\left (x^{3} - x^{2} \log \relax (3) + x^{2}\right )} \log \relax (x)}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2+(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x
)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log(3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="fricas")

[Out]

(x^4 + x^2*log(3)^2 + 25*x^2*log(x)^2 + 2*x^3 + x^2 - 2*(x^3 + x^2)*log(3) + 10*(x^3 - x^2*log(3) + x^2)*log(x
))/log(x)^2

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giac [B]  time = 0.14, size = 94, normalized size = 4.48 \begin {gather*} 25 \, x^{2} + \frac {x^{4}}{\log \relax (x)^{2}} - \frac {2 \, x^{3} \log \relax (3)}{\log \relax (x)^{2}} + \frac {x^{2} \log \relax (3)^{2}}{\log \relax (x)^{2}} + \frac {10 \, x^{3}}{\log \relax (x)} - \frac {10 \, x^{2} \log \relax (3)}{\log \relax (x)} + \frac {2 \, x^{3}}{\log \relax (x)^{2}} - \frac {2 \, x^{2} \log \relax (3)}{\log \relax (x)^{2}} + \frac {10 \, x^{2}}{\log \relax (x)} + \frac {x^{2}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2+(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x
)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log(3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="giac")

[Out]

25*x^2 + x^4/log(x)^2 - 2*x^3*log(3)/log(x)^2 + x^2*log(3)^2/log(x)^2 + 10*x^3/log(x) - 10*x^2*log(3)/log(x) +
2*x^3/log(x)^2 - 2*x^2*log(3)/log(x)^2 + 10*x^2/log(x) + x^2/log(x)^2

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maple [B]  time = 0.05, size = 51, normalized size = 2.43

method	result	size
risch	\(25 x^{2}+\frac {x^{2} \left (\ln \relax (3)^{2}-2 x \ln \relax (3)-10 \ln \relax (3) \ln \relax (x )+x^{2}+10 x \ln \relax (x )-2 \ln \relax (3)+2 x +10 \ln \relax (x )+1\right )}{\ln \relax (x )^{2}}\)	\(51\)
norman	\(\frac {x^{4}+\left (-2 \ln \relax (3)+2\right ) x^{3}+\left (\ln \relax (3)^{2}-2 \ln \relax (3)+1\right ) x^{2}+\left (-10 \ln \relax (3)+10\right ) x^{2} \ln \relax (x )+25 x^{2} \ln \relax (x )^{2}+10 x^{3} \ln \relax (x )}{\ln \relax (x )^{2}}\)	\(62\)
default	\(\frac {10 x^{2}}{\ln \relax (x )}+25 x^{2}+\frac {x^{2}}{\ln \relax (x )^{2}}+\frac {10 x^{3}}{\ln \relax (x )}+\frac {x^{4}}{\ln \relax (x )^{2}}+\frac {2 x^{3}}{\ln \relax (x )^{2}}+20 \ln \relax (3) \expIntegralEi \left (1, -2 \ln \relax (x )\right )+2 \ln \relax (3)^{2} \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )-6 \ln \relax (3) \left (-\frac {x^{3}}{\ln \relax (x )}-3 \expIntegralEi \left (1, -3 \ln \relax (x )\right )\right )+6 \ln \relax (3) \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )-2 \ln \relax (3)^{2} \left (-\frac {x^{2}}{2 \ln \relax (x )^{2}}-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )+4 \ln \relax (3) \left (-\frac {x^{3}}{2 \ln \relax (x )^{2}}-\frac {3 x^{3}}{2 \ln \relax (x )}-\frac {9 \expIntegralEi \left (1, -3 \ln \relax (x )\right )}{2}\right )+4 \ln \relax (3) \left (-\frac {x^{2}}{2 \ln \relax (x )^{2}}-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )\)	\(223\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x*ln(x)^3+(-20*x*ln(3)+30*x^2+20*x)*ln(x)^2+(2*x*ln(3)^2+(-6*x^2+6*x)*ln(3)+4*x^3-4*x^2-8*x)*ln(x)-2*x
*ln(3)^2+(4*x^2+4*x)*ln(3)-2*x^3-4*x^2-2*x)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

25*x^2+x^2*(ln(3)^2-2*x*ln(3)-10*ln(3)*ln(x)+x^2+10*x*ln(x)-2*ln(3)+2*x+10*ln(x)+1)/ln(x)^2

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maxima [C]  time = 0.41, size = 141, normalized size = 6.71 \begin {gather*} 4 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) \log \relax (3)^{2} + 8 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) \log \relax (3)^{2} + 25 \, x^{2} - 20 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) \log \relax (3) + 12 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) \log \relax (3) - 18 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) \log \relax (3) - 16 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) \log \relax (3) - 36 \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) \log \relax (3) + 30 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) + 20 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) - 16 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) - 12 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 16 \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) + 8 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) + 36 \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) + 32 \, \Gamma \left (-2, -4 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*log(x)^3+(-20*x*log(3)+30*x^2+20*x)*log(x)^2+(2*x*log(3)^2+(-6*x^2+6*x)*log(3)+4*x^3-4*x^2-8*x
)*log(x)-2*x*log(3)^2+(4*x^2+4*x)*log(3)-2*x^3-4*x^2-2*x)/log(x)^3,x, algorithm="maxima")

[Out]

4*gamma(-1, -2*log(x))*log(3)^2 + 8*gamma(-2, -2*log(x))*log(3)^2 + 25*x^2 - 20*Ei(2*log(x))*log(3) + 12*gamma
(-1, -2*log(x))*log(3) - 18*gamma(-1, -3*log(x))*log(3) - 16*gamma(-2, -2*log(x))*log(3) - 36*gamma(-2, -3*log
(x))*log(3) + 30*Ei(3*log(x)) + 20*Ei(2*log(x)) - 16*gamma(-1, -2*log(x)) - 12*gamma(-1, -3*log(x)) + 16*gamma
(-1, -4*log(x)) + 8*gamma(-2, -2*log(x)) + 36*gamma(-2, -3*log(x)) + 32*gamma(-2, -4*log(x))

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mupad [B]  time = 5.45, size = 63, normalized size = 3.00 \begin {gather*} 25\,x^2-\frac {x^4\,\left (\ln \relax (9)-2\right )-x^3\,\left ({\ln \relax (3)}^2-\ln \relax (9)+1\right )+\ln \relax (x)\,\left (x^3\,\left (10\,\ln \relax (3)-10\right )-10\,x^4\right )-x^5}{x\,{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 50*x*log(x)^3 - log(3)*(4*x + 4*x^2) - log(x)^2*(20*x - 20*x*log(3) + 30*x^2) + 2*x*log(3)^2 - log
(x)*(log(3)*(6*x - 6*x^2) - 8*x + 2*x*log(3)^2 - 4*x^2 + 4*x^3) + 4*x^2 + 2*x^3)/log(x)^3,x)

[Out]

25*x^2 - (x^4*(log(9) - 2) - x^3*(log(3)^2 - log(9) + 1) + log(x)*(x^3*(10*log(3) - 10) - 10*x^4) - x^5)/(x*lo
g(x)^2)

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sympy [B]  time = 0.14, size = 68, normalized size = 3.24 \begin {gather*} 25 x^{2} + \frac {x^{4} - 2 x^{3} \log {\relax (3 )} + 2 x^{3} - 2 x^{2} \log {\relax (3 )} + x^{2} + x^{2} \log {\relax (3 )}^{2} + \left (10 x^{3} - 10 x^{2} \log {\relax (3 )} + 10 x^{2}\right ) \log {\relax (x )}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x*ln(x)**3+(-20*x*ln(3)+30*x**2+20*x)*ln(x)**2+(2*x*ln(3)**2+(-6*x**2+6*x)*ln(3)+4*x**3-4*x**2-8
*x)*ln(x)-2*x*ln(3)**2+(4*x**2+4*x)*ln(3)-2*x**3-4*x**2-2*x)/ln(x)**3,x)

[Out]

25*x**2 + (x**4 - 2*x**3*log(3) + 2*x**3 - 2*x**2*log(3) + x**2 + x**2*log(3)**2 + (10*x**3 - 10*x**2*log(3) +
10*x**2)*log(x))/log(x)**2

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