3.99.12 \(\int \frac {-36963+11655 x+219 x^2+x^3+(36963-23310 x-657 x^2-4 x^3) \log (x)}{\log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ \frac {(3-x) x (111+x)^2}{\log (x)} \]

________________________________________________________________________________________

Rubi [B]  time = 0.35, antiderivative size = 35, normalized size of antiderivative = 2.19, number of steps used = 25, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6741, 6742, 2356, 2297, 2298, 2306, 2309, 2178} \begin {gather*} -\frac {x^4}{\log (x)}-\frac {219 x^3}{\log (x)}-\frac {11655 x^2}{\log (x)}+\frac {36963 x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36963 + 11655*x + 219*x^2 + x^3 + (36963 - 23310*x - 657*x^2 - 4*x^3)*Log[x])/Log[x]^2,x]

[Out]

(36963*x)/Log[x] - (11655*x^2)/Log[x] - (219*x^3)/Log[x] - x^4/Log[x]

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 2297

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

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

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 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 6741

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

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 {(111+x) \left (-333+108 x+x^2+333 \log (x)-213 x \log (x)-4 x^2 \log (x)\right )}{\log ^2(x)} \, dx\\ &=\int \left (\frac {(-3+x) (111+x)^2}{\log ^2(x)}+\frac {36963-23310 x-657 x^2-4 x^3}{\log (x)}\right ) \, dx\\ &=\int \frac {(-3+x) (111+x)^2}{\log ^2(x)} \, dx+\int \frac {36963-23310 x-657 x^2-4 x^3}{\log (x)} \, dx\\ &=\int \left (-\frac {36963}{\log ^2(x)}+\frac {11655 x}{\log ^2(x)}+\frac {219 x^2}{\log ^2(x)}+\frac {x^3}{\log ^2(x)}\right ) \, dx+\int \left (\frac {36963}{\log (x)}-\frac {23310 x}{\log (x)}-\frac {657 x^2}{\log (x)}-\frac {4 x^3}{\log (x)}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{\log (x)} \, dx\right )+219 \int \frac {x^2}{\log ^2(x)} \, dx-657 \int \frac {x^2}{\log (x)} \, dx+11655 \int \frac {x}{\log ^2(x)} \, dx-23310 \int \frac {x}{\log (x)} \, dx-36963 \int \frac {1}{\log ^2(x)} \, dx+36963 \int \frac {1}{\log (x)} \, dx+\int \frac {x^3}{\log ^2(x)} \, dx\\ &=\frac {36963 x}{\log (x)}-\frac {11655 x^2}{\log (x)}-\frac {219 x^3}{\log (x)}-\frac {x^4}{\log (x)}+36963 \text {li}(x)+4 \int \frac {x^3}{\log (x)} \, dx-4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+657 \int \frac {x^2}{\log (x)} \, dx-657 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+23310 \int \frac {x}{\log (x)} \, dx-23310 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-36963 \int \frac {1}{\log (x)} \, dx\\ &=-23310 \text {Ei}(2 \log (x))-657 \text {Ei}(3 \log (x))-4 \text {Ei}(4 \log (x))+\frac {36963 x}{\log (x)}-\frac {11655 x^2}{\log (x)}-\frac {219 x^3}{\log (x)}-\frac {x^4}{\log (x)}+4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+657 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+23310 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {36963 x}{\log (x)}-\frac {11655 x^2}{\log (x)}-\frac {219 x^3}{\log (x)}-\frac {x^4}{\log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 15, normalized size = 0.94 \begin {gather*} -\frac {(-3+x) x (111+x)^2}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36963 + 11655*x + 219*x^2 + x^3 + (36963 - 23310*x - 657*x^2 - 4*x^3)*Log[x])/Log[x]^2,x]

[Out]

-(((-3 + x)*x*(111 + x)^2)/Log[x])

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 23, normalized size = 1.44 \begin {gather*} -\frac {x^{4} + 219 \, x^{3} + 11655 \, x^{2} - 36963 \, x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-657*x^2-23310*x+36963)*log(x)+x^3+219*x^2+11655*x-36963)/log(x)^2,x, algorithm="fricas")

[Out]

-(x^4 + 219*x^3 + 11655*x^2 - 36963*x)/log(x)

________________________________________________________________________________________

giac [B]  time = 0.18, size = 35, normalized size = 2.19 \begin {gather*} -\frac {x^{4}}{\log \relax (x)} - \frac {219 \, x^{3}}{\log \relax (x)} - \frac {11655 \, x^{2}}{\log \relax (x)} + \frac {36963 \, x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-657*x^2-23310*x+36963)*log(x)+x^3+219*x^2+11655*x-36963)/log(x)^2,x, algorithm="giac")

[Out]

-x^4/log(x) - 219*x^3/log(x) - 11655*x^2/log(x) + 36963*x/log(x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 21, normalized size = 1.31

method	result	size
risch	\(-\frac {x \left (x^{3}+219 x^{2}+11655 x -36963\right )}{\ln \relax (x )}\)	\(21\)
norman	\(\frac {-x^{4}-219 x^{3}-11655 x^{2}+36963 x}{\ln \relax (x )}\)	\(25\)
default	\(-\frac {x^{4}}{\ln \relax (x )}-\frac {219 x^{3}}{\ln \relax (x )}-\frac {11655 x^{2}}{\ln \relax (x )}+\frac {36963 x}{\ln \relax (x )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3-657*x^2-23310*x+36963)*ln(x)+x^3+219*x^2+11655*x-36963)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

-x*(x^3+219*x^2+11655*x-36963)/ln(x)

________________________________________________________________________________________

maxima [C]  time = 0.39, size = 59, normalized size = 3.69 \begin {gather*} -4 \, {\rm Ei}\left (4 \, \log \relax (x)\right ) - 657 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) - 23310 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + 36963 \, {\rm Ei}\left (\log \relax (x)\right ) - 36963 \, \Gamma \left (-1, -\log \relax (x)\right ) + 23310 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) + 657 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 4 \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-657*x^2-23310*x+36963)*log(x)+x^3+219*x^2+11655*x-36963)/log(x)^2,x, algorithm="maxima")

[Out]

-4*Ei(4*log(x)) - 657*Ei(3*log(x)) - 23310*Ei(2*log(x)) + 36963*Ei(log(x)) - 36963*gamma(-1, -log(x)) + 23310*
gamma(-1, -2*log(x)) + 657*gamma(-1, -3*log(x)) + 4*gamma(-1, -4*log(x))

________________________________________________________________________________________

mupad [B]  time = 5.62, size = 15, normalized size = 0.94 \begin {gather*} -\frac {x\,\left (x-3\right )\,{\left (x+111\right )}^2}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((11655*x + 219*x^2 + x^3 - log(x)*(23310*x + 657*x^2 + 4*x^3 - 36963) - 36963)/log(x)^2,x)

[Out]

-(x*(x - 3)*(x + 111)^2)/log(x)

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 19, normalized size = 1.19 \begin {gather*} \frac {- x^{4} - 219 x^{3} - 11655 x^{2} + 36963 x}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3-657*x**2-23310*x+36963)*ln(x)+x**3+219*x**2+11655*x-36963)/ln(x)**2,x)

[Out]

(-x**4 - 219*x**3 - 11655*x**2 + 36963*x)/log(x)

________________________________________________________________________________________