3.19.65 \(\int \frac {-18 x^3+8 x^3 \log (x)}{4-4 \log (x)+\log ^2(x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {2 x^3}{-\frac {2+x}{x}+\frac {x+\log (x)}{x}} \]

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Rubi [C]  time = 0.34, antiderivative size = 79, normalized size of antiderivative = 3.29, number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2561, 6741, 12, 2306, 2309, 2178, 2366, 6482} \begin {gather*} 8 e^8 \text {Ei}(-4 (2-\log (x)))-8 e^8 (9-4 \log (x)) \text {Ei}(-4 (2-\log (x)))+32 e^8 (2-\log (x)) \text {Ei}(-4 (2-\log (x)))+8 x^4-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18*x^3 + 8*x^3*Log[x])/(4 - 4*Log[x] + Log[x]^2),x]

[Out]

8*x^4 + 8*E^8*ExpIntegralEi[-4*(2 - Log[x])] - 8*E^8*ExpIntegralEi[-4*(2 - Log[x])]*(9 - 4*Log[x]) - (2*x^4*(9
- 4*Log[x]))/(2 - Log[x]) + 32*E^8*ExpIntegralEi[-4*(2 - Log[x])]*(2 - 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 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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

Rule 6482

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*ExpIntegralEi[a + b*x])/b, x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 (-18+8 \log (x))}{4-4 \log (x)+\log ^2(x)} \, dx\\ &=\int \frac {2 x^3 (-9+4 \log (x))}{(2-\log (x))^2} \, dx\\ &=2 \int \frac {x^3 (-9+4 \log (x))}{(2-\log (x))^2} \, dx\\ &=-8 e^8 \text {Ei}(-4 (2-\log (x))) (9-4 \log (x))-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)}-8 \int \left (\frac {4 e^8 \text {Ei}(4 (-2+\log (x)))}{x}+\frac {x^3}{2-\log (x)}\right ) \, dx\\ &=-8 e^8 \text {Ei}(-4 (2-\log (x))) (9-4 \log (x))-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)}-8 \int \frac {x^3}{2-\log (x)} \, dx-\left (32 e^8\right ) \int \frac {\text {Ei}(4 (-2+\log (x)))}{x} \, dx\\ &=-8 e^8 \text {Ei}(-4 (2-\log (x))) (9-4 \log (x))-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)}-8 \operatorname {Subst}\left (\int \frac {e^{4 x}}{2-x} \, dx,x,\log (x)\right )-\left (32 e^8\right ) \operatorname {Subst}(\int \text {Ei}(4 (-2+x)) \, dx,x,\log (x))\\ &=8 e^8 \text {Ei}(-4 (2-\log (x)))-8 e^8 \text {Ei}(-4 (2-\log (x))) (9-4 \log (x))-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)}-\left (8 e^8\right ) \operatorname {Subst}(\int \text {Ei}(x) \, dx,x,-8+4 \log (x))\\ &=8 x^4+8 e^8 \text {Ei}(-4 (2-\log (x)))-8 e^8 \text {Ei}(-4 (2-\log (x))) (9-4 \log (x))-\frac {2 x^4 (9-4 \log (x))}{2-\log (x)}+32 e^8 \text {Ei}(-8+4 \log (x)) (2-\log (x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 11, normalized size = 0.46 \begin {gather*} \frac {2 x^4}{-2+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x^3 + 8*x^3*Log[x])/(4 - 4*Log[x] + Log[x]^2),x]

[Out]

(2*x^4)/(-2 + Log[x])

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fricas [A]  time = 0.79, size = 11, normalized size = 0.46 \begin {gather*} \frac {2 \, x^{4}}{\log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3*log(x)-18*x^3)/(log(x)^2-4*log(x)+4),x, algorithm="fricas")

[Out]

2*x^4/(log(x) - 2)

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giac [A]  time = 0.25, size = 11, normalized size = 0.46 \begin {gather*} \frac {2 \, x^{4}}{\log \relax (x) - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3*log(x)-18*x^3)/(log(x)^2-4*log(x)+4),x, algorithm="giac")

[Out]

2*x^4/(log(x) - 2)

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maple [A]  time = 0.02, size = 12, normalized size = 0.50

method	result	size
norman	\(\frac {2 x^{4}}{\ln \relax (x )-2}\)	\(12\)
risch	\(\frac {2 x^{4}}{\ln \relax (x )-2}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3*ln(x)-18*x^3)/(ln(x)^2-4*ln(x)+4),x,method=_RETURNVERBOSE)

[Out]

2*x^4/(ln(x)-2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {16 \, x^{4}}{\log \relax (x) - 2} + \frac {18 \, e^{8} E_{2}\left (-4 \, \log \relax (x) + 8\right )}{\log \relax (x) - 2} + 72 \, \int \frac {x^{3}}{\log \relax (x) - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x^3*log(x)-18*x^3)/(log(x)^2-4*log(x)+4),x, algorithm="maxima")

[Out]

-16*x^4/(log(x) - 2) + 18*e^8*exp_integral_e(2, -4*log(x) + 8)/(log(x) - 2) + 72*integrate(x^3/(log(x) - 2), x
)

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mupad [B]  time = 1.25, size = 11, normalized size = 0.46 \begin {gather*} \frac {2\,x^4}{\ln \relax (x)-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^3*log(x) - 18*x^3)/(log(x)^2 - 4*log(x) + 4),x)

[Out]

(2*x^4)/(log(x) - 2)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.33 \begin {gather*} \frac {2 x^{4}}{\log {\relax (x )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((8*x**3*ln(x)-18*x**3)/(ln(x)**2-4*ln(x)+4),x)

[Out]

2*x**4/(log(x) - 2)

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