3.102.98 \(\int \frac {-960 x^2+1440 x^2 \log (x)}{\log ^3(x)} \, dx\)

Optimal. Leaf size=9 \[ \frac {480 x^3}{\log ^2(x)} \]

________________________________________________________________________________________

Rubi [C]  time = 0.23, antiderivative size = 93, normalized size of antiderivative = 10.33, number of steps used = 11, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2561, 2306, 2309, 2178, 2366, 6482} \begin {gather*} 2160 \text {Ei}(3 \log (x))-2160 (2-3 \log (x)) \text {Ei}(3 \log (x))-12960 \log (x) \text {Ei}(3 \log (x))+2160 (3 \log (x)+1) \text {Ei}(3 \log (x))+4320 x^3+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}-\frac {720 x^3 (3 \log (x)+1)}{\log (x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-960*x^2 + 1440*x^2*Log[x])/Log[x]^3,x]

[Out]

4320*x^3 + 2160*ExpIntegralEi[3*Log[x]] - 2160*ExpIntegralEi[3*Log[x]]*(2 - 3*Log[x]) + (240*x^3*(2 - 3*Log[x]
))/Log[x]^2 + (720*x^3*(2 - 3*Log[x]))/Log[x] - 12960*ExpIntegralEi[3*Log[x]]*Log[x] + 2160*ExpIntegralEi[3*Lo
g[x]]*(1 + 3*Log[x]) - (720*x^3*(1 + 3*Log[x]))/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 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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 (-960+1440 \log (x))}{\log ^3(x)} \, dx\\ &=-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}-1440 \int \left (\frac {9 \text {Ei}(3 \log (x))}{2 x}-\frac {x^2 (1+3 \log (x))}{2 \log ^2(x)}\right ) \, dx\\ &=-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}+720 \int \frac {x^2 (1+3 \log (x))}{\log ^2(x)} \, dx-6480 \int \frac {\text {Ei}(3 \log (x))}{x} \, dx\\ &=-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}+2160 \text {Ei}(3 \log (x)) (1+3 \log (x))-\frac {720 x^3 (1+3 \log (x))}{\log (x)}-2160 \int \left (\frac {3 \text {Ei}(3 \log (x))}{x}-\frac {x^2}{\log (x)}\right ) \, dx-6480 \operatorname {Subst}(\int \text {Ei}(3 x) \, dx,x,\log (x))\\ &=2160 x^3-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}-6480 \text {Ei}(3 \log (x)) \log (x)+2160 \text {Ei}(3 \log (x)) (1+3 \log (x))-\frac {720 x^3 (1+3 \log (x))}{\log (x)}+2160 \int \frac {x^2}{\log (x)} \, dx-6480 \int \frac {\text {Ei}(3 \log (x))}{x} \, dx\\ &=2160 x^3-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}-6480 \text {Ei}(3 \log (x)) \log (x)+2160 \text {Ei}(3 \log (x)) (1+3 \log (x))-\frac {720 x^3 (1+3 \log (x))}{\log (x)}+2160 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-6480 \operatorname {Subst}(\int \text {Ei}(3 x) \, dx,x,\log (x))\\ &=4320 x^3+2160 \text {Ei}(3 \log (x))-2160 \text {Ei}(3 \log (x)) (2-3 \log (x))+\frac {240 x^3 (2-3 \log (x))}{\log ^2(x)}+\frac {720 x^3 (2-3 \log (x))}{\log (x)}-12960 \text {Ei}(3 \log (x)) \log (x)+2160 \text {Ei}(3 \log (x)) (1+3 \log (x))-\frac {720 x^3 (1+3 \log (x))}{\log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 9, normalized size = 1.00 \begin {gather*} \frac {480 x^3}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-960*x^2 + 1440*x^2*Log[x])/Log[x]^3,x]

[Out]

(480*x^3)/Log[x]^2

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 9, normalized size = 1.00 \begin {gather*} \frac {480 \, x^{3}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1440*x^2*log(x)-960*x^2)/log(x)^3,x, algorithm="fricas")

[Out]

480*x^3/log(x)^2

________________________________________________________________________________________

giac [A]  time = 0.12, size = 9, normalized size = 1.00 \begin {gather*} \frac {480 \, x^{3}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1440*x^2*log(x)-960*x^2)/log(x)^3,x, algorithm="giac")

[Out]

480*x^3/log(x)^2

________________________________________________________________________________________

maple [A]  time = 0.02, size = 10, normalized size = 1.11

method	result	size
default	\(\frac {480 x^{3}}{\ln \relax (x )^{2}}\)	\(10\)
norman	\(\frac {480 x^{3}}{\ln \relax (x )^{2}}\)	\(10\)
risch	\(\frac {480 x^{3}}{\ln \relax (x )^{2}}\)	\(10\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1440*x^2*ln(x)-960*x^2)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

480*x^3/ln(x)^2

________________________________________________________________________________________

maxima [C]  time = 0.38, size = 17, normalized size = 1.89 \begin {gather*} 4320 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 8640 \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1440*x^2*log(x)-960*x^2)/log(x)^3,x, algorithm="maxima")

[Out]

4320*gamma(-1, -3*log(x)) + 8640*gamma(-2, -3*log(x))

________________________________________________________________________________________

mupad [B]  time = 6.91, size = 9, normalized size = 1.00 \begin {gather*} \frac {480\,x^3}{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1440*x^2*log(x) - 960*x^2)/log(x)^3,x)

[Out]

(480*x^3)/log(x)^2

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 8, normalized size = 0.89 \begin {gather*} \frac {480 x^{3}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1440*x**2*ln(x)-960*x**2)/ln(x)**3,x)

[Out]

480*x**3/log(x)**2

________________________________________________________________________________________