3.93.67 \(\int \frac {-x^3+(3-x^3-5 x^4) \log (x)+4 x^3 \log (x) \log (\frac {4}{x \log (x)})}{\log (x)} \, dx\)

Optimal. Leaf size=28 \[ -3 e^5+x \left (3+x^3 \left (-x+\log \left (\frac {4}{x \log (x)}\right )\right )\right ) \]

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Rubi [C]  time = 0.21, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 11, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {6742, 6688, 2309, 2178, 30, 2555, 12, 2366, 6482} \begin {gather*} -\text {Ei}(4 \log (x))-\log (x) \text {Ei}(4 \log (x))+(\log (x)+1) \text {Ei}(4 \log (x))-x^5+x^4 \log \left (\frac {4}{x \log (x)}\right )+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x - x^5 - ExpIntegralEi[4*Log[x]] - ExpIntegralEi[4*Log[x]]*Log[x] + ExpIntegralEi[4*Log[x]]*(1 + Log[x]) +
x^4*Log[4/(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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 2555

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*Simplify
[D[u, x]/u], x], x] /; InverseFunctionFreeQ[w, x]] /; ProductQ[u]

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 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 \left (\frac {-x^3+3 \log (x)-x^3 \log (x)-5 x^4 \log (x)}{\log (x)}+4 x^3 \log \left (\frac {4}{x \log (x)}\right )\right ) \, dx\\ &=4 \int x^3 \log \left (\frac {4}{x \log (x)}\right ) \, dx+\int \frac {-x^3+3 \log (x)-x^3 \log (x)-5 x^4 \log (x)}{\log (x)} \, dx\\ &=x^4 \log \left (\frac {4}{x \log (x)}\right )-4 \int -\frac {x^3 (1+\log (x))}{4 \log (x)} \, dx+\int \left (3-x^3-5 x^4-\frac {x^3}{\log (x)}\right ) \, dx\\ &=3 x-\frac {x^4}{4}-x^5+x^4 \log \left (\frac {4}{x \log (x)}\right )-\int \frac {x^3}{\log (x)} \, dx+\int \frac {x^3 (1+\log (x))}{\log (x)} \, dx\\ &=3 x-\frac {x^4}{4}-x^5+\text {Ei}(4 \log (x)) (1+\log (x))+x^4 \log \left (\frac {4}{x \log (x)}\right )-\int \frac {\text {Ei}(4 \log (x))}{x} \, dx-\operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=3 x-\frac {x^4}{4}-x^5-\text {Ei}(4 \log (x))+\text {Ei}(4 \log (x)) (1+\log (x))+x^4 \log \left (\frac {4}{x \log (x)}\right )-\operatorname {Subst}(\int \text {Ei}(4 x) \, dx,x,\log (x))\\ &=3 x-x^5-\text {Ei}(4 \log (x))-\text {Ei}(4 \log (x)) \log (x)+\text {Ei}(4 \log (x)) (1+\log (x))+x^4 \log \left (\frac {4}{x \log (x)}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 23, normalized size = 0.82 \begin {gather*} 3 x-x^5+x^4 \log \left (\frac {4}{x \log (x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

3*x - x^5 + x^4*Log[4/(x*Log[x])]

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fricas [A]  time = 0.91, size = 23, normalized size = 0.82 \begin {gather*} -x^{5} + x^{4} \log \left (\frac {4}{x \log \relax (x)}\right ) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)*log(4/x/log(x))+(-5*x^4-x^3+3)*log(x)-x^3)/log(x),x, algorithm="fricas")

[Out]

-x^5 + x^4*log(4/(x*log(x))) + 3*x

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giac [A]  time = 0.15, size = 31, normalized size = 1.11 \begin {gather*} -x^{5} + 2 \, x^{4} \log \relax (2) - x^{4} \log \relax (x) - x^{4} \log \left (\log \relax (x)\right ) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)*log(4/x/log(x))+(-5*x^4-x^3+3)*log(x)-x^3)/log(x),x, algorithm="giac")

[Out]

-x^5 + 2*x^4*log(2) - x^4*log(x) - x^4*log(log(x)) + 3*x

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maple [C]  time = 0.10, size = 140, normalized size = 5.00

method	result	size
risch	\(-x^{4} \ln \left (\ln \relax (x )\right )-x^{4} \ln \relax (x )+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )}\right )^{2}}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )}\right ) \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )}{2}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \relax (x )}\right )}{2}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (\frac {i}{x \ln \relax (x )}\right )^{3}}{2}+2 x^{4} \ln \relax (2)-x^{5}+3 x\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3*ln(x)*ln(4/x/ln(x))+(-5*x^4-x^3+3)*ln(x)-x^3)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-x^4*ln(ln(x))-x^4*ln(x)+1/2*I*Pi*x^4*csgn(I/x)*csgn(I/x/ln(x))^2-1/2*I*Pi*x^4*csgn(I/x)*csgn(I/x/ln(x))*csgn(
I/ln(x))+1/2*I*Pi*x^4*csgn(I/x/ln(x))^2*csgn(I/ln(x))-1/2*I*Pi*x^4*csgn(I/x/ln(x))^3+2*x^4*ln(2)-x^5+3*x

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -x^{5} + \frac {1}{4} \, x^{4} {\left (8 \, \log \relax (2) + 1\right )} - x^{4} \log \relax (x) - x^{4} \log \left (\log \relax (x)\right ) - \frac {1}{4} \, x^{4} + 3 \, x - {\rm Ei}\left (4 \, \log \relax (x)\right ) + \int \frac {x^{3}}{\log \relax (x)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3*log(x)*log(4/x/log(x))+(-5*x^4-x^3+3)*log(x)-x^3)/log(x),x, algorithm="maxima")

[Out]

-x^5 + 1/4*x^4*(8*log(2) + 1) - x^4*log(x) - x^4*log(log(x)) - 1/4*x^4 + 3*x - Ei(4*log(x)) + integrate(x^3/lo
g(x), x)

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mupad [B]  time = 7.97, size = 23, normalized size = 0.82 \begin {gather*} 3\,x+x^4\,\ln \left (\frac {4}{x\,\ln \relax (x)}\right )-x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(x^3 + 5*x^4 - 3) + x^3 - 4*x^3*log(x)*log(4/(x*log(x))))/log(x),x)

[Out]

3*x + x^4*log(4/(x*log(x))) - x^5

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sympy [A]  time = 0.33, size = 17, normalized size = 0.61 \begin {gather*} - x^{5} + x^{4} \log {\left (\frac {4}{x \log {\relax (x )}} \right )} + 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3*ln(x)*ln(4/x/ln(x))+(-5*x**4-x**3+3)*ln(x)-x**3)/ln(x),x)

[Out]

-x**5 + x**4*log(4/(x*log(x))) + 3*x

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