3.43.78 \(\int \frac {108+108 \log (x)+36 \log ^2(x)+4 \log ^3(x)}{84 x+e^4 x+108 x \log (x)+54 x \log ^2(x)+12 x \log ^3(x)+x \log ^4(x)} \, dx\)

Optimal. Leaf size=12 \[ \log \left (3+e^4+(3+\log (x))^4\right ) \]

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Rubi [B]  time = 0.07, antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 4, number of rules used = 3, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.055, Rules used = {6, 12, 1587} \begin {gather*} \log \left (\log ^4(x)+12 \log ^3(x)+54 \log ^2(x)+108 \log (x)+e^4+84\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(108 + 108*Log[x] + 36*Log[x]^2 + 4*Log[x]^3)/(84*x + E^4*x + 108*x*Log[x] + 54*x*Log[x]^2 + 12*x*Log[x]^3
+ x*Log[x]^4),x]

[Out]

Log[84 + E^4 + 108*Log[x] + 54*Log[x]^2 + 12*Log[x]^3 + Log[x]^4]

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 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {108+108 \log (x)+36 \log ^2(x)+4 \log ^3(x)}{\left (84+e^4\right ) x+108 x \log (x)+54 x \log ^2(x)+12 x \log ^3(x)+x \log ^4(x)} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 \left (27+27 x+9 x^2+x^3\right )}{84+e^4+108 x+54 x^2+12 x^3+x^4} \, dx,x,\log (x)\right )\\ &=4 \operatorname {Subst}\left (\int \frac {27+27 x+9 x^2+x^3}{84+e^4+108 x+54 x^2+12 x^3+x^4} \, dx,x,\log (x)\right )\\ &=\log \left (84+e^4+108 \log (x)+54 \log ^2(x)+12 \log ^3(x)+\log ^4(x)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 12, normalized size = 1.00 \begin {gather*} \log \left (3+e^4+(3+\log (x))^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(108 + 108*Log[x] + 36*Log[x]^2 + 4*Log[x]^3)/(84*x + E^4*x + 108*x*Log[x] + 54*x*Log[x]^2 + 12*x*Lo
g[x]^3 + x*Log[x]^4),x]

[Out]

Log[3 + E^4 + (3 + Log[x])^4]

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fricas [B]  time = 0.53, size = 25, normalized size = 2.08 \begin {gather*} \log \left (\log \relax (x)^{4} + 12 \, \log \relax (x)^{3} + 54 \, \log \relax (x)^{2} + e^{4} + 108 \, \log \relax (x) + 84\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)^3+36*log(x)^2+108*log(x)+108)/(x*log(x)^4+12*x*log(x)^3+54*x*log(x)^2+108*x*log(x)+x*exp(4
)+84*x),x, algorithm="fricas")

[Out]

log(log(x)^4 + 12*log(x)^3 + 54*log(x)^2 + e^4 + 108*log(x) + 84)

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giac [B]  time = 0.13, size = 25, normalized size = 2.08 \begin {gather*} \log \left (\log \relax (x)^{4} + 12 \, \log \relax (x)^{3} + 54 \, \log \relax (x)^{2} + e^{4} + 108 \, \log \relax (x) + 84\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)^3+36*log(x)^2+108*log(x)+108)/(x*log(x)^4+12*x*log(x)^3+54*x*log(x)^2+108*x*log(x)+x*exp(4
)+84*x),x, algorithm="giac")

[Out]

log(log(x)^4 + 12*log(x)^3 + 54*log(x)^2 + e^4 + 108*log(x) + 84)

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maple [B]  time = 0.05, size = 26, normalized size = 2.17

method	result	size
default	\(\ln \left (\ln \relax (x )^{4}+12 \ln \relax (x )^{3}+54 \ln \relax (x )^{2}+108 \ln \relax (x )+{\mathrm e}^{4}+84\right )\)	\(26\)
norman	\(\ln \left (\ln \relax (x )^{4}+12 \ln \relax (x )^{3}+54 \ln \relax (x )^{2}+108 \ln \relax (x )+{\mathrm e}^{4}+84\right )\)	\(26\)
risch	\(\ln \left (\ln \relax (x )^{4}+12 \ln \relax (x )^{3}+54 \ln \relax (x )^{2}+108 \ln \relax (x )+{\mathrm e}^{4}+84\right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*ln(x)^3+36*ln(x)^2+108*ln(x)+108)/(x*ln(x)^4+12*x*ln(x)^3+54*x*ln(x)^2+108*x*ln(x)+x*exp(4)+84*x),x,met
hod=_RETURNVERBOSE)

[Out]

ln(ln(x)^4+12*ln(x)^3+54*ln(x)^2+108*ln(x)+exp(4)+84)

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maxima [B]  time = 0.37, size = 25, normalized size = 2.08 \begin {gather*} \log \left (\log \relax (x)^{4} + 12 \, \log \relax (x)^{3} + 54 \, \log \relax (x)^{2} + e^{4} + 108 \, \log \relax (x) + 84\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*log(x)^3+36*log(x)^2+108*log(x)+108)/(x*log(x)^4+12*x*log(x)^3+54*x*log(x)^2+108*x*log(x)+x*exp(4
)+84*x),x, algorithm="maxima")

[Out]

log(log(x)^4 + 12*log(x)^3 + 54*log(x)^2 + e^4 + 108*log(x) + 84)

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mupad [B]  time = 3.18, size = 25, normalized size = 2.08 \begin {gather*} \ln \left ({\ln \relax (x)}^4+12\,{\ln \relax (x)}^3+54\,{\ln \relax (x)}^2+108\,\ln \relax (x)+{\mathrm {e}}^4+84\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((108*log(x) + 36*log(x)^2 + 4*log(x)^3 + 108)/(84*x + 54*x*log(x)^2 + 12*x*log(x)^3 + x*log(x)^4 + x*exp(4
) + 108*x*log(x)),x)

[Out]

log(exp(4) + 108*log(x) + 54*log(x)^2 + 12*log(x)^3 + log(x)^4 + 84)

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sympy [B]  time = 0.17, size = 29, normalized size = 2.42 \begin {gather*} \log {\left (\log {\relax (x )}^{4} + 12 \log {\relax (x )}^{3} + 54 \log {\relax (x )}^{2} + 108 \log {\relax (x )} + e^{4} + 84 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*ln(x)**3+36*ln(x)**2+108*ln(x)+108)/(x*ln(x)**4+12*x*ln(x)**3+54*x*ln(x)**2+108*x*ln(x)+x*exp(4)+
84*x),x)

[Out]

log(log(x)**4 + 12*log(x)**3 + 54*log(x)**2 + 108*log(x) + exp(4) + 84)

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