3.92.75 \(\int \frac {1+2 x^3+3 x^3 \log (x)}{x^4+(x+x^4) \log (x)+x \log ^2(x)} \, dx\)

Optimal. Leaf size=17 \[ -\log (3 (1+\log (x)))+\log \left (x^3+\log (x)\right ) \]

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Rubi [A]  time = 0.14, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6742, 2302, 29, 6684} \begin {gather*} \log \left (x^3+\log (x)\right )-\log (\log (x)+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^3 + 3*x^3*Log[x])/(x^4 + (x + x^4)*Log[x] + x*Log[x]^2),x]

[Out]

-Log[1 + Log[x]] + Log[x^3 + Log[x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

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

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Mathematica [A]  time = 0.09, size = 15, normalized size = 0.88 \begin {gather*} -\log (1+\log (x))+\log \left (x^3+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^3 + 3*x^3*Log[x])/(x^4 + (x + x^4)*Log[x] + x*Log[x]^2),x]

[Out]

-Log[1 + Log[x]] + Log[x^3 + Log[x]]

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fricas [A]  time = 0.57, size = 15, normalized size = 0.88 \begin {gather*} \log \left (x^{3} + \log \relax (x)\right ) - \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*log(x)+2*x^3+1)/(x*log(x)^2+(x^4+x)*log(x)+x^4),x, algorithm="fricas")

[Out]

log(x^3 + log(x)) - log(log(x) + 1)

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giac [A]  time = 0.23, size = 15, normalized size = 0.88 \begin {gather*} \log \left (x^{3} + \log \relax (x)\right ) - \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*log(x)+2*x^3+1)/(x*log(x)^2+(x^4+x)*log(x)+x^4),x, algorithm="giac")

[Out]

log(x^3 + log(x)) - log(log(x) + 1)

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maple [A]  time = 0.04, size = 16, normalized size = 0.94

method	result	size
norman	\(-\ln \left (\ln \relax (x )+1\right )+\ln \left (\ln \relax (x )+x^{3}\right )\)	\(16\)
risch	\(-\ln \left (\ln \relax (x )+1\right )+\ln \left (\ln \relax (x )+x^{3}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3*ln(x)+2*x^3+1)/(x*ln(x)^2+(x^4+x)*ln(x)+x^4),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(x)+1)+ln(ln(x)+x^3)

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maxima [A]  time = 0.39, size = 15, normalized size = 0.88 \begin {gather*} \log \left (x^{3} + \log \relax (x)\right ) - \log \left (\log \relax (x) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3*log(x)+2*x^3+1)/(x*log(x)^2+(x^4+x)*log(x)+x^4),x, algorithm="maxima")

[Out]

log(x^3 + log(x)) - log(log(x) + 1)

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mupad [B]  time = 7.18, size = 15, normalized size = 0.88 \begin {gather*} \ln \left (\ln \relax (x)+x^3\right )-\ln \left (\ln \relax (x)+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3*log(x) + 2*x^3 + 1)/(x*log(x)^2 + x^4 + log(x)*(x + x^4)),x)

[Out]

log(log(x) + x^3) - log(log(x) + 1)

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sympy [A]  time = 0.22, size = 14, normalized size = 0.82 \begin {gather*} \log {\left (x^{3} + \log {\relax (x )} \right )} - \log {\left (\log {\relax (x )} + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3*ln(x)+2*x**3+1)/(x*ln(x)**2+(x**4+x)*ln(x)+x**4),x)

[Out]

log(x**3 + log(x)) - log(log(x) + 1)

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