3.3.24 \(\int \frac {-1-2 x^2-x^5-2 x^3 \log (x)-x \log ^2(x)}{x^5+2 x^3 \log (x)+x \log ^2(x)} \, dx\)

Optimal. Leaf size=15 \[ -\frac {4}{3}-x+\frac {1}{x^2+\log (x)} \]

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Rubi [A]  time = 0.38, antiderivative size = 12, normalized size of antiderivative = 0.80, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6688, 6742, 6686} \begin {gather*} \frac {1}{x^2+\log (x)}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 2*x^2 - x^5 - 2*x^3*Log[x] - x*Log[x]^2)/(x^5 + 2*x^3*Log[x] + x*Log[x]^2),x]

[Out]

-x + (x^2 + Log[x])^(-1)

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

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 \frac {-1-2 x^2-x^5-2 x^3 \log (x)-x \log ^2(x)}{x \left (x^2+\log (x)\right )^2} \, dx\\ &=\int \left (-1+\frac {-1-2 x^2}{x \left (x^2+\log (x)\right )^2}\right ) \, dx\\ &=-x+\int \frac {-1-2 x^2}{x \left (x^2+\log (x)\right )^2} \, dx\\ &=-x+\frac {1}{x^2+\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 12, normalized size = 0.80 \begin {gather*} -x+\frac {1}{x^2+\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 2*x^2 - x^5 - 2*x^3*Log[x] - x*Log[x]^2)/(x^5 + 2*x^3*Log[x] + x*Log[x]^2),x]

[Out]

-x + (x^2 + Log[x])^(-1)

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fricas [A]  time = 0.80, size = 19, normalized size = 1.27 \begin {gather*} -\frac {x^{3} + x \log \relax (x) - 1}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)^2-2*x^3*log(x)-x^5-2*x^2-1)/(x*log(x)^2+2*x^3*log(x)+x^5),x, algorithm="fricas")

[Out]

-(x^3 + x*log(x) - 1)/(x^2 + log(x))

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giac [A]  time = 0.28, size = 12, normalized size = 0.80 \begin {gather*} -x + \frac {1}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)^2-2*x^3*log(x)-x^5-2*x^2-1)/(x*log(x)^2+2*x^3*log(x)+x^5),x, algorithm="giac")

[Out]

-x + 1/(x^2 + log(x))

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maple [A]  time = 0.02, size = 13, normalized size = 0.87

method	result	size
risch	\(-x +\frac {1}{\ln \relax (x )+x^{2}}\)	\(13\)
norman	\(\frac {1-x^{3}-x \ln \relax (x )}{\ln \relax (x )+x^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*ln(x)^2-2*x^3*ln(x)-x^5-2*x^2-1)/(x*ln(x)^2+2*x^3*ln(x)+x^5),x,method=_RETURNVERBOSE)

[Out]

-x+1/(ln(x)+x^2)

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maxima [A]  time = 0.55, size = 19, normalized size = 1.27 \begin {gather*} -\frac {x^{3} + x \log \relax (x) - 1}{x^{2} + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*log(x)^2-2*x^3*log(x)-x^5-2*x^2-1)/(x*log(x)^2+2*x^3*log(x)+x^5),x, algorithm="maxima")

[Out]

-(x^3 + x*log(x) - 1)/(x^2 + log(x))

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mupad [B]  time = 0.45, size = 12, normalized size = 0.80 \begin {gather*} \frac {1}{\ln \relax (x)+x^2}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x*log(x)^2 + 2*x^3*log(x) + 2*x^2 + x^5 + 1)/(x*log(x)^2 + 2*x^3*log(x) + x^5),x)

[Out]

1/(log(x) + x^2) - x

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sympy [A]  time = 0.10, size = 8, normalized size = 0.53 \begin {gather*} - x + \frac {1}{x^{2} + \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x*ln(x)**2-2*x**3*ln(x)-x**5-2*x**2-1)/(x*ln(x)**2+2*x**3*ln(x)+x**5),x)

[Out]

-x + 1/(x**2 + log(x))

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