3.9.89 \(\int \frac {x+(-x+x^2+200 x^3) \log (x)+\log ^2(x)+(-x-200 x^2) \log ^3(x)}{-x^2 \log (x)+x \log ^3(x)} \, dx\)

Optimal. Leaf size=22 \[ -1-x-100 x^2+\log \left (\frac {x}{\log (x)}-\log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.41, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 6, number of rules used = 5, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6741, 6742, 2302, 29, 2541} \begin {gather*} -100 x^2-x+\log \left (x-\log ^2(x)\right )-\log (\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + (-x + x^2 + 200*x^3)*Log[x] + Log[x]^2 + (-x - 200*x^2)*Log[x]^3)/(-(x^2*Log[x]) + x*Log[x]^3),x]

[Out]

-x - 100*x^2 - Log[Log[x]] + Log[x - Log[x]^2]

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 2541

Int[(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x_)^(m_.))/((x_)*(Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_
)^(m_.))), x_Symbol] :> Simp[(e*Log[a*x^m + b*Log[c*x^n]^q])/(b*n*q), x] /; FreeQ[{a, b, c, d, e, m, n, q, r},
x] && EqQ[r, q - 1] && EqQ[a*e*m - b*d*n*q, 0]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

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 {-x-\left (-x+x^2+200 x^3\right ) \log (x)-\log ^2(x)-\left (-x-200 x^2\right ) \log ^3(x)}{x \log (x) \left (x-\log ^2(x)\right )} \, dx\\ &=\int \left (-1-200 x-\frac {1}{x \log (x)}+\frac {x-2 \log (x)}{x \left (x-\log ^2(x)\right )}\right ) \, dx\\ &=-x-100 x^2-\int \frac {1}{x \log (x)} \, dx+\int \frac {x-2 \log (x)}{x \left (x-\log ^2(x)\right )} \, dx\\ &=-x-100 x^2+\log \left (x-\log ^2(x)\right )-\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right )\\ &=-x-100 x^2-\log (\log (x))+\log \left (x-\log ^2(x)\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 23, normalized size = 1.05 \begin {gather*} -x-100 x^2-\log (\log (x))+\log \left (x-\log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + (-x + x^2 + 200*x^3)*Log[x] + Log[x]^2 + (-x - 200*x^2)*Log[x]^3)/(-(x^2*Log[x]) + x*Log[x]^3),
x]

[Out]

-x - 100*x^2 - Log[Log[x]] + Log[x - Log[x]^2]

________________________________________________________________________________________

fricas [A]  time = 1.00, size = 23, normalized size = 1.05 \begin {gather*} -100 \, x^{2} - x + \log \left (\log \relax (x)^{2} - x\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-200*x^2-x)*log(x)^3+log(x)^2+(200*x^3+x^2-x)*log(x)+x)/(x*log(x)^3-x^2*log(x)),x, algorithm="fric
as")

[Out]

-100*x^2 - x + log(log(x)^2 - x) - log(log(x))

________________________________________________________________________________________

giac [A]  time = 0.36, size = 23, normalized size = 1.05 \begin {gather*} -100 \, x^{2} - x + \log \left (-\log \relax (x)^{2} + x\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-200*x^2-x)*log(x)^3+log(x)^2+(200*x^3+x^2-x)*log(x)+x)/(x*log(x)^3-x^2*log(x)),x, algorithm="giac
")

[Out]

-100*x^2 - x + log(-log(x)^2 + x) - log(log(x))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 24, normalized size = 1.09

method	result	size
norman	\(-x -100 x^{2}-\ln \left (\ln \relax (x )\right )+\ln \left (-\ln \relax (x )^{2}+x \right )\)	\(24\)
risch	\(-100 x^{2}-x -\ln \left (\ln \relax (x )\right )+\ln \left (\ln \relax (x )^{2}-x \right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-200*x^2-x)*ln(x)^3+ln(x)^2+(200*x^3+x^2-x)*ln(x)+x)/(x*ln(x)^3-x^2*ln(x)),x,method=_RETURNVERBOSE)

[Out]

-x-100*x^2-ln(ln(x))+ln(-ln(x)^2+x)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 23, normalized size = 1.05 \begin {gather*} -100 \, x^{2} - x + \log \left (\log \relax (x)^{2} - x\right ) - \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-200*x^2-x)*log(x)^3+log(x)^2+(200*x^3+x^2-x)*log(x)+x)/(x*log(x)^3-x^2*log(x)),x, algorithm="maxi
ma")

[Out]

-100*x^2 - x + log(log(x)^2 - x) - log(log(x))

________________________________________________________________________________________

mupad [B]  time = 0.73, size = 23, normalized size = 1.05 \begin {gather*} \ln \left (x-{\ln \relax (x)}^2\right )-\ln \left (\ln \relax (x)\right )-x-100\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + log(x)^2 + log(x)*(x^2 - x + 200*x^3) - log(x)^3*(x + 200*x^2))/(x^2*log(x) - x*log(x)^3),x)

[Out]

log(x - log(x)^2) - log(log(x)) - x - 100*x^2

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 19, normalized size = 0.86 \begin {gather*} - 100 x^{2} - x + \log {\left (- x + \log {\relax (x )}^{2} \right )} - \log {\left (\log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-200*x**2-x)*ln(x)**3+ln(x)**2+(200*x**3+x**2-x)*ln(x)+x)/(x*ln(x)**3-x**2*ln(x)),x)

[Out]

-100*x**2 - x + log(-x + log(x)**2) - log(log(x))

________________________________________________________________________________________