3.2.57 \(\int \frac {x^2}{9-10 x^3+x^6} \, dx\) [157]

Optimal. Leaf size=25 \[ -\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right ) \]

[Out]

-1/24*ln(-x^3+1)+1/24*ln(-x^3+9)

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1366, 630, 31} \begin {gather*} \frac {1}{24} \log \left (9-x^3\right )-\frac {1}{24} \log \left (1-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(9 - 10*x^3 + x^6),x]

[Out]

-1/24*Log[1 - x^3] + Log[9 - x^3]/24

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 630

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*x +
c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]

Rubi steps

\begin {align*} \int \frac {x^2}{9-10 x^3+x^6} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{9-10 x+x^2} \, dx,x,x^3\right )\\ &=\frac {1}{24} \text {Subst}\left (\int \frac {1}{-9+x} \, dx,x,x^3\right )-\frac {1}{24} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^3\right )\\ &=-\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right )\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {1}{24} \log \left (1-x^3\right )+\frac {1}{24} \log \left (9-x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(9 - 10*x^3 + x^6),x]

[Out]

-1/24*Log[1 - x^3] + Log[9 - x^3]/24

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Maple [A]
time = 0.02, size = 18, normalized size = 0.72

method	result	size
default	\(-\frac {\ln \left (x^{3}-1\right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}\)	\(18\)
risch	\(-\frac {\ln \left (x^{3}-1\right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}\)	\(18\)
norman	\(-\frac {\ln \left (-1+x \right )}{24}+\frac {\ln \left (x^{3}-9\right )}{24}-\frac {\ln \left (x^{2}+x +1\right )}{24}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^6-10*x^3+9),x,method=_RETURNVERBOSE)

[Out]

-1/24*ln(x^3-1)+1/24*ln(x^3-9)

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Maxima [A]
time = 1.90, size = 17, normalized size = 0.68 \begin {gather*} -\frac {1}{24} \, \log \left (x^{3} - 1\right ) + \frac {1}{24} \, \log \left (x^{3} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6-10*x^3+9),x, algorithm="maxima")

[Out]

-1/24*log(x^3 - 1) + 1/24*log(x^3 - 9)

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Fricas [A]
time = 0.37, size = 17, normalized size = 0.68 \begin {gather*} -\frac {1}{24} \, \log \left (x^{3} - 1\right ) + \frac {1}{24} \, \log \left (x^{3} - 9\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6-10*x^3+9),x, algorithm="fricas")

[Out]

-1/24*log(x^3 - 1) + 1/24*log(x^3 - 9)

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Sympy [A]
time = 0.03, size = 15, normalized size = 0.60 \begin {gather*} \frac {\log {\left (x^{3} - 9 \right )}}{24} - \frac {\log {\left (x^{3} - 1 \right )}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**6-10*x**3+9),x)

[Out]

log(x**3 - 9)/24 - log(x**3 - 1)/24

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Giac [A]
time = 0.48, size = 19, normalized size = 0.76 \begin {gather*} -\frac {1}{24} \, \log \left ({\left | x^{3} - 1 \right |}\right ) + \frac {1}{24} \, \log \left ({\left | x^{3} - 9 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^6-10*x^3+9),x, algorithm="giac")

[Out]

-1/24*log(abs(x^3 - 1)) + 1/24*log(abs(x^3 - 9))

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Mupad [B]
time = 0.58, size = 16, normalized size = 0.64 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {81}{320\,\left (\frac {5\,x^3}{4}-\frac {9}{8}\right )}-\frac {41}{40}\right )}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^6 - 10*x^3 + 9),x)

[Out]

atanh(81/(320*((5*x^3)/4 - 9/8)) - 41/40)/12

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