3.72.92 \(\int \frac {e^{6/5} (-6+x^3)}{3 x-4 x^3+x^4} \, dx\)

Optimal. Leaf size=15 \[ e^{6/5} \log \left (-4+\frac {3}{x^2}+x\right ) \]

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Rubi [B]  time = 0.14, antiderivative size = 39, normalized size of antiderivative = 2.60, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 1594, 6742, 628} \begin {gather*} e^{6/5} \log \left (-x^2+3 x+3\right )+e^{6/5} \log (1-x)-2 e^{6/5} \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(6/5)*(-6 + x^3))/(3*x - 4*x^3 + x^4),x]

[Out]

E^(6/5)*Log[1 - x] - 2*E^(6/5)*Log[x] + E^(6/5)*Log[3 + 3*x - x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{6/5} \int \frac {-6+x^3}{3 x-4 x^3+x^4} \, dx\\ &=e^{6/5} \int \frac {-6+x^3}{x \left (3-4 x^2+x^3\right )} \, dx\\ &=e^{6/5} \int \left (\frac {1}{-1+x}-\frac {2}{x}+\frac {-3+2 x}{-3-3 x+x^2}\right ) \, dx\\ &=e^{6/5} \log (1-x)-2 e^{6/5} \log (x)+e^{6/5} \int \frac {-3+2 x}{-3-3 x+x^2} \, dx\\ &=e^{6/5} \log (1-x)-2 e^{6/5} \log (x)+e^{6/5} \log \left (3+3 x-x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.47 \begin {gather*} e^{6/5} \left (-2 \log (x)+\log \left (3-4 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(6/5)*(-6 + x^3))/(3*x - 4*x^3 + x^4),x]

[Out]

E^(6/5)*(-2*Log[x] + Log[3 - 4*x^2 + x^3])

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fricas [A]  time = 0.57, size = 21, normalized size = 1.40 \begin {gather*} e^{\frac {6}{5}} \log \left (x^{3} - 4 \, x^{2} + 3\right ) - 2 \, e^{\frac {6}{5}} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6)*exp(3/5)^2/(x^4-4*x^3+3*x),x, algorithm="fricas")

[Out]

e^(6/5)*log(x^3 - 4*x^2 + 3) - 2*e^(6/5)*log(x)

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giac [A]  time = 0.15, size = 24, normalized size = 1.60 \begin {gather*} {\left (\log \left ({\left | x^{2} - 3 \, x - 3 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) - 2 \, \log \left ({\left | x \right |}\right )\right )} e^{\frac {6}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6)*exp(3/5)^2/(x^4-4*x^3+3*x),x, algorithm="giac")

[Out]

(log(abs(x^2 - 3*x - 3)) + log(abs(x - 1)) - 2*log(abs(x)))*e^(6/5)

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maple [A]  time = 0.03, size = 22, normalized size = 1.47

method	result	size
risch	\(-2 \,{\mathrm e}^{\frac {6}{5}} \ln \relax (x )+{\mathrm e}^{\frac {6}{5}} \ln \left (x^{3}-4 x^{2}+3\right )\)	\(22\)
default	\({\mathrm e}^{\frac {6}{5}} \left (-2 \ln \relax (x )+\ln \left (x -1\right )+\ln \left (x^{2}-3 x -3\right )\right )\)	\(24\)
norman	\({\mathrm e}^{\frac {6}{5}} \ln \left (x -1\right )+{\mathrm e}^{\frac {6}{5}} \ln \left (x^{2}-3 x -3\right )-2 \,{\mathrm e}^{\frac {6}{5}} \ln \relax (x )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-6)*exp(3/5)^2/(x^4-4*x^3+3*x),x,method=_RETURNVERBOSE)

[Out]

-2*exp(6/5)*ln(x)+exp(6/5)*ln(x^3-4*x^2+3)

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maxima [A]  time = 0.36, size = 21, normalized size = 1.40 \begin {gather*} {\left (\log \left (x^{2} - 3 \, x - 3\right ) + \log \left (x - 1\right ) - 2 \, \log \relax (x)\right )} e^{\frac {6}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-6)*exp(3/5)^2/(x^4-4*x^3+3*x),x, algorithm="maxima")

[Out]

(log(x^2 - 3*x - 3) + log(x - 1) - 2*log(x))*e^(6/5)

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mupad [B]  time = 4.33, size = 21, normalized size = 1.40 \begin {gather*} {\mathrm {e}}^{6/5}\,\ln \left (x^3-4\,x^2+3\right )-2\,{\mathrm {e}}^{6/5}\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(6/5)*(x^3 - 6))/(3*x - 4*x^3 + x^4),x)

[Out]

exp(6/5)*log(x^3 - 4*x^2 + 3) - 2*exp(6/5)*log(x)

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sympy [A]  time = 0.29, size = 26, normalized size = 1.73 \begin {gather*} - 2 e^{\frac {6}{5}} \log {\relax (x )} + e^{\frac {6}{5}} \log {\left (x^{3} - 4 x^{2} + 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-6)*exp(3/5)**2/(x**4-4*x**3+3*x),x)

[Out]

-2*exp(6/5)*log(x) + exp(6/5)*log(x**3 - 4*x**2 + 3)

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