3.138 \(\int \frac{-7 x+4 x^3}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

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Rubi [A]  time = 0.0293958, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 1247, 632, 31} \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-7*x + 4*x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

Rule 1593

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

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 632

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

Rule 31

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

Rubi steps

\begin{align*} \int \frac{-7 x+4 x^3}{4-5 x^2+x^4} \, dx &=\int \frac{x \left (-7+4 x^2\right )}{4-5 x^2+x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-7+4 x}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{-4+x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0054111, size = 25, normalized size = 1. \[ \frac{1}{2} \log \left (1-x^2\right )+\frac{3}{2} \log \left (4-x^2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-7*x + 4*x^3)/(4 - 5*x^2 + x^4),x]

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

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Maple [A]  time = 0.005, size = 18, normalized size = 0.7 \begin{align*}{\frac{\ln \left ({x}^{2}-1 \right ) }{2}}+{\frac{3\,\ln \left ({x}^{2}-4 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-7*x)/(x^4-5*x^2+4),x)

[Out]

1/2*ln(x^2-1)+3/2*ln(x^2-4)

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Maxima [A]  time = 0.945906, size = 34, normalized size = 1.36 \begin{align*} \frac{3}{2} \, \log \left (x + 2\right ) + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) + \frac{3}{2} \, \log \left (x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-7*x)/(x^4-5*x^2+4),x, algorithm="maxima")

[Out]

3/2*log(x + 2) + 1/2*log(x + 1) + 1/2*log(x - 1) + 3/2*log(x - 2)

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Fricas [A]  time = 1.47928, size = 50, normalized size = 2. \begin{align*} \frac{1}{2} \, \log \left (x^{2} - 1\right ) + \frac{3}{2} \, \log \left (x^{2} - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-7*x)/(x^4-5*x^2+4),x, algorithm="fricas")

[Out]

1/2*log(x^2 - 1) + 3/2*log(x^2 - 4)

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Sympy [A]  time = 0.110134, size = 17, normalized size = 0.68 \begin{align*} \frac{3 \log{\left (x^{2} - 4 \right )}}{2} + \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-7*x)/(x**4-5*x**2+4),x)

[Out]

3*log(x**2 - 4)/2 + log(x**2 - 1)/2

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Giac [A]  time = 1.08159, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \frac{3}{2} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-7*x)/(x^4-5*x^2+4),x, algorithm="giac")

[Out]

1/2*log(abs(x^2 - 1)) + 3/2*log(abs(x^2 - 4))