3.2.37 \(\int \frac {x (-7+4 x^2)}{4-5 x^2+x^4} \, dx\) [137]

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

[Out]

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

________________________________________________________________________________________

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 = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1261, 646, 31} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

Rule 31

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

Rule 646

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 1261

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]

Rubi steps

\begin {align*} \int \frac {x \left (-7+4 x^2\right )}{4-5 x^2+x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {-7+4 x}{4-5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x} \, dx,x,x^2\right )+\frac {3}{2} \text {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 = 0.00, size = 25, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )+\frac {3}{2} \log \left (4-x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[1 - x^2]/2 + (3*Log[4 - x^2])/2

________________________________________________________________________________________

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

method	result	size
default	\(\frac {\ln \left (x^{2}-1\right )}{2}+\frac {3 \ln \left (x^{2}-4\right )}{2}\)	\(18\)
risch	\(\frac {\ln \left (x^{2}-1\right )}{2}+\frac {3 \ln \left (x^{2}-4\right )}{2}\)	\(18\)
norman	\(\frac {3 \ln \left (x -2\right )}{2}+\frac {\ln \left (-1+x \right )}{2}+\frac {\ln \left (1+x \right )}{2}+\frac {3 \ln \left (x +2\right )}{2}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(4*x^2-7)/(x^4-5*x^2+4),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {3}{2} \, \log \left (x^{2} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 - 1) + 3/2*log(x^2 - 4)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) + \frac {3}{2} \, \log \left (x^{2} - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(x^2 - 1) + 3/2*log(x^2 - 4)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 17, normalized size = 0.68 \begin {gather*} \frac {3 \log {\left (x^{2} - 4 \right )}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*log(x**2 - 4)/2 + log(x**2 - 1)/2

________________________________________________________________________________________

Giac [A]
time = 4.19, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) + \frac {3}{2} \, \log \left ({\left | x^{2} - 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.06, size = 17, normalized size = 0.68 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}+\frac {3\,\ln \left (x^2-4\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x^2 - 1)/2 + (3*log(x^2 - 4))/2

________________________________________________________________________________________