3.312 \(\int \frac {2-4 x^2+x^3}{(1+x^2) (2+x^2)} \, dx\)

Optimal. Leaf size=36 \[ -\frac {1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]

[Out]

6*arctan(x)-1/2*ln(x^2+1)+ln(x^2+2)-5*arctan(1/2*x*2^(1/2))*2^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6725, 635, 203, 260} \[ -\frac {1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 - 4*x^2 + x^3)/((1 + x^2)*(2 + x^2)),x]

[Out]

6*ArcTan[x] - 5*Sqrt[2]*ArcTan[x/Sqrt[2]] - Log[1 + x^2]/2 + Log[2 + x^2]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {2-4 x^2+x^3}{\left (1+x^2\right ) \left (2+x^2\right )} \, dx &=\int \left (\frac {6-x}{1+x^2}+\frac {2 (-5+x)}{2+x^2}\right ) \, dx\\ &=2 \int \frac {-5+x}{2+x^2} \, dx+\int \frac {6-x}{1+x^2} \, dx\\ &=2 \int \frac {x}{2+x^2} \, dx+6 \int \frac {1}{1+x^2} \, dx-10 \int \frac {1}{2+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=6 \tan ^{-1}(x)-5 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{2} \log \left (1+x^2\right )+\log \left (2+x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 36, normalized size = 1.00 \[ -\frac {1}{2} \log \left (x^2+1\right )+\log \left (x^2+2\right )+6 \tan ^{-1}(x)-5 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 - 4*x^2 + x^3)/((1 + x^2)*(2 + x^2)),x]

[Out]

6*ArcTan[x] - 5*Sqrt[2]*ArcTan[x/Sqrt[2]] - Log[1 + x^2]/2 + Log[2 + x^2]

________________________________________________________________________________________

fricas [A]  time = 0.84, size = 31, normalized size = 0.86 \[ -5 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 6 \, \arctan \relax (x) + \log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="fricas")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 31, normalized size = 0.86 \[ -5 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 6 \, \arctan \relax (x) + \log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="giac")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 32, normalized size = 0.89 \[ 6 \arctan \relax (x )-5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {\ln \left (x^{2}+1\right )}{2}+\ln \left (x^{2}+2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*arctan(x)-1/2*ln(x^2+1)+ln(x^2+2)-5*2^(1/2)*arctan(1/2*2^(1/2)*x)

________________________________________________________________________________________

maxima [A]  time = 2.10, size = 31, normalized size = 0.86 \[ -5 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + 6 \, \arctan \relax (x) + \log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-4*x^2+2)/(x^2+1)/(x^2+2),x, algorithm="maxima")

[Out]

-5*sqrt(2)*arctan(1/2*sqrt(2)*x) + 6*arctan(x) + log(x^2 + 2) - 1/2*log(x^2 + 1)

________________________________________________________________________________________

mupad [B]  time = 0.11, size = 56, normalized size = 1.56 \[ \ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{2}-3{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+3{}\mathrm {i}\right )+\ln \left (x-\sqrt {2}\,1{}\mathrm {i}\right )\,\left (1+\frac {\sqrt {2}\,5{}\mathrm {i}}{2}\right )-\ln \left (x+\sqrt {2}\,1{}\mathrm {i}\right )\,\left (-1+\frac {\sqrt {2}\,5{}\mathrm {i}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 36, normalized size = 1.00 \[ - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \log {\left (x^{2} + 2 \right )} + 6 \operatorname {atan}{\relax (x )} - 5 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-4*x**2+2)/(x**2+1)/(x**2+2),x)

[Out]

-log(x**2 + 1)/2 + log(x**2 + 2) + 6*atan(x) - 5*sqrt(2)*atan(sqrt(2)*x/2)

________________________________________________________________________________________