Integrand size = 62, antiderivative size = 74 \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=-\frac {2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (-1-x^2+2 x^4\right )}{3 \left (-1+x^2+2 x^4\right )}+3 \text {arctanh}\left (\sqrt {\frac {1-2 x^2}{1+2 x^2}}\right ) \]
-2*((-2*x^2+1)/(2*x^2+1))^(1/2)*(2*x^4-x^2-1)/(6*x^4+3*x^2-3)+3*arctanh((( -2*x^2+1)/(2*x^2+1))^(1/2))
Time = 3.84 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=\frac {2 \left (-1+x^2\right ) \sqrt {1+2 x^2}+9 \sqrt {1-2 x^2} \left (1+x^2\right ) \text {arctanh}\left (\frac {\sqrt {1-2 x^2}}{\sqrt {1+2 x^2}}\right )}{3 \left (1+x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}} \]
Integrate[(3 - 9*x^4 + 2*x^6)/(x*(1 + x^2)^2*(-1 + 2*x^2)*Sqrt[(1 - 2*x^2) /(1 + 2*x^2)]*(1 + 2*x^2)),x]
(2*(-1 + x^2)*Sqrt[1 + 2*x^2] + 9*Sqrt[1 - 2*x^2]*(1 + x^2)*ArcTanh[Sqrt[1 - 2*x^2]/Sqrt[1 + 2*x^2]])/(3*(1 + x^2)*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqr t[1 + 2*x^2])
Time = 1.60 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2058, 25, 7282, 2109, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {2 x^6-9 x^4+3}{x \left (x^2+1\right )^2 \left (2 x^2-1\right ) \sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {1-2 x^2} \int -\frac {2 x^6-9 x^4+3}{x \left (1-2 x^2\right )^{3/2} \left (x^2+1\right )^2 \sqrt {2 x^2+1}}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\sqrt {1-2 x^2} \int \frac {2 x^6-9 x^4+3}{x \left (1-2 x^2\right )^{3/2} \left (x^2+1\right )^2 \sqrt {2 x^2+1}}dx}{\sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
\(\Big \downarrow \) 7282 |
\(\displaystyle -\frac {\sqrt {1-2 x^2} \int \frac {2 x^6-9 x^4+3}{x^2 \left (1-2 x^2\right )^{3/2} \left (x^2+1\right )^2 \sqrt {2 x^2+1}}dx^2}{2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
\(\Big \downarrow \) 2109 |
\(\displaystyle -\frac {\sqrt {1-2 x^2} \int \left (-\frac {16}{\left (1-2 x^2\right )^{3/2} \left (x^2+1\right ) \sqrt {2 x^2+1}}+\frac {3}{x^2 \left (1-2 x^2\right )^{3/2} \sqrt {2 x^2+1}}+\frac {2}{\left (1-2 x^2\right )^{3/2} \sqrt {2 x^2+1}}+\frac {8}{\left (1-2 x^2\right )^{3/2} \left (x^2+1\right )^2 \sqrt {2 x^2+1}}\right )dx^2}{2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt {1-2 x^2} \left (-3 \text {arctanh}\left (\sqrt {1-2 x^2} \sqrt {2 x^2+1}\right )-\frac {4 \sqrt {2 x^2+1}}{3 \sqrt {1-2 x^2}}+\frac {8 \sqrt {2 x^2+1}}{3 \sqrt {1-2 x^2} \left (x^2+1\right )}\right )}{2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}}\) |
Int[(3 - 9*x^4 + 2*x^6)/(x*(1 + x^2)^2*(-1 + 2*x^2)*Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*(1 + 2*x^2)),x]
-1/2*(Sqrt[1 - 2*x^2]*((-4*Sqrt[1 + 2*x^2])/(3*Sqrt[1 - 2*x^2]) + (8*Sqrt[ 1 + 2*x^2])/(3*Sqrt[1 - 2*x^2]*(1 + x^2)) - 3*ArcTanh[Sqrt[1 - 2*x^2]*Sqrt [1 + 2*x^2]]))/(Sqrt[(1 - 2*x^2)/(1 + 2*x^2)]*Sqrt[1 + 2*x^2])
3.10.81.3.1 Defintions of rubi rules used
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegran d[Px*(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b , c, d, e, f, g, h, m, n, p, q}, x] && PolyQ[Px, x] && IntegersQ[m, n]
Int[(u_)/(x_), x_Symbol] :> With[{lst = PowerVariableExpn[u, 0, x]}, Simp[1 /lst[[2]] Subst[Int[NormalizeIntegrand[Simplify[lst[[1]]/x], x], x], x, ( lst[[3]]*x)^lst[[2]]], x] /; !FalseQ[lst] && NeQ[lst[[2]], 0]] /; NonsumQ[ u] && !RationalFunctionQ[u, x]
Time = 1.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.39
method | result | size |
trager | \(-\frac {2 \left (2 x^{2}+1\right ) \left (x^{2}-1\right ) \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}}{3 \left (2 x^{4}+x^{2}-1\right )}-\frac {3 \ln \left (-\frac {2 \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}\, x^{2}+\sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}-1}{x^{2}}\right )}{2}\) | \(103\) |
default | \(-\frac {64 \sqrt {-4 x^{4}+1}\, x^{6}+16 \left (-4 x^{4}+1\right )^{\frac {3}{2}} x^{2}-32 \sqrt {-4 x^{4}+1}\, x^{4}+162 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-4 x^{4}+1}}\right ) x^{4}-8 \left (-4 x^{4}+1\right )^{\frac {3}{2}}-52 x^{2} \sqrt {-4 x^{4}+1}+81 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-4 x^{4}+1}}\right ) x^{2}+44 \sqrt {-4 x^{4}+1}-81 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-4 x^{4}+1}}\right )}{54 \left (x^{2}+1\right ) \sqrt {-\left (2 x^{2}+1\right ) \left (2 x^{2}-1\right )}\, \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}}\) | \(169\) |
int((2*x^6-9*x^4+3)/x/(x^2+1)^2/(2*x^2-1)/((-2*x^2+1)/(2*x^2+1))^(1/2)/(2* x^2+1),x,method=_RETURNVERBOSE)
-2/3*(2*x^2+1)*(x^2-1)/(2*x^4+x^2-1)*(-(2*x^2-1)/(2*x^2+1))^(1/2)-3/2*ln(- (2*(-(2*x^2-1)/(2*x^2+1))^(1/2)*x^2+(-(2*x^2-1)/(2*x^2+1))^(1/2)-1)/x^2)
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.45 \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=-\frac {8 \, x^{4} + 4 \, x^{2} + 9 \, {\left (2 \, x^{4} + x^{2} - 1\right )} \log \left (\frac {{\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 1}{x^{2}}\right ) + 4 \, {\left (2 \, x^{4} - x^{2} - 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 4}{6 \, {\left (2 \, x^{4} + x^{2} - 1\right )}} \]
integrate((2*x^6-9*x^4+3)/x/(x^2+1)^2/(2*x^2-1)/((-2*x^2+1)/(2*x^2+1))^(1/ 2)/(2*x^2+1),x, algorithm="fricas")
-1/6*(8*x^4 + 4*x^2 + 9*(2*x^4 + x^2 - 1)*log(((2*x^2 + 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) - 1)/x^2) + 4*(2*x^4 - x^2 - 1)*sqrt(-(2*x^2 - 1)/(2*x^2 + 1)) - 4)/(2*x^4 + x^2 - 1)
Timed out. \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=\text {Timed out} \]
integrate((2*x**6-9*x**4+3)/x/(x**2+1)**2/(2*x**2-1)/((-2*x**2+1)/(2*x**2+ 1))**(1/2)/(2*x**2+1),x)
\[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=\int { \frac {2 \, x^{6} - 9 \, x^{4} + 3}{{\left (2 \, x^{2} + 1\right )} {\left (2 \, x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{2} x \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}} \,d x } \]
integrate((2*x^6-9*x^4+3)/x/(x^2+1)^2/(2*x^2-1)/((-2*x^2+1)/(2*x^2+1))^(1/ 2)/(2*x^2+1),x, algorithm="maxima")
integrate((2*x^6 - 9*x^4 + 3)/((2*x^2 + 1)*(2*x^2 - 1)*(x^2 + 1)^2*x*sqrt( -(2*x^2 - 1)/(2*x^2 + 1))), x)
Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.58 \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=-\frac {32}{3 \, {\left (\frac {{\left (\sqrt {-4 \, x^{4} + 1} - 1\right )}^{3}}{x^{6}} + 8\right )}} - \frac {3}{2} \, \log \left (-\frac {\sqrt {-4 \, x^{4} + 1} - 1}{2 \, x^{2}}\right ) \]
integrate((2*x^6-9*x^4+3)/x/(x^2+1)^2/(2*x^2-1)/((-2*x^2+1)/(2*x^2+1))^(1/ 2)/(2*x^2+1),x, algorithm="giac")
Time = 6.63 (sec) , antiderivative size = 270, normalized size of antiderivative = 3.65 \[ \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx=3\,\mathrm {atanh}\left (\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\right )-\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}}{3}\right )+\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {1-2\,x^2}\,\sqrt {\frac {1}{2\,x^2+1}}}{3}\right )}{2\,x^2+2}-\frac {\left (x^2+\frac {1}{2}\right )\,\left (\frac {x^2}{3}-\frac {1}{3}\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}}{2\,x^4+x^2-1}+\frac {3\,x^2}{\sqrt {1-2\,x^2}\,\left (2\,x^2+2\right )\,\sqrt {\frac {1}{2\,x^2+1}}}+\frac {2\,\sqrt {3}\,x^2\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {1-2\,x^2}\,\sqrt {\frac {1}{2\,x^2+1}}}{3}\right )}{2\,x^2+2}-\frac {2{}\mathrm {i}}{\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,3{}\mathrm {i}+{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{3/2}\,1{}\mathrm {i}} \]
int((2*x^6 - 9*x^4 + 3)/(x*(x^2 + 1)^2*(2*x^2 - 1)*(2*x^2 + 1)*(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)),x)
3*atanh((-(2*x^2 - 1)/(2*x^2 + 1))^(1/2)) - 2i/((-(2*x^2 - 1)/(2*x^2 + 1)) ^(1/2)*3i + (-(2*x^2 - 1)/(2*x^2 + 1))^(3/2)*1i) - 3^(1/2)*atan((3^(1/2)*( -(2*x^2 - 1)/(2*x^2 + 1))^(1/2))/3) + (2*3^(1/2)*atan((3^(1/2)*(1 - 2*x^2) ^(1/2)*(1/(2*x^2 + 1))^(1/2))/3))/(2*x^2 + 2) - ((x^2 + 1/2)*(x^2/3 - 1/3) *(-(2*x^2 - 1)/(2*x^2 + 1))^(1/2))/(x^2 + 2*x^4 - 1) + (3*x^2)/((1 - 2*x^2 )^(1/2)*(2*x^2 + 2)*(1/(2*x^2 + 1))^(1/2)) + (2*3^(1/2)*x^2*atan((3^(1/2)* (1 - 2*x^2)^(1/2)*(1/(2*x^2 + 1))^(1/2))/3))/(2*x^2 + 2)