Integrand size = 28, antiderivative size = 47 \[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {3}{2} \text {arcsinh}\left (\frac {1+2 x^2}{\sqrt {3}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {1-x^2}{2 \sqrt {1+x^2+x^4}}\right ) \] Output:
3/2*arcsinh(1/3*(2*x^2+1)*3^(1/2))+1/2*arctanh(1/2*(-x^2+1)/(x^4+x^2+1)^(1 /2))
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.04 \[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\text {arctanh}\left (1+x^2-\sqrt {1+x^2+x^4}\right )-\frac {3}{2} \log \left (-1-2 x^2+2 \sqrt {1+x^2+x^4}\right ) \] Input:
Integrate[(x*(2 + 3*x^2))/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
-ArcTanh[1 + x^2 - Sqrt[1 + x^2 + x^4]] - (3*Log[-1 - 2*x^2 + 2*Sqrt[1 + x ^2 + x^4]])/2
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2238, 1269, 1090, 222, 1154, 219}
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.
\(\displaystyle \int \frac {x \left (3 x^2+2\right )}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}} \, dx\) |
\(\Big \downarrow \) 2238 |
\(\displaystyle \frac {1}{2} \int \frac {3 x^2+2}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx^2\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {1}{2} \left (3 \int \frac {1}{\sqrt {x^4+x^2+1}}dx^2-\int \frac {1}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx^2\right )\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {1}{2} \left (\sqrt {3} \int \frac {1}{\sqrt {\frac {x^4}{3}+1}}d\left (2 x^2+1\right )-\int \frac {1}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx^2\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (3 \text {arcsinh}\left (\frac {2 x^2+1}{\sqrt {3}}\right )-\int \frac {1}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}dx^2\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{4-x^4}d\frac {1-x^2}{\sqrt {x^4+x^2+1}}+3 \text {arcsinh}\left (\frac {2 x^2+1}{\sqrt {3}}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (3 \text {arcsinh}\left (\frac {2 x^2+1}{\sqrt {3}}\right )+\text {arctanh}\left (\frac {1-x^2}{2 \sqrt {x^4+x^2+1}}\right )\right )\) |
Input:
Int[(x*(2 + 3*x^2))/((1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
(3*ArcSinh[(1 + 2*x^2)/Sqrt[3]] + ArcTanh[(1 - x^2)/(2*Sqrt[1 + x^2 + x^4] )])/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Px_)*(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(Px /. x -> Sqrt[x])*(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] && PolyQ[Px, x^2]
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {3 \,\operatorname {arcsinh}\left (\frac {\left (2 x^{2}+1\right ) \sqrt {3}}{3}\right )}{2}-\frac {\operatorname {arctanh}\left (\frac {x^{2}-1}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}\) | \(37\) |
default | \(\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x^{2}+\frac {1}{2}\right )}{3}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {-x^{2}+1}{2 \sqrt {\left (x^{2}+1\right )^{2}-x^{2}}}\right )}{2}\) | \(42\) |
elliptic | \(\frac {3 \,\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x^{2}+\frac {1}{2}\right )}{3}\right )}{2}+\frac {\operatorname {arctanh}\left (\frac {-x^{2}+1}{2 \sqrt {\left (x^{2}+1\right )^{2}-x^{2}}}\right )}{2}\) | \(42\) |
trager | \(-\frac {\ln \left (-\frac {-32 x^{8}+32 \sqrt {x^{4}+x^{2}+1}\, x^{6}-112 x^{6}+96 \sqrt {x^{4}+x^{2}+1}\, x^{4}-162 x^{4}+102 x^{2} \sqrt {x^{4}+x^{2}+1}-121 x^{2}+40 \sqrt {x^{4}+x^{2}+1}-41}{x^{2}+1}\right )}{2}\) | \(92\) |
Input:
int(x*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
3/2*arcsinh(1/3*(2*x^2+1)*3^(1/2))-1/2*arctanh(1/2*(x^2-1)/(x^4+x^2+1)^(1/ 2))
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.32 \[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1}\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1} - 2\right ) - \frac {3}{2} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} + 1} - 1\right ) \] Input:
integrate(x*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")
Output:
1/2*log(-x^2 + sqrt(x^4 + x^2 + 1)) - 1/2*log(-x^2 + sqrt(x^4 + x^2 + 1) - 2) - 3/2*log(-2*x^2 + 2*sqrt(x^4 + x^2 + 1) - 1)
\[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {x \left (3 x^{2} + 2\right )}{\sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:
integrate(x*(3*x**2+2)/(x**2+1)/(x**4+x**2+1)**(1/2),x)
Output:
Integral(x*(3*x**2 + 2)/(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1)), x)
\[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {{\left (3 \, x^{2} + 2\right )} x}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(x*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x^2 + 2)*x/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)), x)
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.32 \[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} + x^{2} + 1} + 2\right ) + \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1}\right ) - \frac {3}{2} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + x^{2} + 1} - 1\right ) \] Input:
integrate(x*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="giac")
Output:
-1/2*log(x^2 - sqrt(x^4 + x^2 + 1) + 2) + 1/2*log(-x^2 + sqrt(x^4 + x^2 + 1)) - 3/2*log(-2*x^2 + 2*sqrt(x^4 + x^2 + 1) - 1)
Timed out. \[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {x\,\left (3\,x^2+2\right )}{\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \] Input:
int((x*(3*x^2 + 2))/((x^2 + 1)*(x^2 + x^4 + 1)^(1/2)),x)
Output:
int((x*(3*x^2 + 2))/((x^2 + 1)*(x^2 + x^4 + 1)^(1/2)), x)
\[ \int \frac {x \left (2+3 x^2\right )}{\left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=3 \left (\int \frac {x^{3}}{\sqrt {x^{4}+x^{2}+1}\, x^{2}+\sqrt {x^{4}+x^{2}+1}}d x \right )+2 \left (\int \frac {x}{\sqrt {x^{4}+x^{2}+1}\, x^{2}+\sqrt {x^{4}+x^{2}+1}}d x \right ) \] Input:
int(x*(3*x^2+2)/(x^2+1)/(x^4+x^2+1)^(1/2),x)
Output:
3*int(x**3/(sqrt(x**4 + x**2 + 1)*x**2 + sqrt(x**4 + x**2 + 1)),x) + 2*int (x/(sqrt(x**4 + x**2 + 1)*x**2 + sqrt(x**4 + x**2 + 1)),x)