Integrand size = 30, antiderivative size = 53 \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\frac {1}{2} \text {arctanh}\left (\frac {1-x^2}{2 \sqrt {1+x^2+x^4}}\right )-\text {arctanh}\left (\frac {2+x^2}{2 \sqrt {1+x^2+x^4}}\right ) \] Output:
-1/2*arctanh(1/2*(-x^2+1)/(x^4+x^2+1)^(1/2))-arctanh(1/2*(x^2+2)/(x^4+x^2+ 1)^(1/2))
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=2 \text {arctanh}\left (x^2-\sqrt {1+x^2+x^4}\right )+\text {arctanh}\left (1+x^2-\sqrt {1+x^2+x^4}\right ) \] Input:
Integrate[(2 + 3*x^2)/(x*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
2*ArcTanh[x^2 - Sqrt[1 + x^2 + x^4]] + ArcTanh[1 + x^2 - Sqrt[1 + x^2 + x^ 4]]
Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2248, 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.
\(\displaystyle \int \frac {3 x^2+2}{x \left (x^2+1\right ) \sqrt {x^4+x^2+1}} \, dx\) |
\(\Big \downarrow \) 2248 |
\(\displaystyle \int \left (\frac {x}{\left (x^2+1\right ) \sqrt {x^4+x^2+1}}+\frac {2}{\sqrt {x^4+x^2+1} x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \text {arctanh}\left (\frac {1-x^2}{2 \sqrt {x^4+x^2+1}}\right )-\text {arctanh}\left (\frac {x^2+2}{2 \sqrt {x^4+x^2+1}}\right )\) |
Input:
Int[(2 + 3*x^2)/(x*(1 + x^2)*Sqrt[1 + x^2 + x^4]),x]
Output:
-1/2*ArcTanh[(1 - x^2)/(2*Sqrt[1 + x^2 + x^4])] - ArcTanh[(2 + x^2)/(2*Sqr t[1 + x^2 + x^4])]
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_) ^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && In tegerQ[q]
Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {x^{2}-1}{2 \sqrt {x^{4}+x^{2}+1}}\right )}{2}-\operatorname {arctanh}\left (\frac {x^{2}+2}{2 \sqrt {x^{4}+x^{2}+1}}\right )\) | \(42\) |
default | \(-\frac {\operatorname {arctanh}\left (\frac {-x^{2}+1}{2 \sqrt {\left (x^{2}+1\right )^{2}-x^{2}}}\right )}{2}-\operatorname {arctanh}\left (\frac {x^{2}+2}{2 \sqrt {x^{4}+x^{2}+1}}\right )\) | \(49\) |
elliptic | \(-\frac {\operatorname {arctanh}\left (\frac {-x^{2}+1}{2 \sqrt {\left (x^{2}+1\right )^{2}-x^{2}}}\right )}{2}-\operatorname {arctanh}\left (\frac {x^{2}+2}{2 \sqrt {x^{4}+x^{2}+1}}\right )\) | \(49\) |
trager | \(-\frac {\ln \left (\frac {x^{6}+2 \sqrt {x^{4}+x^{2}+1}\, x^{4}+7 x^{4}+4 x^{2} \sqrt {x^{4}+x^{2}+1}+8 x^{2}+8 \sqrt {x^{4}+x^{2}+1}+8}{x^{4} \left (x^{2}+1\right )}\right )}{2}\) | \(72\) |
Input:
int((3*x^2+2)/x/(x^2+1)/(x^4+x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*arctanh(1/2*(x^2-1)/(x^4+x^2+1)^(1/2))-arctanh(1/2*(x^2+2)/(x^4+x^2+1) ^(1/2))
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1}\right ) + \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1} - 1\right ) + \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1} - 2\right ) \] Input:
integrate((3*x^2+2)/x/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="fricas")
Output:
-log(-x^2 + sqrt(x^4 + x^2 + 1) + 1) - 1/2*log(-x^2 + sqrt(x^4 + x^2 + 1)) + log(-x^2 + sqrt(x^4 + x^2 + 1) - 1) + 1/2*log(-x^2 + sqrt(x^4 + x^2 + 1 ) - 2)
\[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3 x^{2} + 2}{x \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )}\, dx \] Input:
integrate((3*x**2+2)/x/(x**2+1)/(x**4+x**2+1)**(1/2),x)
Output:
Integral((3*x**2 + 2)/(x*sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1)), x)
\[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int { \frac {3 \, x^{2} + 2}{\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )} x} \,d x } \] Input:
integrate((3*x^2+2)/x/(x^2+1)/(x^4+x^2+1)^(1/2),x, algorithm="maxima")
Output:
integrate((3*x^2 + 2)/(sqrt(x^4 + x^2 + 1)*(x^2 + 1)*x), x)
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.47 \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} + x^{2} + 1} + 2\right ) + \log \left (x^{2} - \sqrt {x^{4} + x^{2} + 1} + 1\right ) - \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + x^{2} + 1}\right ) \] Input:
integrate((3*x^2+2)/x/(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) + log(x^2 - sqrt(x^4 + x^2 + 1) + 1 ) - log(-x^2 + sqrt(x^4 + x^2 + 1) + 1) - 1/2*log(-x^2 + sqrt(x^4 + x^2 + 1))
Timed out. \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=\int \frac {3\,x^2+2}{x\,\left (x^2+1\right )\,\sqrt {x^4+x^2+1}} \,d x \] Input:
int((3*x^2 + 2)/(x*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)),x)
Output:
int((3*x^2 + 2)/(x*(x^2 + 1)*(x^2 + x^4 + 1)^(1/2)), x)
Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.92 \[ \int \frac {2+3 x^2}{x \left (1+x^2\right ) \sqrt {1+x^2+x^4}} \, dx=-\mathrm {log}\left (\frac {6 \sqrt {x^{4}+x^{2}+1}+6 x^{2}+6}{\sqrt {3}}\right )+\frac {\mathrm {log}\left (\frac {6 \sqrt {x^{4}+x^{2}+1}+6 x^{2}}{\sqrt {3}}\right )}{2}+\mathrm {log}\left (\frac {2 \sqrt {x^{4}+x^{2}+1}+2 x^{2}-2}{\sqrt {3}}\right )-\frac {\mathrm {log}\left (\frac {2 \sqrt {x^{4}+x^{2}+1}+2 x^{2}+4}{\sqrt {3}}\right )}{2} \] Input:
int((3*x^2+2)/x/(x^2+1)/(x^4+x^2+1)^(1/2),x)
Output:
( - 2*log((6*sqrt(x**4 + x**2 + 1) + 6*x**2 + 6)/sqrt(3)) + log((6*sqrt(x* *4 + x**2 + 1) + 6*x**2)/sqrt(3)) + 2*log((2*sqrt(x**4 + x**2 + 1) + 2*x** 2 - 2)/sqrt(3)) - log((2*sqrt(x**4 + x**2 + 1) + 2*x**2 + 4)/sqrt(3)))/2