Integrand size = 35, antiderivative size = 62 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {\log (x)}{\sqrt {a}}+\frac {\log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \] Output:
-ln(x)/a^(1/2)+ln(-2*a-b*x-2*a*x^2+2*a^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/ 2))/a^(1/2)
Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {-\log (x)+\log \left (-b x-2 a \left (1+x^2\right )+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \] Input:
Integrate[(1 - x^2)/(x*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
Output:
(-Log[x] + Log[-(b*x) - 2*a*(1 + x^2) + 2*Sqrt[a]*Sqrt[a + b*x + c*x^2 + b *x^3 + a*x^4]])/Sqrt[a]
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.
\(\displaystyle \int \frac {1-x^2}{x \sqrt {a x^4+a+b x^3+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{x \sqrt {a x^4+a+b x^3+b x+c x^2}}-\frac {x}{\sqrt {a x^4+a+b x^3+b x+c x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {1}{x \sqrt {a x^4+b x^3+c x^2+b x+a}}dx-\int \frac {x}{\sqrt {a x^4+b x^3+c x^2+b x+a}}dx\) |
Input:
Int[(1 - x^2)/(x*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]),x]
Output:
$Aborted
Time = 2.59 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\ln \left (2\right )-\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(58\) |
default | \(-\frac {-\ln \left (2\right )+\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(59\) |
elliptic | \(\text {Expression too large to display}\) | \(3365\) |
Input:
int((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(ln(2)-ln((2*a*x^2+2*a^(1/2)*(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)+b*x+2*a)/x/a^ (1/2)))/a^(1/2)
Time = 0.67 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.82 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a^{2} x^{4} + a b x^{3} + a c x^{2} + a b x + a^{2}\right )}}\right )}{a}\right ] \] Input:
integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="fricas" )
Output:
[1/2*log((8*a^2*x^4 + 8*a*b*x^3 + 8*a*b*x + (8*a^2 + b^2 + 4*a*c)*x^2 - 4* sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*(2*a*x^2 + b*x + 2*a)*sqrt(a) + 8*a^ 2)/x^2)/sqrt(a), sqrt(-a)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a) *(2*a*x^2 + b*x + 2*a)*sqrt(-a)/(a^2*x^4 + a*b*x^3 + a*c*x^2 + a*b*x + a^2 ))/a]
\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \int \left (- \frac {1}{x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx - \int \frac {x}{\sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \] Input:
integrate((-x**2+1)/x/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)
Output:
-Integral(-1/(x*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x) - Integral(x /sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2), x)
\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x} \,d x } \] Input:
integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="maxima" )
Output:
-integrate((x^2 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x), x)
\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x} \,d x } \] Input:
integrate((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(-(x^2 - 1)/(sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*x), x)
Timed out. \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int -\frac {x^2-1}{x\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \] Input:
int(-(x^2 - 1)/(x*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)
Output:
int(-(x^2 - 1)/(x*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.81 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {\sqrt {a}\, \left (\mathrm {log}\left (-2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+2 a \,x^{2}+2 a +b x \right )-\mathrm {log}\left (x \right )\right )}{a} \] Input:
int((-x^2+1)/x/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2),x)
Output:
(sqrt(a)*(log( - 2*sqrt(a)*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) + 2*a* x**2 + 2*a + b*x) - log(x)))/a