Integrand size = 20, antiderivative size = 196 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=-\frac {\sqrt {b+d-\sqrt {b^2-4 a c+2 b d+d^2}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+d-\sqrt {b^2-4 a c+2 b d+d^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c+2 b d+d^2}}+\frac {\sqrt {b+d+\sqrt {b^2-4 a c+2 b d+d^2}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+d+\sqrt {b^2-4 a c+2 b d+d^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c+2 b d+d^2}} \] Output:
-1/2*(b+d-(-4*a*c+b^2+2*b*d+d^2)^(1/2))^(1/2)*arctan(2^(1/2)*c^(1/2)*x/(b+ d-(-4*a*c+b^2+2*b*d+d^2)^(1/2))^(1/2))*2^(1/2)/c^(1/2)/(-4*a*c+b^2+2*b*d+d ^2)^(1/2)+1/2*(b+d+(-4*a*c+b^2+2*b*d+d^2)^(1/2))^(1/2)*arctan(2^(1/2)*c^(1 /2)*x/(b+d+(-4*a*c+b^2+2*b*d+d^2)^(1/2))^(1/2))*2^(1/2)/c^(1/2)/(-4*a*c+b^ 2+2*b*d+d^2)^(1/2)
Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.13 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=\frac {\left (-b-d+\sqrt {b^2-4 a c+2 b d+d^2}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+d-\sqrt {b^2-4 a c+2 b d+d^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c+2 b d+d^2} \sqrt {b+d-\sqrt {b^2-4 a c+2 b d+d^2}}}+\frac {\sqrt {b+d+\sqrt {b^2-4 a c+2 b d+d^2}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+d+\sqrt {b^2-4 a c+2 b d+d^2}}}\right )}{\sqrt {2} \sqrt {c} \sqrt {b^2-4 a c+2 b d+d^2}} \] Input:
Integrate[(d + (a + b*x^2 + c*x^4)/x^2)^(-1),x]
Output:
((-b - d + Sqrt[b^2 - 4*a*c + 2*b*d + d^2])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqr t[b + d - Sqrt[b^2 - 4*a*c + 2*b*d + d^2]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4 *a*c + 2*b*d + d^2]*Sqrt[b + d - Sqrt[b^2 - 4*a*c + 2*b*d + d^2]]) + (Sqrt [b + d + Sqrt[b^2 - 4*a*c + 2*b*d + d^2]]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[ b + d + Sqrt[b^2 - 4*a*c + 2*b*d + d^2]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b^2 - 4*a *c + 2*b*d + d^2])
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.
| \(\displaystyle \int \frac {1}{\frac {a+b x^2+c x^4}{x^2}+d} \, dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \frac {1}{\frac {a+b x^2+c x^4}{x^2}+d}dx\) |
Input:
Int[(d + (a + b*x^2 + c*x^4)/x^2)^(-1),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.23
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\left (b +d \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b +\textit {\_R} d}\right )}{2}\) | \(46\) |
| default | \(4 c \left (-\frac {\left (\sqrt {-4 a c +b^{2}+2 b d +d^{2}}-b -d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +b^{2}+2 b d +d^{2}}-b -d \right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}+2 b d +d^{2}}\, \sqrt {\left (\sqrt {-4 a c +b^{2}+2 b d +d^{2}}-b -d \right ) c}}+\frac {\left (b +d +\sqrt {-4 a c +b^{2}+2 b d +d^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +d +\sqrt {-4 a c +b^{2}+2 b d +d^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}+2 b d +d^{2}}\, c \sqrt {\left (b +d +\sqrt {-4 a c +b^{2}+2 b d +d^{2}}\right ) c}}\right )\) | \(217\) |
Input:
int(1/(d+(c*x^4+b*x^2+a)/x^2),x,method=_RETURNVERBOSE)
Output:
1/2*sum(_R^2/(2*_R^3*c+_R*b+_R*d)*ln(x-_R),_R=RootOf(c*_Z^4+(b+d)*_Z^2+a))
Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (161) = 322\).
Time = 0.08 (sec) , antiderivative size = 935, normalized size of antiderivative = 4.77 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(d+(c*x^4+b*x^2+a)/x^2),x, algorithm="fricas")
Output:
1/2*sqrt(1/2)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2* c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)) *log(sqrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2)) /(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2) + x) - 1/2*sqrt(1/2)*sqrt(-(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))*log(-sqrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)*sqrt( -(b + d + (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b *c^2*d + c^2*d^2))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))/sqrt(b^2*c^2 - 4*a *c^3 + 2*b*c^2*d + c^2*d^2) + x) - 1/2*sqrt(1/2)*sqrt(-(b + d - (b^2*c - 4 *a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2))/( b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))*log(sqrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c *d + c*d^2)*sqrt(-(b + d - (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^ 2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))/s qrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2) + x) + 1/2*sqrt(1/2)*sqrt(-(b + d - (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^ 2*d + c^2*d^2))/(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2))*log(-sqrt(1/2)*(b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)*sqrt(-(b + d - (b^2*c - 4*a*c^2 + 2*b*c*d + c*d^2)/sqrt(b^2*c^2 - 4*a*c^3 + 2*b*c^2*d + c^2*d^2))/(b^2*c - 4*a*c^2 ...
Time = 1.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.92 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{2} c^{3} - 128 a b^{2} c^{2} - 256 a b c^{2} d - 128 a c^{2} d^{2} + 16 b^{4} c + 64 b^{3} c d + 96 b^{2} c d^{2} + 64 b c d^{3} + 16 c d^{4}\right ) + t^{2} \left (- 16 a b c - 16 a c d + 4 b^{3} + 12 b^{2} d + 12 b d^{2} + 4 d^{3}\right ) + a, \left ( t \mapsto t \log {\left (64 t^{3} a c^{2} - 16 t^{3} b^{2} c - 32 t^{3} b c d - 16 t^{3} c d^{2} - 2 t b - 2 t d + x \right )} \right )\right )} \] Input:
integrate(1/(d+(c*x**4+b*x**2+a)/x**2),x)
Output:
RootSum(_t**4*(256*a**2*c**3 - 128*a*b**2*c**2 - 256*a*b*c**2*d - 128*a*c* *2*d**2 + 16*b**4*c + 64*b**3*c*d + 96*b**2*c*d**2 + 64*b*c*d**3 + 16*c*d* *4) + _t**2*(-16*a*b*c - 16*a*c*d + 4*b**3 + 12*b**2*d + 12*b*d**2 + 4*d** 3) + a, Lambda(_t, _t*log(64*_t**3*a*c**2 - 16*_t**3*b**2*c - 32*_t**3*b*c *d - 16*_t**3*c*d**2 - 2*_t*b - 2*_t*d + x)))
\[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=\int { \frac {1}{d + \frac {c x^{4} + b x^{2} + a}{x^{2}}} \,d x } \] Input:
integrate(1/(d+(c*x^4+b*x^2+a)/x^2),x, algorithm="maxima")
Output:
integrate(1/(d + (c*x^4 + b*x^2 + a)/x^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (161) = 322\).
Time = 1.26 (sec) , antiderivative size = 1160, normalized size of antiderivative = 5.92 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(d+(c*x^4+b*x^2+a)/x^2),x, algorithm="giac")
Output:
1/2*(2*b^2*c^2 - 8*a*c^3 + 4*b*c^2*d + 2*c^2*d^2 - sqrt(2)*sqrt(b^2 - 4*a* c + 2*b*d + d^2)*sqrt(b*c + c*d + sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d + sqrt(b^2 - 4*a *c + 2*b*d + d^2)*c)*a*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt( b*c + c*d + sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*b*c - sqrt(2)*sqrt(b^2 - 4* a*c + 2*b*d + d^2)*sqrt(b*c + c*d + sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*c^2 - 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d + sqrt(b^2 - 4 *a*c + 2*b*d + d^2)*c)*b*d + 2*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqr t(b*c + c*d + sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*c*d - sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d + sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*d ^2 - 2*(b^2 - 4*a*c + 2*b*d + d^2)*c^2)*arctan(2*sqrt(1/2)*x/sqrt((b + d + sqrt((b + d)^2 - 4*a*c))/c))/((b^4 - 8*a*b^2*c - 2*b^3*c + 16*a^2*c^2 + 8 *a*b*c^2 + b^2*c^2 - 4*a*c^3 + 4*b^3*d - 16*a*b*c*d - 6*b^2*c*d + 8*a*c^2* d + 2*b*c^2*d + 6*b^2*d^2 - 8*a*c*d^2 - 6*b*c*d^2 + c^2*d^2 + 4*b*d^3 - 2* c*d^3 + d^4)*abs(c)) - 1/2*(2*b^2*c^2 - 8*a*c^3 + 4*b*c^2*d + 2*c^2*d^2 - sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d - sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*b^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d - sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*a*c + 2*sqrt(2)*sqrt(b^2 - 4*a *c + 2*b*d + d^2)*sqrt(b*c + c*d - sqrt(b^2 - 4*a*c + 2*b*d + d^2)*c)*b*c - sqrt(2)*sqrt(b^2 - 4*a*c + 2*b*d + d^2)*sqrt(b*c + c*d - sqrt(b^2 - 4...
Time = 17.35 (sec) , antiderivative size = 921, normalized size of antiderivative = 4.70 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=2\,\mathrm {atanh}\left (\frac {\left (x\,\left (2\,b^2\,c+4\,b\,c\,d-4\,a\,c^2+2\,c\,d^2\right )-\frac {x\,\left (8\,b^3\,c^2+24\,b^2\,c^2\,d-32\,a\,b\,c^3+24\,b\,c^2\,d^2-32\,a\,c^3\,d+8\,c^2\,d^3\right )\,\left (3\,b\,d^2+3\,b^2\,d+b^3+d^3+\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}\right )\,\sqrt {-\frac {3\,b\,d^2+3\,b^2\,d+b^3+d^3+\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}}}{a\,c}\right )\,\sqrt {-\frac {3\,b\,d^2+3\,b^2\,d+b^3+d^3+\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}}+2\,\mathrm {atanh}\left (\frac {\left (x\,\left (2\,b^2\,c+4\,b\,c\,d-4\,a\,c^2+2\,c\,d^2\right )-\frac {x\,\left (8\,b^3\,c^2+24\,b^2\,c^2\,d-32\,a\,b\,c^3+24\,b\,c^2\,d^2-32\,a\,c^3\,d+8\,c^2\,d^3\right )\,\left (3\,b\,d^2+3\,b^2\,d+b^3+d^3-\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d\right )}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}\right )\,\sqrt {-\frac {3\,b\,d^2+3\,b^2\,d+b^3+d^3-\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}}}{a\,c}\right )\,\sqrt {-\frac {3\,b\,d^2+3\,b^2\,d+b^3+d^3-\sqrt {{\left (b^2+2\,b\,d+d^2-4\,a\,c\right )}^3}-4\,a\,b\,c-4\,a\,c\,d}{8\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2-16\,a\,b\,c^2\,d-8\,a\,c^2\,d^2+b^4\,c+4\,b^3\,c\,d+6\,b^2\,c\,d^2+4\,b\,c\,d^3+c\,d^4\right )}} \] Input:
int(1/(d + (a + b*x^2 + c*x^4)/x^2),x)
Output:
2*atanh(((x*(2*b^2*c - 4*a*c^2 + 2*c*d^2 + 4*b*c*d) - (x*(8*b^3*c^2 + 8*c^ 2*d^3 + 24*b*c^2*d^2 + 24*b^2*c^2*d - 32*a*b*c^3 - 32*a*c^3*d)*(3*b*d^2 + 3*b^2*d + b^3 + d^3 + ((2*b*d - 4*a*c + b^2 + d^2)^3)^(1/2) - 4*a*b*c - 4* a*c*d))/(8*(b^4*c + c*d^4 + 16*a^2*c^3 - 8*a*b^2*c^2 - 8*a*c^2*d^2 + 6*b^2 *c*d^2 + 4*b*c*d^3 + 4*b^3*c*d - 16*a*b*c^2*d)))*(-(3*b*d^2 + 3*b^2*d + b^ 3 + d^3 + ((2*b*d - 4*a*c + b^2 + d^2)^3)^(1/2) - 4*a*b*c - 4*a*c*d)/(8*(b ^4*c + c*d^4 + 16*a^2*c^3 - 8*a*b^2*c^2 - 8*a*c^2*d^2 + 6*b^2*c*d^2 + 4*b* c*d^3 + 4*b^3*c*d - 16*a*b*c^2*d)))^(1/2))/(a*c))*(-(3*b*d^2 + 3*b^2*d + b ^3 + d^3 + ((2*b*d - 4*a*c + b^2 + d^2)^3)^(1/2) - 4*a*b*c - 4*a*c*d)/(8*( b^4*c + c*d^4 + 16*a^2*c^3 - 8*a*b^2*c^2 - 8*a*c^2*d^2 + 6*b^2*c*d^2 + 4*b *c*d^3 + 4*b^3*c*d - 16*a*b*c^2*d)))^(1/2) + 2*atanh(((x*(2*b^2*c - 4*a*c^ 2 + 2*c*d^2 + 4*b*c*d) - (x*(8*b^3*c^2 + 8*c^2*d^3 + 24*b*c^2*d^2 + 24*b^2 *c^2*d - 32*a*b*c^3 - 32*a*c^3*d)*(3*b*d^2 + 3*b^2*d + b^3 + d^3 - ((2*b*d - 4*a*c + b^2 + d^2)^3)^(1/2) - 4*a*b*c - 4*a*c*d))/(8*(b^4*c + c*d^4 + 1 6*a^2*c^3 - 8*a*b^2*c^2 - 8*a*c^2*d^2 + 6*b^2*c*d^2 + 4*b*c*d^3 + 4*b^3*c* d - 16*a*b*c^2*d)))*(-(3*b*d^2 + 3*b^2*d + b^3 + d^3 - ((2*b*d - 4*a*c + b ^2 + d^2)^3)^(1/2) - 4*a*b*c - 4*a*c*d)/(8*(b^4*c + c*d^4 + 16*a^2*c^3 - 8 *a*b^2*c^2 - 8*a*c^2*d^2 + 6*b^2*c*d^2 + 4*b*c*d^3 + 4*b^3*c*d - 16*a*b*c^ 2*d)))^(1/2))/(a*c))*(-(3*b*d^2 + 3*b^2*d + b^3 + d^3 - ((2*b*d - 4*a*c + b^2 + d^2)^3)^(1/2) - 4*a*b*c - 4*a*c*d)/(8*(b^4*c + c*d^4 + 16*a^2*c^3...
Time = 0.30 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.02 \[ \int \frac {1}{d+\frac {a+b x^2+c x^4}{x^2}} \, dx=\frac {-4 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) c +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) b +2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) d +4 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) c -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) b -2 \sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+b +d}}\right ) d +2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c -2 \sqrt {a}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) c +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b +\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) b -\sqrt {c}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, \mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-b -d}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d}{4 c \left (4 a c -b^{2}-2 b d -d^{2}\right )} \] Input:
int(1/(d+(c*x^4+b*x^2+a)/x^2),x)
Output:
( - 4*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b + d)*atan((sqrt(2*sqrt(c)*sqrt(a) - b - d) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b + d))*c + 2*sqrt(c)*sq rt(2*sqrt(c)*sqrt(a) + b + d)*atan((sqrt(2*sqrt(c)*sqrt(a) - b - d) - 2*sq rt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b + d))*b + 2*sqrt(c)*sqrt(2*sqrt(c)*sqr t(a) + b + d)*atan((sqrt(2*sqrt(c)*sqrt(a) - b - d) - 2*sqrt(c)*x)/sqrt(2* sqrt(c)*sqrt(a) + b + d))*d + 4*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b + d)*at an((sqrt(2*sqrt(c)*sqrt(a) - b - d) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b + d))*c - 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b + d)*atan((sqrt(2*sqrt( c)*sqrt(a) - b - d) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b + d))*b - 2* sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) + b + d)*atan((sqrt(2*sqrt(c)*sqrt(a) - b - d) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b + d))*d + 2*sqrt(a)*sqrt(2*s qrt(c)*sqrt(a) - b - d)*log( - sqrt(2*sqrt(c)*sqrt(a) - b - d)*x + sqrt(a) + sqrt(c)*x**2)*c - 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) - b - d)*log(sqrt(2* sqrt(c)*sqrt(a) - b - d)*x + sqrt(a) + sqrt(c)*x**2)*c + sqrt(c)*sqrt(2*sq rt(c)*sqrt(a) - b - d)*log( - sqrt(2*sqrt(c)*sqrt(a) - b - d)*x + sqrt(a) + sqrt(c)*x**2)*b + sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - b - d)*log( - sqrt(2* sqrt(c)*sqrt(a) - b - d)*x + sqrt(a) + sqrt(c)*x**2)*d - sqrt(c)*sqrt(2*sq rt(c)*sqrt(a) - b - d)*log(sqrt(2*sqrt(c)*sqrt(a) - b - d)*x + sqrt(a) + s qrt(c)*x**2)*b - sqrt(c)*sqrt(2*sqrt(c)*sqrt(a) - b - d)*log(sqrt(2*sqrt(c )*sqrt(a) - b - d)*x + sqrt(a) + sqrt(c)*x**2)*d)/(4*c*(4*a*c - b**2 - ...