Integrand size = 29, antiderivative size = 290 \[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=-\frac {1}{4 a \left (1-\sqrt {1-x^2}\right )}+\frac {1}{4 a \left (1+\sqrt {1-x^2}\right )}+\frac {(a+2 b) \text {arctanh}\left (\sqrt {1-x^2}\right )}{2 a^2}-\frac {\sqrt {c} \left (a+b+\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a+b-\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]
1/2*(a+2*b)*arctanh((-x^2+1)^(1/2))/a^2-1/4/a/(1-(-x^2+1)^(1/2))+1/4/a/(1+ (-x^2+1)^(1/2))-1/2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1)^(1/2)/(b+2*c-(-4*a*c+ b^2)^(1/2))^(1/2))*c^(1/2)*(a+b+(b^2+a*(b-2*c))/(-4*a*c+b^2)^(1/2))/a^2*2^ (1/2)/(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctanh(2^(1/2)*c^(1/2)*(-x^2+1 )^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(a+b+(-b^2-a*(b-2*c))/(- 4*a*c+b^2)^(1/2))/a^2*2^(1/2)/(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 1.45 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\frac {-\frac {a \sqrt {1-x^2}}{x^2}+\frac {\sqrt {2} \sqrt {c} \left (b \left (-b+\sqrt {b^2-4 a c}\right )+a \left (-b+2 c+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-2 c-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (b \left (b+\sqrt {b^2-4 a c}\right )+a \left (b-2 c+\sqrt {b^2-4 a c}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-b-2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {-b-2 c+\sqrt {b^2-4 a c}}}+(a+2 b) \text {arctanh}\left (\sqrt {1-x^2}\right )}{2 a^2} \]
(-((a*Sqrt[1 - x^2])/x^2) + (Sqrt[2]*Sqrt[c]*(b*(-b + Sqrt[b^2 - 4*a*c]) + a*(-b + 2*c + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/ Sqrt[-b - 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*c - Sq rt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(b*(b + Sqrt[b^2 - 4*a*c]) + a*(b - 2 *c + Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[-b - 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[-b - 2*c + Sqrt[b^2 - 4 *a*c]]) + (a + 2*b)*ArcTanh[Sqrt[1 - x^2]])/(2*a^2)
Time = 1.33 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 1199, 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.
| \(\displaystyle \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int \frac {\sqrt {1-x^2}}{x^4 \left (c x^4+b x^2+a\right )}dx^2\) |
| \(\Big \downarrow \) 1199 |
| \(\displaystyle -\int \left (-\frac {a+2 b}{2 a^2 \left (1-x^4\right )}+\frac {b (a+b+c)-(a+b) c x^4}{a^2 \left (c x^8-(b+2 c) x^4+a+b+c\right )}+\frac {1}{4 a \left (1-\sqrt {1-x^2}\right )^2}+\frac {1}{4 a \left (\sqrt {1-x^2}+1\right )^2}\right )d\sqrt {1-x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c} \left (\frac {a (b-2 c)+b^2}{\sqrt {b^2-4 a c}}+a+b\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {-\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} a^2 \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {\sqrt {c} \left (-\frac {a (b-2 c)+b^2}{\sqrt {b^2-4 a c}}+a+b\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {\sqrt {b^2-4 a c}+b+2 c}}\right )}{\sqrt {2} a^2 \sqrt {\sqrt {b^2-4 a c}+b+2 c}}+\frac {(a+2 b) \text {arctanh}\left (\sqrt {1-x^2}\right )}{2 a^2}-\frac {1}{4 a \left (1-\sqrt {1-x^2}\right )}+\frac {1}{4 a \left (\sqrt {1-x^2}+1\right )}\) |
-1/4*1/(a*(1 - Sqrt[1 - x^2])) + 1/(4*a*(1 + Sqrt[1 - x^2])) + ((a + 2*b)* ArcTanh[Sqrt[1 - x^2]])/(2*a^2) - (Sqrt[c]*(a + b + (b^2 + a*(b - 2*c))/Sq rt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/Sqrt[b + 2*c - Sq rt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (Sqrt [c]*(a + b - (b^2 + a*(b - 2*c))/Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[ c]*Sqrt[1 - x^2])/Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a^2*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])
3.4.80.3.1 Defintions of rubi rules used
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x _) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e Subs t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer Q[n] && FractionQ[m]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && Int egerQ[(m - 1)/2]
Time = 0.67 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.14
| method | result | size |
| pseudoelliptic | \(-\frac {-2 \sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, \sqrt {2}\, c \,x^{2} \left (\left (a +b \right ) \sqrt {-4 a c +b^{2}}+a \left (b -2 c \right )+b^{2}\right ) \arctan \left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}}\right )+\left (-2 \left (\left (-a -b \right ) \sqrt {-4 a c +b^{2}}+a \left (b -2 c \right )+b^{2}\right ) \sqrt {2}\, c \,x^{2} \operatorname {arctanh}\left (\frac {c \sqrt {-x^{2}+1}\, \sqrt {2}}{\sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}}\right )+\sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, \sqrt {-4 a c +b^{2}}\, \left (-x^{2} \left (a +2 b \right ) \ln \left (1+\sqrt {-x^{2}+1}\right )+x^{2} \left (a +2 b \right ) \ln \left (\sqrt {-x^{2}+1}-1\right )+2 a \sqrt {-x^{2}+1}\right )\right ) \sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}}{4 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (\sqrt {-4 a c +b^{2}}-b -2 c \right ) c}\, \sqrt {\left (b +2 c +\sqrt {-4 a c +b^{2}}\right ) c}\, a^{2} x^{2}}\) | \(330\) |
| risch | \(\frac {x^{2}-1}{2 a \,x^{2} \sqrt {-x^{2}+1}}+\frac {-\frac {\left (-a -2 b \right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{a}+\frac {2 \left (2 \sqrt {-4 a c +b^{2}}\, a^{2} c -\sqrt {-4 a c +b^{2}}\, a \,b^{2}+3 \sqrt {-4 a c +b^{2}}\, a b c -\sqrt {-4 a c +b^{2}}\, b^{3}+4 c b \,a^{2}-4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {2 \left (-2 \sqrt {-4 a c +b^{2}}\, a^{2} c +\sqrt {-4 a c +b^{2}}\, a \,b^{2}-3 \sqrt {-4 a c +b^{2}}\, a b c +\sqrt {-4 a c +b^{2}}\, b^{3}+4 c b \,a^{2}-4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}}{2 a}\) | \(526\) |
| default | \(\frac {-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {\sqrt {-x^{2}+1}}{2}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{2}}{a}-\frac {b \left (\sqrt {-x^{2}+1}-\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right )}{a^{2}}-\frac {-2 a \left (\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a^{2} c -\sqrt {-4 a c +b^{2}}\, a \,b^{2}+3 \sqrt {-4 a c +b^{2}}\, a b c -\sqrt {-4 a c +b^{2}}\, b^{3}+4 c b \,a^{2}-4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (-2 \sqrt {-4 a c +b^{2}}\, a^{2} c +\sqrt {-4 a c +b^{2}}\, a \,b^{2}-3 \sqrt {-4 a c +b^{2}}\, a b c +\sqrt {-4 a c +b^{2}}\, b^{3}+4 c b \,a^{2}-4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )-\frac {2 b}{\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+1}}{a^{2}}\) | \(580\) |
-1/4/(-4*a*c+b^2)^(1/2)*(-2*((b+2*c+(-4*a*c+b^2)^(1/2))*c)^(1/2)*2^(1/2)*c *x^2*((a+b)*(-4*a*c+b^2)^(1/2)+a*(b-2*c)+b^2)*arctan(c*(-x^2+1)^(1/2)*2^(1 /2)/(((-4*a*c+b^2)^(1/2)-b-2*c)*c)^(1/2))+(-2*((-a-b)*(-4*a*c+b^2)^(1/2)+a *(b-2*c)+b^2)*2^(1/2)*c*x^2*arctanh(c*(-x^2+1)^(1/2)*2^(1/2)/((b+2*c+(-4*a *c+b^2)^(1/2))*c)^(1/2))+((b+2*c+(-4*a*c+b^2)^(1/2))*c)^(1/2)*(-4*a*c+b^2) ^(1/2)*(-x^2*(a+2*b)*ln(1+(-x^2+1)^(1/2))+x^2*(a+2*b)*ln((-x^2+1)^(1/2)-1) +2*a*(-x^2+1)^(1/2)))*(((-4*a*c+b^2)^(1/2)-b-2*c)*c)^(1/2))/(((-4*a*c+b^2) ^(1/2)-b-2*c)*c)^(1/2)/((b+2*c+(-4*a*c+b^2)^(1/2))*c)^(1/2)/a^2/x^2
Leaf count of result is larger than twice the leaf count of optimal. 2799 vs. \(2 (236) = 472\).
Time = 7.50 (sec) , antiderivative size = 2799, normalized size of antiderivative = 9.65 \[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
-1/2*(sqrt(1/2)*a^2*x^2*sqrt((a*b^3 + b^4 + 2*a^2*c^2 - (3*a^2*b + 4*a*b^2 )*c - (a^4*b^2 - 4*a^5*c)*sqrt((a^2*b^4 + 2*a*b^5 + b^6 + (a^4 + 4*a^3*b + 4*a^2*b^2)*c^2 - 2*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*c)/(a^8*b^2 - 4*a^9*c) ))/(a^4*b^2 - 4*a^5*c))*log(((a^4*b^2*c - 4*a^5*c^2)*x^2*sqrt((a^2*b^4 + 2 *a*b^5 + b^6 + (a^4 + 4*a^3*b + 4*a^2*b^2)*c^2 - 2*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*c)/(a^8*b^2 - 4*a^9*c)) + 2*(a^3 + 2*a^2*b)*c^2 + ((a^2*b + 2*a*b ^2)*c^2 - (a*b^3 + b^4)*c)*x^2 - 2*(a^2*b^2 + a*b^3)*c + sqrt(1/2)*((a^5*b ^3 - 4*a^6*b*c)*x^2*sqrt((a^2*b^4 + 2*a*b^5 + b^6 + (a^4 + 4*a^3*b + 4*a^2 *b^2)*c^2 - 2*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*c)/(a^8*b^2 - 4*a^9*c)) + (a ^2*b^4 + a*b^5 + 4*(a^4 + 2*a^3*b)*c^2 - (5*a^3*b^2 + 6*a^2*b^3)*c)*x^2)*s qrt((a*b^3 + b^4 + 2*a^2*c^2 - (3*a^2*b + 4*a*b^2)*c - (a^4*b^2 - 4*a^5*c) *sqrt((a^2*b^4 + 2*a*b^5 + b^6 + (a^4 + 4*a^3*b + 4*a^2*b^2)*c^2 - 2*(a^3* b^2 + 3*a^2*b^3 + 2*a*b^4)*c)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^2 - 4*a^5*c)) - 2*((a^3 + 2*a^2*b)*c^2 - (a^2*b^2 + a*b^3)*c)*sqrt(-x^2 + 1))/x^2) - sqrt (1/2)*a^2*x^2*sqrt((a*b^3 + b^4 + 2*a^2*c^2 - (3*a^2*b + 4*a*b^2)*c - (a^4 *b^2 - 4*a^5*c)*sqrt((a^2*b^4 + 2*a*b^5 + b^6 + (a^4 + 4*a^3*b + 4*a^2*b^2 )*c^2 - 2*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*c)/(a^8*b^2 - 4*a^9*c)))/(a^4*b^ 2 - 4*a^5*c))*log(((a^4*b^2*c - 4*a^5*c^2)*x^2*sqrt((a^2*b^4 + 2*a*b^5 + b ^6 + (a^4 + 4*a^3*b + 4*a^2*b^2)*c^2 - 2*(a^3*b^2 + 3*a^2*b^3 + 2*a*b^4)*c )/(a^8*b^2 - 4*a^9*c)) + 2*(a^3 + 2*a^2*b)*c^2 + ((a^2*b + 2*a*b^2)*c^2...
\[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{x^{3} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {-x^{2} + 1}}{{\left (c x^{4} + b x^{2} + a\right )} x^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1675 vs. \(2 (236) = 472\).
Time = 1.57 (sec) , antiderivative size = 1675, normalized size of antiderivative = 5.78 \[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
-1/4*(sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^5 - 8*sqrt(2)*sqr t(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(-b*c - 2*c^ 2 - sqrt(b^2 - 4*a*c)*c)*b^4*c + 2*b^5*c + 16*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 5*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b ^3*c^2 - 16*a*b^3*c^2 + 2*b^4*c^2 - 20*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^ 2 - 4*a*c)*c)*a*b*c^3 + 32*a^2*b*c^3 - 12*a*b^2*c^3 + 16*a^2*c^4 - sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^4 + 6*sqrt(2 )*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*s qrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*b^3*c - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*c ^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)* a*b*c^2 - 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c )*c)*b^2*c^2 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*c^3 - 2*(b^2 - 4*a*c)*b^3*c + 8*(b^2 - 4*a*c)*a*b*c^2 - 2*(b^ 2 - 4*a*c)*b^2*c^2 + 4*(b^2 - 4*a*c)*a*c^3)*arctan(2*sqrt(1/2)*sqrt(-x^2 + 1)/sqrt(-(a^2*b + 2*a^2*c + sqrt(-4*(a^3 + a^2*b + a^2*c)*a^2*c + (a^2*b + 2*a^2*c)^2))/(a^2*c)))/((a^2*b^4 - 8*a^3*b^2*c + 2*a^2*b^3*c + 16*a^4*c^ 2 - 8*a^3*b*c^2 + 5*a^2*b^2*c^2 - 20*a^3*c^3)*abs(c)) - 1/4*(sqrt(2)*sqrt( -b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*b^5 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 +...
Time = 8.24 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.84 \[ \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx=\frac {\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )}{2\,a}-\frac {\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )\,\left (a+b\right )}{a^2}-\frac {\sqrt {1-x^2}}{2\,a\,x^2}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a^2\,c-a\,b^2-b^3+b^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b\,c+a\,b\,\sqrt {b^2-4\,a\,c}-2\,a\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (a\,b^2-4\,a^2\,c+b^3+b^2\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c+a\,b\,\sqrt {b^2-4\,a\,c}-2\,a\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (4\,a^2\,c-a\,b^2-b^3+b^2\,\sqrt {b^2-4\,a\,c}+4\,a\,b\,c+a\,b\,\sqrt {b^2-4\,a\,c}-2\,a\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (a\,b^2-4\,a^2\,c+b^3+b^2\,\sqrt {b^2-4\,a\,c}-4\,a\,b\,c+a\,b\,\sqrt {b^2-4\,a\,c}-2\,a\,c\,\sqrt {b^2-4\,a\,c}\right )}{4\,a^2\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )} \]
log((1/x^2 - 1)^(1/2) - (1/x^2)^(1/2))/(2*a) - (log((1/x^2 - 1)^(1/2) - (1 /x^2)^(1/2))*(a + b))/a^2 - (1 - x^2)^(1/2)/(2*a*x^2) - (log((((x*(-(b + ( b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^ (1/2)))*(4*a^2*c - a*b^2 - b^3 + b^2*(b^2 - 4*a*c)^(1/2) + 4*a*b*c + a*b*( b^2 - 4*a*c)^(1/2) - 2*a*c*(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)) + (log((((x*(-(b - (b^2 - 4*a*c)^ (1/2))/(2*c))^(1/2) + 1)*1i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(a*b^ 2 - 4*a^2*c + b^3 + b^2*(b^2 - 4*a*c)^(1/2) - 4*a*b*c + a*b*(b^2 - 4*a*c)^ (1/2) - 2*a*c*(b^2 - 4*a*c)^(1/2)))/(4*a^2*((b - (b^2 - 4*a*c)^(1/2))/(2*c ) + 1)^(1/2)*(4*a*c - b^2)) - (log((((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c)) ^(1/2) - 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1 /2)*1i)/(x - (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(4*a^2*c - a*b^2 - b^3 + b^2*(b^2 - 4*a*c)^(1/2) + 4*a*b*c + a*b*(b^2 - 4*a*c)^(1/2) - 2*a*c *(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*((b + (b^2 - 4*a*c)^(1/2))/(2* c) + 1)^(1/2)) + (log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 1)*1 i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(a*b^2 - 4*a^2*c + b^3 + b^2*(b ^2 - 4*a*c)^(1/2) - 4*a*b*c + a*b*(b^2 - 4*a*c)^(1/2) - 2*a*c*(b^2 - 4*...