3.4.83 \(\int \frac {\sqrt {1-x^2}}{a+b x^2+c x^4} \, dx\) [383]

Optimal. Leaf size=220 \[ \frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]

[Out]

arctan(x*(b+2*c-(-4*a*c+b^2)^(1/2))^(1/2)/(-x^2+1)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b+2*c-(-4*a*c+b^2)^(1/
2))^(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-arctan(x*(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)/(-x^2+1)^(
1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*(b+2*c+(-4*a*c+b^2)^(1/2))^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^
(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1188, 399, 222, 385, 211} \begin {gather*} \frac {\sqrt {-\sqrt {b^2-4 a c}+b+2 c} \text {ArcTan}\left (\frac {x \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {\sqrt {b^2-4 a c}+b+2 c} \text {ArcTan}\left (\frac {x \sqrt {\sqrt {b^2-4 a c}+b+2 c}}{\sqrt {1-x^2} \sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(a + b*x^2 + c*x^4),x]

[Out]

(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*ArcTan[(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*S
qrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*ArcTan[(S
qrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b +
Sqrt[b^2 - 4*a*c]])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 1188

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]
}, Dist[2*(c/r), Int[(d + e*x^2)^q/(b - r + 2*c*x^2), x], x] - Dist[2*(c/r), Int[(d + e*x^2)^q/(b + r + 2*c*x^
2), x], x]] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !Integ
erQ[q]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^2}}{a+b x^2+c x^4} \, dx &=\frac {(2 c) \int \frac {\sqrt {1-x^2}}{b-\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {(2 c) \int \frac {\sqrt {1-x^2}}{b+\sqrt {b^2-4 a c}+2 c x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (b+2 c-\sqrt {b^2-4 a c}\right ) \int \frac {1}{\sqrt {1-x^2} \left (b-\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{\sqrt {b^2-4 a c}}-\frac {\left (b+2 c+\sqrt {b^2-4 a c}\right ) \int \frac {1}{\sqrt {1-x^2} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\left (b+2 c-\sqrt {b^2-4 a c}\right ) \text {Subst}\left (\int \frac {1}{b-\sqrt {b^2-4 a c}-\left (-b-2 c+\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c}}-\frac {\left (b+2 c+\sqrt {b^2-4 a c}\right ) \text {Subst}\left (\int \frac {1}{b+\sqrt {b^2-4 a c}-\left (-b-2 c-\sqrt {b^2-4 a c}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c}}\\ &=\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+2 c-\sqrt {b^2-4 a c}} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} \tan ^{-1}\left (\frac {\sqrt {b+2 c+\sqrt {b^2-4 a c}} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {1-x^2}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.12, size = 155, normalized size = 0.70 \begin {gather*} \frac {1}{2} i \text {RootSum}\left [16 a+16 b+16 c-32 a \text {$\#$1}-32 b \text {$\#$1}-32 c \text {$\#$1}+16 a \text {$\#$1}^2+20 b \text {$\#$1}^2+24 c \text {$\#$1}^2-4 b \text {$\#$1}^3-8 c \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {\log \left (2-2 x^2-2 i x \sqrt {1-x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 a-8 b-8 c+8 a \text {$\#$1}+10 b \text {$\#$1}+12 c \text {$\#$1}-3 b \text {$\#$1}^2-6 c \text {$\#$1}^2+c \text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/(a + b*x^2 + c*x^4),x]

[Out]

(I/2)*RootSum[16*a + 16*b + 16*c - 32*a*#1 - 32*b*#1 - 32*c*#1 + 16*a*#1^2 + 20*b*#1^2 + 24*c*#1^2 - 4*b*#1^3
- 8*c*#1^3 + c*#1^4 & , (Log[2 - 2*x^2 - (2*I)*x*Sqrt[1 - x^2] - #1]*#1^2)/(-8*a - 8*b - 8*c + 8*a*#1 + 10*b*#
1 + 12*c*#1 - 3*b*#1^2 - 6*c*#1^2 + c*#1^3) & ]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.12, size = 130, normalized size = 0.59

method	result	size
default	\(-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{8}+\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{6}-\textit {\_R}^{4}-\textit {\_R}^{2}+1\right ) \ln \left (\frac {\sqrt {-x^{2}+1}-1}{x}-\textit {\_R} \right )}{\textit {\_R}^{7} a +3 \textit {\_R}^{5} a +3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a +4 \textit {\_R}^{3} b +8 c \,\textit {\_R}^{3}+\textit {\_R} a +\textit {\_R} b}\right )}{4}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/4*sum((_R^6-_R^4-_R^2+1)/(_R^7*a+3*_R^5*a+3*_R^5*b+3*_R^3*a+4*_R^3*b+8*_R^3*c+_R*a+_R*b)*ln(((-x^2+1)^(1/2)
-1)/x-_R),_R=RootOf(a*_Z^8+(4*a+4*b)*_Z^6+(6*a+8*b+16*c)*_Z^4+(4*a+4*b)*_Z^2+a))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)/(c*x^4 + b*x^2 + a), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (180) = 360\).
time = 0.45, size = 759, normalized size = 3.45 \begin {gather*} \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {2 \, a + b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {x^{2} + \frac {\sqrt {\frac {1}{2}} {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {-x^{2} + 1} x - {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-\frac {2 \, a + b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {2 \, a + b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {x^{2} - \frac {\sqrt {\frac {1}{2}} {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {-x^{2} + 1} x - {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-\frac {2 \, a + b + \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {2 \, a + b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {x^{2} + \frac {\sqrt {\frac {1}{2}} {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {-x^{2} + 1} x - {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-\frac {2 \, a + b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {2 \, a + b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}} \log \left (-\frac {x^{2} - \frac {\sqrt {\frac {1}{2}} {\left ({\left (a b^{2} - 4 \, a^{2} c\right )} \sqrt {-x^{2} + 1} x - {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-\frac {2 \, a + b - \frac {a b^{2} - 4 \, a^{2} c}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}}}{a b^{2} - 4 \, a^{2} c}}}{\sqrt {a^{2} b^{2} - 4 \, a^{3} c}} + \sqrt {-x^{2} + 1} - 1}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

1/2*sqrt(1/2)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 + sqrt(
1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^
2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*sqrt(-(2*a
+ b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 - sqrt(1/2)*((a*b^2 - 4*a^2*c)*
sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4
*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)
/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 + sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^
2 - 4*a^2*c)*x)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 -
4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) + 1/2*sqrt(1/2)*sqrt(-(2*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c)
)/(a*b^2 - 4*a^2*c))*log(-(x^2 - sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2
*a + b - (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1
) - 1)/x^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(a + b*x**2 + c*x**4), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (180) = 360\).
time = 5.62, size = 641, normalized size = 2.91 \begin {gather*} -\frac {{\left (2 \, a^{2} b^{2} - 8 \, a^{3} c + 3 \, \sqrt {2} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a^{2} + 2 \, \sqrt {2} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a b - \sqrt {2} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} b^{2} + 4 \, \sqrt {2} \sqrt {2 \, a^{2} + a b + \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a c - 2 \, {\left (b^{2} - 4 \, a c\right )} a^{2}\right )} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}}{2 \, \sqrt {\frac {2 \, a + b + \sqrt {{\left (2 \, a + b\right )}^{2} - 4 \, {\left (a + b + c\right )} a}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - a^{2} b^{4} - 12 \, a^{5} c - 8 \, a^{4} b c + 8 \, a^{3} b^{2} c - 16 \, a^{4} c^{2}\right )}} - \frac {{\left (2 \, a^{2} b^{2} - 8 \, a^{3} c + 3 \, \sqrt {2} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a^{2} + 2 \, \sqrt {2} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a b - \sqrt {2} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} b^{2} + 4 \, \sqrt {2} \sqrt {2 \, a^{2} + a b - \sqrt {b^{2} - 4 \, a c} a} \sqrt {b^{2} - 4 \, a c} a c - 2 \, {\left (b^{2} - 4 \, a c\right )} a^{2}\right )} {\left | a \right |} \arctan \left (-\frac {\sqrt {2} {\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}}{2 \, \sqrt {\frac {2 \, a + b - \sqrt {{\left (2 \, a + b\right )}^{2} - 4 \, {\left (a + b + c\right )} a}}{a}}}\right )}{2 \, {\left (3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - a^{2} b^{4} - 12 \, a^{5} c - 8 \, a^{4} b c + 8 \, a^{3} b^{2} c - 16 \, a^{4} c^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/2*(2*a^2*b^2 - 8*a^3*c + 3*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2 + 2*sqrt(2
)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)
*a)*sqrt(b^2 - 4*a*c)*b^2 + 4*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*c - 2*(b^2 -
4*a*c)*a^2)*abs(a)*arctan(-1/2*sqrt(2)*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*a + b + sqrt
((2*a + b)^2 - 4*(a + b + c)*a))/a))/(3*a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - 12*a^5*c - 8*a^4*b*c + 8*a^3*b^2*c - 1
6*a^4*c^2) - 1/2*(2*a^2*b^2 - 8*a^3*c + 3*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^
2 + 2*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b - sqrt(2)*sqrt(2*a^2 + a*b - sqrt(
b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^2 + 4*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*
c - 2*(b^2 - 4*a*c)*a^2)*abs(a)*arctan(-1/2*sqrt(2)*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*
a + b - sqrt((2*a + b)^2 - 4*(a + b + c)*a))/a))/(3*a^4*b^2 + 2*a^3*b^3 - a^2*b^4 - 12*a^5*c - 8*a^4*b*c + 8*a
^3*b^2*c - 16*a^4*c^2)

________________________________________________________________________________________

Mupad [B]
time = 1.27, size = 989, normalized size = 4.50 \begin {gather*} -\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 (b^2\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a\,\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 (b^2\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a\,\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 (b^2\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a\,\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 (b^2\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+2\,a\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x^2)^(1/2)/(a + b*x^2 + c*x^4),x)

[Out]

(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)))*(b^2*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)
+ a*b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*a*c*(-(b +
(b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a*(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)))*(b^2*(
-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c*(-(b - (b^2 - 4
*a*c)^(1/2))/(2*c))^(1/2) + 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(
2*c))^(3/2)))/(2*a*((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)))*(b^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b*(-(b - (b^2 - 4*a*c)^(1/
2))/(2*c))^(1/2) - 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*a*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(
3/2) + b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a*((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)))*(b^2*(-(b + (b^2 - 4*a*c)^(1/2))/(2*
c))^(1/2) + a*b*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 2*
a*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a*(4*a*c - b^
2)*((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2))

________________________________________________________________________________________