3.4.80 \(\int \frac {\sqrt {1-x^2}}{x^3 (a+b x^2+c x^4)} \, dx\) [380]

Optimal. Leaf size=290 \[ -\frac {1}{4 a \left (1-\sqrt {1-x^2}\right )}+\frac {1}{4 a \left (1+\sqrt {1-x^2}\right )}+\frac {(a+2 b) \tanh ^{-1}\left (\sqrt {1-x^2}\right )}{2 a^2}-\frac {\sqrt {c} \left (a+b+\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (a+b-\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a^2 \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

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)

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Rubi [A]
time = 1.52, antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1265, 911, 1301, 213, 1180, 214} \begin {gather*} -\frac {\sqrt {c} \left (\frac {a (b-2 c)+b^2}{\sqrt {b^2-4 a c}}+a+b\right ) \tanh ^{-1}\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 ) \tanh ^{-1}\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) \tanh ^{-1}\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 )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

-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) - (S
qrt[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]]) - (Sqrt[c]*(a + b - (b^2 + a*(b - 2*c))/S
qrt[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*Sqr
t[b + 2*c + Sqrt[b^2 - 4*a*c]])

Rule 213

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

Rule 214

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[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))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[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] &&
IntegerQ[(m - 1)/2]

Rule 1301

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[(f*x)^m*((d + e*x^2)^q/(a + b*x^2 + c*x^4)), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x^2}}{x^3 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2 \left (a+b+c+(-b-2 c) x^2+c x^4\right )} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\text {Subst}\left (\int \left (\frac {1}{4 a (-1+x)^2}+\frac {1}{4 a (1+x)^2}+\frac {a+2 b}{2 a^2 \left (-1+x^2\right )}+\frac {b (a+b+c)-(a+b) c x^2}{a^2 \left (a+b+c-(b+2 c) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {1}{4 a \left (1-\sqrt {1-x^2}\right )}+\frac {1}{4 a \left (1+\sqrt {1-x^2}\right )}-\frac {\text {Subst}\left (\int \frac {b (a+b+c)-(a+b) c x^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )}{a^2}-\frac {(a+2 b) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-x^2}\right )}{2 a^2}\\ &=-\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) \tanh ^{-1}\left (\sqrt {1-x^2}\right )}{2 a^2}+\frac {\left (c \left (a+b-\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} (-b-2 c)-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {1-x^2}\right )}{2 a^2}+\frac {\left (c \left (a+b+\frac {b^2+a (b-2 c)}{\sqrt {b^2-4 a c}}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} (-b-2 c)+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx,x,\sqrt {1-x^2}\right )}{2 a^2}\\ &=-\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) \tanh ^{-1}\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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}}}\\ \end {align*}

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Mathematica [A]
time = 1.54, size = 307, normalized size = 1.06 \begin {gather*} \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 ) \tan ^{-1}\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 ) \tan ^{-1}\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) \tanh ^{-1}\left (\sqrt {1-x^2}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - x^2]/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

(-((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]))*A
rcTan[(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]]) + (Sqrt[2]*Sqrt[c]*(b*(b + Sqrt[b^2 - 4*a*c]) + a*(b - 2*c + Sqrt[b^2 - 4*a*c]))*ArcTan[(S
qrt[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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(579\) vs. \(2(240)=480\).
time = 0.17, size = 580, normalized size = 2.00

method	result	size
default	\(-\frac {-2 a \left (\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a^{2} c -\sqrt {-4 a c +b^{2}}\, a \,b^{2}+3 \sqrt {-4 a c +b^{2}}\, a b c -\sqrt {-4 a c +b^{2}}\, b^{3}+4 a^{2} b c -4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}-\frac {\left (-2 \sqrt {-4 a c +b^{2}}\, a^{2} c +\sqrt {-4 a c +b^{2}}\, a \,b^{2}-3 \sqrt {-4 a c +b^{2}}\, a b c +\sqrt {-4 a c +b^{2}}\, b^{3}+4 a^{2} b c -4 a^{2} c^{2}-a \,b^{3}+5 a \,b^{2} c -b^{4}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )-\frac {2 b}{\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+1}}{a^{2}}+\frac {-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {\sqrt {-x^{2}+1}}{2}+\frac {\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{2}}{a}-\frac {b \left (\sqrt {-x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right )}{a^{2}}\)	\(580\)
risch	\(\text {Expression too large to display}\)	\(2718\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^2+1)^(1/2)/x^3/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

-1/a^2*(-2*a*(1/2*(2*(-4*a*c+b^2)^(1/2)*a^2*c-(-4*a*c+b^2)^(1/2)*a*b^2+3*(-4*a*c+b^2)^(1/2)*a*b*c-(-4*a*c+b^2)
^(1/2)*b^3+4*a^2*b*c-4*a^2*c^2-a*b^3+5*a*b^2*c-b^4)/a/(4*a*c-b^2)/(4*a*c-2*b^2-2*(-4*a*c+b^2)^(1/2)*a-2*b*(-4*
a*c+b^2)^(1/2)-2*a*b)^(1/2)*arctan(1/2*(2*a*((-x^2+1)^(1/2)-1)^2/x^2+2*(-4*a*c+b^2)^(1/2)+2*a+2*b)/(4*a*c-2*b^
2-2*(-4*a*c+b^2)^(1/2)*a-2*b*(-4*a*c+b^2)^(1/2)-2*a*b)^(1/2))-1/2*(-2*(-4*a*c+b^2)^(1/2)*a^2*c+(-4*a*c+b^2)^(1
/2)*a*b^2-3*(-4*a*c+b^2)^(1/2)*a*b*c+(-4*a*c+b^2)^(1/2)*b^3+4*a^2*b*c-4*a^2*c^2-a*b^3+5*a*b^2*c-b^4)/a/(4*a*c-
b^2)/(4*a*c-2*b^2+2*(-4*a*c+b^2)^(1/2)*a+2*b*(-4*a*c+b^2)^(1/2)-2*a*b)^(1/2)*arctan(1/2*(-2*a*((-x^2+1)^(1/2)-
1)^2/x^2+2*(-4*a*c+b^2)^(1/2)-2*a-2*b)/(4*a*c-2*b^2+2*(-4*a*c+b^2)^(1/2)*a+2*b*(-4*a*c+b^2)^(1/2)-2*a*b)^(1/2)
))-2*b/(((-x^2+1)^(1/2)-1)^2/x^2+1))+1/a*(-1/2/x^2*(-x^2+1)^(3/2)-1/2*(-x^2+1)^(1/2)+1/2*arctanh(1/(-x^2+1)^(1
/2)))-b/a^2*((-x^2+1)^(1/2)-arctanh(1/(-x^2+1)^(1/2)))

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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)/x^3/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2799 vs. \(2 (236) = 472\).
time = 7.20, size = 2799, normalized size = 9.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/x^3/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{x^{3} \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x**2+1)**(1/2)/x**3/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(x**3*(a + b*x**2 + c*x**4)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1675 vs. \(2 (236) = 472\).
time = 6.68, size = 1675, normalized size = 5.78 \begin {gather*} -\frac {{\left (\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} - \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 \, \sqrt {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 \, {\left (b^{2} - 4 \, a c\right )} b^{3} c + 8 \, {\left (b^{2} - 4 \, a c\right )} a b c^{2} - 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c^{2} + 4 \, {\left (b^{2} - 4 \, a c\right )} a c^{3}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {a^{2} b + 2 \, a^{2} c + \sqrt {-4 \, {\left (a^{3} + a^{2} b + a^{2} c\right )} a^{2} c + {\left (a^{2} b + 2 \, a^{2} c\right )}^{2}}}{a^{2} c}}}\right )}{4 \, {\left (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}\right )} {\left | c \right |}} - \frac {{\left (\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} + \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 \, \sqrt {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 \, {\left (b^{2} - 4 \, a c\right )} b^{3} c - 8 \, {\left (b^{2} - 4 \, a c\right )} a b c^{2} + 2 \, {\left (b^{2} - 4 \, a c\right )} b^{2} c^{2} - 4 \, {\left (b^{2} - 4 \, a c\right )} a c^{3}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-x^{2} + 1}}{\sqrt {-\frac {a^{2} b + 2 \, a^{2} c - \sqrt {-4 \, {\left (a^{3} + a^{2} b + a^{2} c\right )} a^{2} c + {\left (a^{2} b + 2 \, a^{2} c\right )}^{2}}}{a^{2} c}}}\right )}{4 \, {\left (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}\right )} {\left | c \right |}} + \frac {{\left (a + 2 \, b\right )} \log \left (\sqrt {-x^{2} + 1} + 1\right )}{4 \, a^{2}} - \frac {{\left (a + 2 \, b\right )} \log \left (-\sqrt {-x^{2} + 1} + 1\right )}{4 \, a^{2}} - \frac {\sqrt {-x^{2} + 1}}{2 \, a x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/x^3/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-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 - 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)*sq
rt(-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 - sqr
t(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*sqrt(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)*sq
rt(-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)*sq
rt(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 + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 2*sqrt(2)*sqrt(-b*c - 2*c^2 + s
qrt(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*sqrt(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*(a + 2*b)*log(sqrt(-x^2 + 1) + 1)/a^2 - 1/4*(a + 2*b)*log(-sqrt(-x^2 + 1) + 1)/a^
2 - 1/2*sqrt(-x^2 + 1)/(a*x^2)

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Mupad [B]
time = 1.41, size = 825, normalized size = 2.84 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*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))

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