3.4.79 \(\int \frac {\sqrt {1-x^2}}{x (a+b x^2+c x^4)} \, dx\) [379]

Optimal. Leaf size=241 \[ -\frac {\tanh ^{-1}\left (\sqrt {1-x^2}\right )}{a}+\frac {\sqrt {c} \left (2 a+b+\sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (2 a+b-\sqrt {b^2-4 a c}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {1-x^2}}{\sqrt {b+2 c+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {b+2 c+\sqrt {b^2-4 a c}}} \]

[Out]

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

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Rubi [A]
time = 1.03, antiderivative size = 241, 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 (\sqrt {b^2-4 a c}+2 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 \sqrt {b^2-4 a c} \sqrt {-\sqrt {b^2-4 a c}+b+2 c}}-\frac {\sqrt {c} \left (-\sqrt {b^2-4 a c}+2 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 \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b+2 c}}-\frac {\tanh ^{-1}\left (\sqrt {1-x^2}\right )}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcTanh[Sqrt[1 - x^2]]/a) + (Sqrt[c]*(2*a + b + Sqrt[b^2 - 4*a*c])*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[1 - x^2])/S
qrt[b + 2*c - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*
(2*a + b - 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]]])/(Sqr
t[2]*a*Sqrt[b^2 - 4*a*c]*Sqrt[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 \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1-x}}{x \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 ) \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}{a \left (-1+x^2\right )}+\frac {-a-b-c+c x^2}{a \left (a+b+c-(b+2 c) x^2+c x^4\right )}\right ) \, dx,x,\sqrt {1-x^2}\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1-x^2}\right )}{a}-\frac {\text {Subst}\left (\int \frac {-a-b-c+c x^2}{a+b+c+(-b-2 c) x^2+c x^4} \, dx,x,\sqrt {1-x^2}\right )}{a}\\ &=-\frac {\tanh ^{-1}\left (\sqrt {1-x^2}\right )}{a}+\frac {\left (c \left (2 a+b-\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 \sqrt {b^2-4 a c}}-\frac {\left (c \left (2 a+b+\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 \sqrt {b^2-4 a c}}\\ &=-\frac {\tanh ^{-1}\left (\sqrt {1-x^2}\right )}{a}+\frac {\sqrt {c} \left (2 a+b+\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 \sqrt {b^2-4 a c} \sqrt {b+2 c-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (2 a+b-\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 \sqrt {b^2-4 a c} \sqrt {b+2 c+\sqrt {b^2-4 a c}}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 264, normalized size = 1.10 \begin {gather*} -\frac {\frac {\sqrt {2} \sqrt {c} \left (-2 a-b+\sqrt {b^2-4 a c}\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 (2 a+b+\sqrt {b^2-4 a c}\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}}}-\log \left (-1+\sqrt {1-x^2}\right )+\log \left (a \left (1+\sqrt {1-x^2}\right )\right )}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(474\) vs. \(2(196)=392\).
time = 0.14, size = 475, normalized size = 1.97

method	result	size
default	\(-\frac {2 a \left (-\frac {\left (\sqrt {-4 a c +b^{2}}\, a b -2 a c \sqrt {-4 a c +b^{2}}+b^{2} \sqrt {-4 a c +b^{2}}+4 a^{2} c -a \,b^{2}+4 a b c -b^{3}\right ) \arctan \left (\frac {-\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}-2 a -2 b}{2 \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}+2 \sqrt {-4 a c +b^{2}}\, a +2 b \sqrt {-4 a c +b^{2}}-2 a b}}+\frac {\left (-\sqrt {-4 a c +b^{2}}\, a b +2 a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}+4 a^{2} c -a \,b^{2}+4 a b c -b^{3}\right ) \arctan \left (\frac {\frac {2 a \left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+2 \sqrt {-4 a c +b^{2}}+2 a +2 b}{2 \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )}{2 a \left (4 a c -b^{2}\right ) \sqrt {4 a c -2 b^{2}-2 \sqrt {-4 a c +b^{2}}\, a -2 b \sqrt {-4 a c +b^{2}}-2 a b}}\right )+\frac {2}{\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+1}}{a}+\frac {\sqrt {-x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )}{a}\)	\(475\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1232 vs. \(2 (196) = 392\).
time = 2.65, size = 1232, normalized size = 5.11 \begin {gather*} \frac {\sqrt {\frac {1}{2}} a \sqrt {\frac {a b + b^{2} - 2 \, a c + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} \sqrt {\frac {a b + b^{2} - 2 \, a c + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} + {\left (a b + b^{2}\right )} x^{2} + 2 \, a^{2} + 2 \, a b - 2 \, {\left (a^{2} + a b\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} a \sqrt {\frac {a b + b^{2} - 2 \, a c + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} \log \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} \sqrt {\frac {a b + b^{2} - 2 \, a c + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} - {\left (a b + b^{2}\right )} x^{2} - 2 \, a^{2} - 2 \, a b + 2 \, {\left (a^{2} + a b\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) + \sqrt {\frac {1}{2}} a \sqrt {\frac {a b + b^{2} - 2 \, a c - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} \log \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} \sqrt {\frac {a b + b^{2} - 2 \, a c - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} + {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} - {\left (a b + b^{2}\right )} x^{2} - 2 \, a^{2} - 2 \, a b + 2 \, {\left (a^{2} + a b\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) - \sqrt {\frac {1}{2}} a \sqrt {\frac {a b + b^{2} - 2 \, a c - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} \sqrt {\frac {a b + b^{2} - 2 \, a c - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}}}{a^{2} b^{2} - 4 \, a^{3} c}} - {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{a^{4} b^{2} - 4 \, a^{5} c}} + {\left (a b + b^{2}\right )} x^{2} + 2 \, a^{2} + 2 \, a b - 2 \, {\left (a^{2} + a b\right )} \sqrt {-x^{2} + 1}}{x^{2}}\right ) + 2 \, \log \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(1/2)*a*sqrt((a*b + b^2 - 2*a*c + (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/
(a^2*b^2 - 4*a^3*c))*log((2*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c))*sq
rt((a*b + b^2 - 2*a*c + (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)
) + (a^2*b^2 - 4*a^3*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)) + (a*b + b^2)*x^2 + 2*a^2 + 2*a*b -
2*(a^2 + a*b)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*a*sqrt((a*b + b^2 - 2*a*c + (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a
*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*x^2*sqrt((a^2 + 2*a
*b + b^2)/(a^4*b^2 - 4*a^5*c))*sqrt((a*b + b^2 - 2*a*c + (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2
- 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - (a^2*b^2 - 4*a^3*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)) - (
a*b + b^2)*x^2 - 2*a^2 - 2*a*b + 2*(a^2 + a*b)*sqrt(-x^2 + 1))/x^2) + sqrt(1/2)*a*sqrt((a*b + b^2 - 2*a*c - (a
^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c))*log(-(2*sqrt(1/2)*(a^3*b
^2 - 4*a^4*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c))*sqrt((a*b + b^2 - 2*a*c - (a^2*b^2 - 4*a^3*c)*
sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) + (a^2*b^2 - 4*a^3*c)*x^2*sqrt((a^2 + 2*a*
b + b^2)/(a^4*b^2 - 4*a^5*c)) - (a*b + b^2)*x^2 - 2*a^2 - 2*a*b + 2*(a^2 + a*b)*sqrt(-x^2 + 1))/x^2) - sqrt(1/
2)*a*sqrt((a*b + b^2 - 2*a*c - (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4
*a^3*c))*log((2*sqrt(1/2)*(a^3*b^2 - 4*a^4*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c))*sqrt((a*b + b^
2 - 2*a*c - (a^2*b^2 - 4*a^3*c)*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - (a^2*b^2
- 4*a^3*c)*x^2*sqrt((a^2 + 2*a*b + b^2)/(a^4*b^2 - 4*a^5*c)) + (a*b + b^2)*x^2 + 2*a^2 + 2*a*b - 2*(a^2 + a*b
)*sqrt(-x^2 + 1))/x^2) + 2*log((sqrt(-x^2 + 1) - 1)/x))/a

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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 \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/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(x*(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. 3639 vs. \(2 (196) = 392\).
time = 8.10, size = 3639, normalized size = 15.10 \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/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/2*log(sqrt(-x^2 + 1) + 1)/a + 1/2*log(-sqrt(-x^2 + 1) + 1)/a + 1/8*(4*a^3*b^3*c^2 + 2*a^2*b^4*c^2 - 16*a^4*
b*c^3 + 4*a^2*b^3*c^3 - 32*a^4*c^4 - 16*a^3*b*c^4 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 -
4*a*c)*c)*a^3*b^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^4 + 8*sqrt(2)*sq
rt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^4*b*c - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c
^2 + sqrt(b^2 - 4*a*c)*c)*a^3*b^2*c - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2
*b^3*c + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^4*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^3*b*c^2 - 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + s
qrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^3*c^
3 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 4*(b^2 - 4*a*c)*a^3*b*c^
2 - 2*(b^2 - 4*a*c)*a^2*b^2*c^2 - 8*(b^2 - 4*a*c)*a^3*c^3 - 4*(b^2 - 4*a*c)*a^2*b*c^3 + (2*b^4*c^2 - 16*a*b^2*
c^3 + 32*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 + 8*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 + s
qrt(b^2 - 4*a*c)*c)*b^3*c - 16*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 + 8*
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 + 20*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^2*c^2 + 8*(b^2 - 4*a*c)*a*c^3)*a^2 + 2*(sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 -
4*a*c)*c)*a^2*b^4 + sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^5 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 + sq
rt(b^2 - 4*a*c)*c)*a^3*b^2*c - 6*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c + 3*sqrt(2)*sqrt(-
b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^4*c - 2*a^2*b^4*c - 2*a*b^5*c + 16*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2
- 4*a*c)*c)*a^4*c^2 + 8*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^3*b*c^2 - 11*sqrt(2)*sqrt(-b*c - 2
*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 + 16*a^3*b^2*c^2 + 7*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*
a*b^3*c^2 + 16*a^2*b^3*c^2 - 2*a*b^4*c^2 - 4*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^3*c^3 - 32*a^4
*c^3 - 28*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 32*a^3*b*c^3 + 5*sqrt(2)*sqrt(-b*c - 2*
c^2 + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^3 + 16*a^2*b^2*c^3 - 20*sqrt(2)*sqrt(-b*c - 2*c^2 + sqrt(b^2 - 4*a*c)*c)*a^
2*c^4 - 32*a^3*c^4 + 2*(b^2 - 4*a*c)*a^2*b^2*c + 2*(b^2 - 4*a*c)*a*b^3*c - 8*(b^2 - 4*a*c)*a^3*c^2 - 8*(b^2 -
4*a*c)*a^2*b*c^2 + 2*(b^2 - 4*a*c)*a*b^2*c^2 - 8*(b^2 - 4*a*c)*a^2*c^3)*abs(a))*arctan(2*sqrt(1/2)*sqrt(-x^2 +
1)/sqrt(-(a*b + 2*a*c + sqrt(-4*(a^2 + a*b + a*c)*a*c + (a*b + 2*a*c)^2))/(a*c)))/((a^3*b^4 + a^2*b^5 - 8*a^4
*b^2*c - 6*a^3*b^3*c + 3*a^2*b^4*c + 16*a^5*c^2 + 8*a^4*b*c^2 - 11*a^3*b^2*c^2 + 7*a^2*b^3*c^2 - 4*a^4*c^3 - 2
8*a^3*b*c^3 + 5*a^2*b^2*c^3 - 20*a^3*c^4)*abs(a)*abs(c)) - 1/8*(4*a^3*b^3*c^2 + 2*a^2*b^4*c^2 - 16*a^4*b*c^3 +
4*a^2*b^3*c^3 - 32*a^4*c^4 - 16*a^3*b*c^4 - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)
*c)*a^3*b^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^4 + 8*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^4*b*c - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sq
rt(b^2 - 4*a*c)*c)*a^3*b^2*c - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c
+ 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^4*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^3*b*c^2 - 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2
- 4*a*c)*c)*a^2*b^2*c^2 - 20*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^3*c^3 - 10*
sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 4*(b^2 - 4*a*c)*a^3*b*c^2 - 2*(
b^2 - 4*a*c)*a^2*b^2*c^2 - 8*(b^2 - 4*a*c)*a^3*c^3 - 4*(b^2 - 4*a*c)*a^2*b*c^3 + (2*b^4*c^2 - 16*a*b^2*c^3 + 3
2*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 + 8*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 - 16*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 + 8*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 + 20*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^2*c^2 + 8*(b^2 - 4*a*c)*a*c^3)*a^2 - 2*(sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)
*c)*a^2*b^4 + sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2 - 4*a*c)*c)*a*b^5 - 8*sqrt(2)*sqrt(-b*c - 2*c^2 - sqrt(b^2
- 4*a*c)*c)*a^3*b^2*c - 6*sqrt(2)*sqrt(-b*c - 2...

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Mupad [B]
time = 1.30, size = 669, normalized size = 2.78 \begin {gather*} \frac {\ln \left (\sqrt {\frac {1}{x^2}-1}-\sqrt {\frac {1}{x^2}}\right )}{a}+\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\,c+2\,a\,\sqrt {b^2-4\,a\,c}+b\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,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 (2\,a\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,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 (4\,a\,c+2\,a\,\sqrt {b^2-4\,a\,c}+b\,\sqrt {b^2-4\,a\,c}-b^2\right )}{4\,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 (2\,a\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b\,\sqrt {b^2-4\,a\,c}+b^2\right )}{4\,a\,\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*(a + b*x^2 + c*x^4)),x)

[Out]

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

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