3.52 \(\int \frac{-a-\sqrt{1+a^2}+x}{(-a+\sqrt{1+a^2}+x) \sqrt{(-a+x) (1+x^2)}} \, dx\)

Optimal. Leaf size=66 \[ -\sqrt{2} \sqrt{\sqrt{a^2+1}+a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{\sqrt{a^2+1}-a} (x-a)}{\sqrt{\left (x^2+1\right ) (x-a)}}\right ) \]

[Out]

-(Sqrt[2]*Sqrt[a + Sqrt[1 + a^2]]*ArcTan[(Sqrt[2]*Sqrt[-a + Sqrt[1 + a^2]]*(-a + x))/Sqrt[(-a + x)*(1 + x^2)]]
)

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Rubi [C]  time = 1.23427, antiderivative size = 204, normalized size of antiderivative = 3.09, number of steps used = 9, number of rules used = 8, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6719, 6742, 719, 419, 932, 168, 538, 537} \[ \frac{4 \sqrt{a^2+1} \sqrt{x^2+1} \sqrt{\frac{a-x}{a+i}} \Pi \left (\frac{2}{1-i \left (a-\sqrt{a^2+1}\right )};\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\left (1-i \left (a-\sqrt{a^2+1}\right )\right ) \sqrt{\left (x^2+1\right ) (-(a-x))}}+\frac{2 i \sqrt{x^2+1} \sqrt{\frac{a-x}{a+i}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\sqrt{\left (x^2+1\right ) (-(a-x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*I)*Sqrt[(a - x)/(I + a)]*Sqrt[1 + x^2]*EllipticF[ArcSin[Sqrt[1 - I*x]/Sqrt[2]], 2/(1 - I*a)])/Sqrt[-((a -
x)*(1 + x^2))] + (4*Sqrt[1 + a^2]*Sqrt[(a - x)/(I + a)]*Sqrt[1 + x^2]*EllipticPi[2/(1 - I*(a - Sqrt[1 + a^2]))
, ArcSin[Sqrt[1 - I*x]/Sqrt[2]], 2/(1 - I*a)])/((1 - I*(a - Sqrt[1 + a^2]))*Sqrt[-((a - x)*(1 + x^2))])

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 419

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

Rule 932

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c,
d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/d, 0]

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rubi steps

\begin{align*} \int \frac{-a-\sqrt{1+a^2}+x}{\left (-a+\sqrt{1+a^2}+x\right ) \sqrt{(-a+x) \left (1+x^2\right )}} \, dx &=\frac{\left (\sqrt{-a+x} \sqrt{1+x^2}\right ) \int \frac{-a-\sqrt{1+a^2}+x}{\sqrt{-a+x} \left (-a+\sqrt{1+a^2}+x\right ) \sqrt{1+x^2}} \, dx}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=\frac{\left (\sqrt{-a+x} \sqrt{1+x^2}\right ) \int \left (\frac{1}{\sqrt{-a+x} \sqrt{1+x^2}}-\frac{2 \sqrt{1+a^2}}{\sqrt{-a+x} \left (-a+\sqrt{1+a^2}+x\right ) \sqrt{1+x^2}}\right ) \, dx}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=\frac{\left (\sqrt{-a+x} \sqrt{1+x^2}\right ) \int \frac{1}{\sqrt{-a+x} \sqrt{1+x^2}} \, dx}{\sqrt{(-a+x) \left (1+x^2\right )}}-\frac{\left (2 \sqrt{1+a^2} \sqrt{-a+x} \sqrt{1+x^2}\right ) \int \frac{1}{\sqrt{-a+x} \left (-a+\sqrt{1+a^2}+x\right ) \sqrt{1+x^2}} \, dx}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=-\frac{\left (2 \sqrt{1+a^2} \sqrt{-a+x} \sqrt{1+x^2}\right ) \int \frac{1}{\sqrt{1-i x} \sqrt{1+i x} \sqrt{-a+x} \left (-a+\sqrt{1+a^2}+x\right )} \, dx}{\sqrt{(-a+x) \left (1+x^2\right )}}+\frac{\left (2 i \sqrt{\frac{-a+x}{-i-a}} \sqrt{1+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 i x^2}{-i-a}}} \, dx,x,\frac{\sqrt{1-i x}}{\sqrt{2}}\right )}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=\frac{2 i \sqrt{\frac{a-x}{i+a}} \sqrt{1+x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\sqrt{-(a-x) \left (1+x^2\right )}}+\frac{\left (4 \sqrt{1+a^2} \sqrt{-a+x} \sqrt{1+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (1-i \left (a-\sqrt{1+a^2}\right )-x^2\right ) \sqrt{-i-a+i x^2}} \, dx,x,\sqrt{1-i x}\right )}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=\frac{2 i \sqrt{\frac{a-x}{i+a}} \sqrt{1+x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\sqrt{-(a-x) \left (1+x^2\right )}}+\frac{\left (4 \sqrt{1+a^2} \sqrt{\frac{a-x}{i+a}} \sqrt{1+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2} \left (1-i \left (a-\sqrt{1+a^2}\right )-x^2\right ) \sqrt{1+\frac{i x^2}{-i-a}}} \, dx,x,\sqrt{1-i x}\right )}{\sqrt{(-a+x) \left (1+x^2\right )}}\\ &=\frac{2 i \sqrt{\frac{a-x}{i+a}} \sqrt{1+x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\sqrt{-(a-x) \left (1+x^2\right )}}+\frac{4 \sqrt{1+a^2} \sqrt{\frac{a-x}{i+a}} \sqrt{1+x^2} \Pi \left (\frac{2}{1-i \left (a-\sqrt{1+a^2}\right )};\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\left (1-i \left (a-\sqrt{1+a^2}\right )\right ) \sqrt{-(a-x) \left (1+x^2\right )}}\\ \end{align*}

Mathematica [C]  time = 1.14754, size = 213, normalized size = 3.23 \[ \frac{2 \sqrt{\frac{a-x}{a+i}} \left (2 i \sqrt{a^2+1} \sqrt{1-i x} \sqrt{x^2+1} \Pi \left (\frac{2 i}{a-\sqrt{a^2+1}+i};\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2 i}{a+i}\right )-\left (\sqrt{a^2+1}-a-i\right ) \sqrt{1+i x} (x+i) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right ),\frac{2 i}{a+i}\right )\right )}{\left (-\sqrt{a^2+1}+a+i\right ) \sqrt{1-i x} \sqrt{\left (x^2+1\right ) (x-a)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a - x)/(I + a)]*(-((-I - a + Sqrt[1 + a^2])*Sqrt[1 + I*x]*(I + x)*EllipticF[ArcSin[Sqrt[1 - I*x]/Sqrt
[2]], (2*I)/(I + a)]) + (2*I)*Sqrt[1 + a^2]*Sqrt[1 - I*x]*Sqrt[1 + x^2]*EllipticPi[(2*I)/(I + a - Sqrt[1 + a^2
]), ArcSin[Sqrt[1 - I*x]/Sqrt[2]], (2*I)/(I + a)]))/((I + a - Sqrt[1 + a^2])*Sqrt[1 - I*x]*Sqrt[(-a + x)*(1 +
x^2)])

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Maple [C]  time = 0.094, size = 1275, normalized size = 19.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-a-I)*((-a+x)/(-a-I))^(1/2)*((x-I)/(a-I))^(1/2)*((x+I)/(a+I))^(1/2)/(-a*x^2+x^3-a+x)^(1/2)*EllipticF(((-a+x
)/(-a-I))^(1/2),((a+I)/(a-I))^(1/2))-2*(a^2+1)^(1/2)*((-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1/(a-I)*x-I/(a-I))^(1/2)
*(1/(a+I)*x+I/(a+I))^(1/2)/(-a*x^2+x^3-a+x)^(1/2)/(a^2+1)^(1/2)*EllipticPi(((-a+x)/(-a-I))^(1/2),-(a+I)/(a^2+1
)^(1/2),((a+I)/(a-I))^(1/2))*a+I*(-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1/(a-I)*x-I/(a-I))^(1/2)*(1/(a+I)*x+I/(a+I))^
(1/2)/(-a*x^2+x^3-a+x)^(1/2)/(a^2+1)^(1/2)*EllipticPi(((-a+x)/(-a-I))^(1/2),-(a+I)/(a^2+1)^(1/2),((a+I)/(a-I))
^(1/2))-(-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1/(a-I)*x-I/(a-I))^(1/2)*(1/(a+I)*x+I/(a+I))^(1/2)/(-a*x^2+x^3-a+x)^(1
/2)/(a^2+1)^(1/2)*EllipticPi(((-a+x)/(-a-I))^(1/2),(a+I)/(a^2+1)^(1/2),((a+I)/(a-I))^(1/2))*a-I*(-1/(-a-I)*a+1
/(-a-I)*x)^(1/2)*(1/(a-I)*x-I/(a-I))^(1/2)*(1/(a+I)*x+I/(a+I))^(1/2)/(-a*x^2+x^3-a+x)^(1/2)/(a^2+1)^(1/2)*Elli
pticPi(((-a+x)/(-a-I))^(1/2),(a+I)/(a^2+1)^(1/2),((a+I)/(a-I))^(1/2))-I/(a^2+1)^(1/2)*(1-I*x)^(1/2)*(-1/(-a-I)
*a+1/(-a-I)*x)^(1/2)*(1+I*x)^(1/2)/(-a^3*x^2+a^2*x^3-a^3+a^2*x-a*x^2+x^3-a+x)^(1/2)/(-I-a-(a^2+1)^(1/2))*Ellip
ticPi(1/2*2^(1/2)*(-I*(x+I))^(1/2),-2*I/(-I-a-(a^2+1)^(1/2)),2^(1/2)*(-I/(-a-I))^(1/2))*a^2-I/(a^2+1)^(1/2)*(1
-I*x)^(1/2)*(-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1+I*x)^(1/2)/(-a^3*x^2+a^2*x^3-a^3+a^2*x-a*x^2+x^3-a+x)^(1/2)/(-I-
a-(a^2+1)^(1/2))*EllipticPi(1/2*2^(1/2)*(-I*(x+I))^(1/2),-2*I/(-I-a-(a^2+1)^(1/2)),2^(1/2)*(-I/(-a-I))^(1/2))+
I/(a^2+1)^(1/2)*(1-I*x)^(1/2)*(-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1+I*x)^(1/2)/(-a^3*x^2+a^2*x^3-a^3+a^2*x-a*x^2+x
^3-a+x)^(1/2)/(-I-a+(a^2+1)^(1/2))*EllipticPi(1/2*2^(1/2)*(-I*(x+I))^(1/2),-2*I/(-I-a+(a^2+1)^(1/2)),2^(1/2)*(
-I/(-a-I))^(1/2))*a^2+I/(a^2+1)^(1/2)*(1-I*x)^(1/2)*(-1/(-a-I)*a+1/(-a-I)*x)^(1/2)*(1+I*x)^(1/2)/(-a^3*x^2+a^2
*x^3-a^3+a^2*x-a*x^2+x^3-a+x)^(1/2)/(-I-a+(a^2+1)^(1/2))*EllipticPi(1/2*2^(1/2)*(-I*(x+I))^(1/2),-2*I/(-I-a+(a
^2+1)^(1/2)),2^(1/2)*(-I/(-a-I))^(1/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a - x + \sqrt{a^{2} + 1}}{\sqrt{-{\left (x^{2} + 1\right )}{\left (a - x\right )}}{\left (a - x - \sqrt{a^{2} + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.06192, size = 1272, normalized size = 19.27 \begin{align*} \left [\frac{1}{4} \, \sqrt{-2 \, a - 2 \, \sqrt{a^{2} + 1}} \log \left (-\frac{8 \, a x^{7} + x^{8} + 4 \,{\left (2 \, a^{2} + 15\right )} x^{6} - 8 \,{\left (4 \, a^{3} + 15 \, a\right )} x^{5} + 2 \,{\left (8 \, a^{4} + 80 \, a^{2} + 67\right )} x^{4} + 64 \, a^{4} - 8 \,{\left (20 \, a^{3} + 37 \, a\right )} x^{3} + 4 \,{\left (16 \, a^{4} + 74 \, a^{2} + 15\right )} x^{2} + 48 \, a^{2} - 4 \,{\left (a x^{6} + 2 \,{\left (2 \, a^{2} + 3\right )} x^{5} -{\left (4 \, a^{3} - a\right )} x^{4} - 8 \, a^{3} -{\left (4 \, a^{3} + 29 \, a\right )} x^{2} + 20 \, x^{3} + 2 \,{\left (10 \, a^{2} + 3\right )} x -{\left (4 \, a x^{5} + x^{6} -{\left (4 \, a^{2} - 15\right )} x^{4} - 16 \, a x^{3} +{\left (4 \, a^{2} + 15\right )} x^{2} + 8 \, a^{2} - 20 \, a x + 1\right )} \sqrt{a^{2} + 1} - 5 \, a\right )} \sqrt{-a x^{2} + x^{3} - a + x} \sqrt{-2 \, a - 2 \, \sqrt{a^{2} + 1}} - 8 \,{\left (24 \, a^{3} + 13 \, a\right )} x + 16 \,{\left (a x^{6} - x^{7} + 15 \, a x^{4} - 7 \, x^{5} -{\left (12 \, a^{2} + 7\right )} x^{3} + 4 \, a^{3} +{\left (4 \, a^{3} + 15 \, a\right )} x^{2} -{\left (12 \, a^{2} + 1\right )} x + a\right )} \sqrt{a^{2} + 1} + 1}{8 \, a x^{7} - x^{8} - 4 \,{\left (6 \, a^{2} - 1\right )} x^{6} + 8 \,{\left (4 \, a^{3} - 3 \, a\right )} x^{5} - 2 \,{\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} x^{4} - 8 \,{\left (4 \, a^{3} - 3 \, a\right )} x^{3} - 4 \,{\left (6 \, a^{2} - 1\right )} x^{2} - 8 \, a x - 1}\right ), -\frac{1}{2} \, \sqrt{2 \, a + 2 \, \sqrt{a^{2} + 1}} \arctan \left (-\frac{\sqrt{-a x^{2} + x^{3} - a + x}{\left (2 \, a^{2} - 2 \, a x - x^{2} - 2 \, \sqrt{a^{2} + 1}{\left (a - x\right )} - 1\right )} \sqrt{2 \, a + 2 \, \sqrt{a^{2} + 1}}}{4 \,{\left (a x^{2} - x^{3} + a - x\right )}}\right )\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(-2*a - 2*sqrt(a^2 + 1))*log(-(8*a*x^7 + x^8 + 4*(2*a^2 + 15)*x^6 - 8*(4*a^3 + 15*a)*x^5 + 2*(8*a^4 +
80*a^2 + 67)*x^4 + 64*a^4 - 8*(20*a^3 + 37*a)*x^3 + 4*(16*a^4 + 74*a^2 + 15)*x^2 + 48*a^2 - 4*(a*x^6 + 2*(2*a
^2 + 3)*x^5 - (4*a^3 - a)*x^4 - 8*a^3 - (4*a^3 + 29*a)*x^2 + 20*x^3 + 2*(10*a^2 + 3)*x - (4*a*x^5 + x^6 - (4*a
^2 - 15)*x^4 - 16*a*x^3 + (4*a^2 + 15)*x^2 + 8*a^2 - 20*a*x + 1)*sqrt(a^2 + 1) - 5*a)*sqrt(-a*x^2 + x^3 - a +
x)*sqrt(-2*a - 2*sqrt(a^2 + 1)) - 8*(24*a^3 + 13*a)*x + 16*(a*x^6 - x^7 + 15*a*x^4 - 7*x^5 - (12*a^2 + 7)*x^3
+ 4*a^3 + (4*a^3 + 15*a)*x^2 - (12*a^2 + 1)*x + a)*sqrt(a^2 + 1) + 1)/(8*a*x^7 - x^8 - 4*(6*a^2 - 1)*x^6 + 8*(
4*a^3 - 3*a)*x^5 - 2*(8*a^4 - 24*a^2 + 3)*x^4 - 8*(4*a^3 - 3*a)*x^3 - 4*(6*a^2 - 1)*x^2 - 8*a*x - 1)), -1/2*sq
rt(2*a + 2*sqrt(a^2 + 1))*arctan(-1/4*sqrt(-a*x^2 + x^3 - a + x)*(2*a^2 - 2*a*x - x^2 - 2*sqrt(a^2 + 1)*(a - x
) - 1)*sqrt(2*a + 2*sqrt(a^2 + 1))/(a*x^2 - x^3 + a - x))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a - x + \sqrt{a^{2} + 1}}{\sqrt{-{\left (x^{2} + 1\right )}{\left (a - x\right )}}{\left (a - x - \sqrt{a^{2} + 1}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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