3.1.52 \(\int \frac {-a-\sqrt {1+a^2}+x}{(-a+\sqrt {1+a^2}+x) \sqrt {(-a+x) (1+x^2)}} \, dx\) [52]

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

[Out]

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

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.76, 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 = {6851, 6874, 733, 430, 946, 174, 552, 551} \begin {gather*} \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 (\left (x^2+1\right ) (a-x)\right )}}+\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 (\left (x^2+1\right ) (a-x)\right )}} \end {gather*}

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 174

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 430

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

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], 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])

Rule 552

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

Rule 733

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 946

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 6851

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a*v^m*w^n)^FracPart[p]/(v^(m*Fr
acPart[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 6874

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

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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.25, size = 99, normalized size = 1.50 \begin {gather*} -\frac {\sqrt {2} \sqrt {-a+x} \sqrt {1+x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} \sqrt {-a+x}}{\sqrt {1+x^2}}\right )}{\sqrt {-a+\sqrt {1+a^2}} \sqrt {(-a+x) \left (1+x^2\right )}} \end {gather*}

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]

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

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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.17, size = 787, normalized size = 11.92

method	result	size
default	\(\frac {2 i \sqrt {-i \left (x +i\right )}\, \sqrt {\frac {-a +x}{-i-a}}\, \sqrt {i \left (x -i\right )}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, \sqrt {2}\, \sqrt {-\frac {i}{-i-a}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}}-\frac {2 \sqrt {a^{2}+1}\, \sqrt {-\left (a -x \right ) \left (x^{2}+1\right ) \left (a^{2}+1\right )}\, \left (2 a x -x^{2}+1\right ) \left (-\frac {i \sqrt {a^{2}+1}\, \sqrt {-i x +1}\, \sqrt {-\frac {a}{-i-a}+\frac {x}{-i-a}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-i-a -\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-i-a}}\right )}{\sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-i-a -\sqrt {a^{2}+1}\right )}+\frac {i \sqrt {a^{2}+1}\, \sqrt {-i x +1}\, \sqrt {-\frac {a}{-i-a}+\frac {x}{-i-a}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-i-a +\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-i-a}}\right )}{\sqrt {-a^{3} x^{2}+a^{2} x^{3}-a^{3}+a^{2} x -a \,x^{2}+x^{3}-a +x}\, \left (-i-a +\sqrt {a^{2}+1}\right )}+\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-i-a}+\frac {x}{-i-a}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-i-a -\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-i-a}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \left (-i-a -\sqrt {a^{2}+1}\right )}+\frac {i \sqrt {-i x +1}\, \sqrt {-\frac {a}{-i-a}+\frac {x}{-i-a}}\, \sqrt {i x +1}\, \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {-i \left (x +i\right )}}{2}, -\frac {2 i}{-i-a +\sqrt {a^{2}+1}}, \sqrt {2}\, \sqrt {-\frac {i}{-i-a}}\right )}{\sqrt {-a \,x^{2}+x^{3}-a +x}\, \left (-i-a +\sqrt {a^{2}+1}\right )}\right )}{\left (-a +x +\sqrt {a^{2}+1}\right ) \left (\sqrt {-\left (a -x \right ) \left (x^{2}+1\right )}\, a^{2}+\sqrt {-\left (a -x \right ) \left (x^{2}+1\right ) \left (a^{2}+1\right )}\, a -\sqrt {-\left (a -x \right ) \left (x^{2}+1\right ) \left (a^{2}+1\right )}\, x +\sqrt {-\left (a -x \right ) \left (x^{2}+1\right )}\right )}\)	\(787\)
elliptic	\(\text {Expression too large to display}\)	\(1463\)

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,method=_RETURNVERBOSE)

[Out]

2*I*(-I*(x+I))^(1/2)*((-a+x)/(-I-a))^(1/2)*(I*(x-I))^(1/2)/(-a*x^2+x^3-a+x)^(1/2)*EllipticF(1/2*2^(1/2)*(-I*(x
+I))^(1/2),2^(1/2)*(-I/(-I-a))^(1/2))-2*(a^2+1)^(1/2)*(-(a-x)*(x^2+1)*(a^2+1))^(1/2)*(2*a*x-x^2+1)/(-a+x+(a^2+
1)^(1/2))/((-(a-x)*(x^2+1))^(1/2)*a^2+(-(a-x)*(x^2+1)*(a^2+1))^(1/2)*a-(-(a-x)*(x^2+1)*(a^2+1))^(1/2)*x+(-(a-x
)*(x^2+1))^(1/2))*(-I*(a^2+1)^(1/2)*(1-I*x)^(1/2)*(-1/(-I-a)*a+1/(-I-a)*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/(-I-a))^(1/2))+I*(a^2+1)^(1/2)*(1-I*x)^(1/2)*(-1/(-I-a)*a+1/(-I-a)*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/(-I-a))^(1/2))+I*(1-I*x)^(1/2)*(-1/(-I-a)*a+1/(-I-a)*x)^(1/2)*(1+I*x)
^(1/2)/(-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/(-I-a))^(1/2))+I*(1-I*x)^(1/2)*(-1/(-I-a)*a+1/(-I-a)*x)^(1/2)*(1+I*x)^(1/2)/(-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/(-I
-a))^(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((-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 = 0.35, size = 546, normalized size = 8.27 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

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]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {a-x+\sqrt {a^2+1}}{\sqrt {-\left (x^2+1\right )\,\left (a-x\right )}\,\left (x-a+\sqrt {a^2+1}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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