Integrand size = 48, antiderivative size = 66 \[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=-\sqrt {2} \sqrt {a+\sqrt {1+a^2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} (-a+x)}{\sqrt {(-a+x) \left (1+x^2\right )}}\right ) \]
-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)
Time = 2.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.50 \[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=-\frac {\sqrt {2} \sqrt {-a+x} \sqrt {1+x^2} \arctan \left (\frac {\sqrt {2} \sqrt {-a+\sqrt {1+a^2}} \sqrt {-a+x}}{\sqrt {1+x^2}}\right )}{\sqrt {-a+\sqrt {1+a^2}} \sqrt {(-a+x) \left (1+x^2\right )}} \]
-((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)]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.22 (sec) , antiderivative size = 429, normalized size of antiderivative = 6.50, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {7270, 2349, 510, 729, 25, 1416, 1534, 1416, 2212, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-\sqrt {a^2+1}-a+x}{\left (\sqrt {a^2+1}-a+x\right ) \sqrt {\left (x^2+1\right ) (x-a)}} \, dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \int \frac {a-x+\sqrt {a^2+1}}{\left (a-x-\sqrt {a^2+1}\right ) \sqrt {x-a} \sqrt {x^2+1}}dx}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 2349 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (2 \sqrt {a^2+1} \int \frac {1}{\left (a-x-\sqrt {a^2+1}\right ) \sqrt {x-a} \sqrt {x^2+1}}dx+\int \frac {1}{\sqrt {x-a} \sqrt {x^2+1}}dx\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 510 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (2 \sqrt {a^2+1} \int \frac {1}{\left (a-x-\sqrt {a^2+1}\right ) \sqrt {x-a} \sqrt {x^2+1}}dx+2 \int \frac {1}{\sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 729 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (2 \int \frac {1}{\sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}+4 \sqrt {a^2+1} \int -\frac {1}{\left (-a+x+\sqrt {a^2+1}\right ) \sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (2 \int \frac {1}{\sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}-4 \sqrt {a^2+1} \int \frac {1}{\left (-a+x+\sqrt {a^2+1}\right ) \sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (\frac {\sqrt [4]{a^2+1} \left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}-4 \sqrt {a^2+1} \int \frac {1}{\left (-a+x+\sqrt {a^2+1}\right ) \sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 1534 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (4 \sqrt {a^2+1} \left (-\frac {\int \frac {1}{\sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}}{2 \sqrt {a^2+1}}-\frac {\int \frac {a-x+\sqrt {a^2+1}}{\left (-a+x+\sqrt {a^2+1}\right ) \sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}}{2 \sqrt {a^2+1}}\right )+\frac {\sqrt [4]{a^2+1} \left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (4 \sqrt {a^2+1} \left (-\frac {\int \frac {a-x+\sqrt {a^2+1}}{\left (-a+x+\sqrt {a^2+1}\right ) \sqrt {a^2+2 (x-a) a+(x-a)^2+1}}d\sqrt {x-a}}{2 \sqrt {a^2+1}}-\frac {\left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{4 \sqrt [4]{a^2+1} \sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )+\frac {\sqrt [4]{a^2+1} \left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 2212 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (4 \sqrt {a^2+1} \left (-\frac {1}{2} \int \frac {1}{\sqrt {a^2+1}-2 \sqrt {a^2+1} \left (a-\sqrt {a^2+1}\right ) (x-a)}d\frac {\sqrt {x-a}}{\sqrt {a^2+2 (x-a) a+(x-a)^2+1}}-\frac {\left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{4 \sqrt [4]{a^2+1} \sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )+\frac {\sqrt [4]{a^2+1} \left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {x^2+1} \sqrt {x-a} \left (4 \sqrt {a^2+1} \left (-\frac {\left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{4 \sqrt [4]{a^2+1} \sqrt {a^2+2 a (x-a)+(x-a)^2+1}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {a-\sqrt {a^2+1}} \sqrt {x-a}}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )}{2 \sqrt {2} \sqrt {a^2+1} \sqrt {a-\sqrt {a^2+1}}}\right )+\frac {\sqrt [4]{a^2+1} \left (\frac {x-a}{\sqrt {a^2+1}}+1\right ) \sqrt {\frac {a^2+2 a (x-a)+(x-a)^2+1}{\left (a^2+1\right ) \left (\frac {x-a}{\sqrt {a^2+1}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {x-a}}{\sqrt [4]{a^2+1}}\right ),\frac {1}{2} \left (1-\frac {a}{\sqrt {a^2+1}}\right )\right )}{\sqrt {a^2+2 a (x-a)+(x-a)^2+1}}\right )}{\sqrt {-\left (\left (x^2+1\right ) (a-x)\right )}}\) |
(Sqrt[-a + x]*Sqrt[1 + x^2]*(((1 + a^2)^(1/4)*(1 + (-a + x)/Sqrt[1 + a^2]) *Sqrt[(1 + a^2 + 2*a*(-a + x) + (-a + x)^2)/((1 + a^2)*(1 + (-a + x)/Sqrt[ 1 + a^2])^2)]*EllipticF[2*ArcTan[Sqrt[-a + x]/(1 + a^2)^(1/4)], (1 - a/Sqr t[1 + a^2])/2])/Sqrt[1 + a^2 + 2*a*(-a + x) + (-a + x)^2] + 4*Sqrt[1 + a^2 ]*(-1/2*ArcTanh[(Sqrt[2]*Sqrt[a - Sqrt[1 + a^2]]*Sqrt[-a + x])/Sqrt[1 + a^ 2 + 2*a*(-a + x) + (-a + x)^2]]/(Sqrt[2]*Sqrt[1 + a^2]*Sqrt[a - Sqrt[1 + a ^2]]) - ((1 + (-a + x)/Sqrt[1 + a^2])*Sqrt[(1 + a^2 + 2*a*(-a + x) + (-a + x)^2)/((1 + a^2)*(1 + (-a + x)/Sqrt[1 + a^2])^2)]*EllipticF[2*ArcTan[Sqrt [-a + x]/(1 + a^2)^(1/4)], (1 - a/Sqrt[1 + a^2])/2])/(4*(1 + a^2)^(1/4)*Sq rt[1 + a^2 + 2*a*(-a + x) + (-a + x)^2]))))/Sqrt[-((a - x)*(1 + x^2))]
3.1.52.3.1 Defintions of rubi rules used
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]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[2/d Subst[Int[1/Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2 )], x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) ^2]), x_Symbol] :> Simp[2 Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> Simp[1/(2*d) Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[1/(2* d) Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; FreeQ [{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0 ] && EqQ[c*d^2 - a*e^2, 0]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Simp[A Subst[Int[1/(d - (b*d - 2*a*e)*x^2), x], x, x/Sqrt[a + b*x^2 + c*x^4]], x] /; FreeQ[{a, b, c, d, e, A, B}, x] & & EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d *x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] && !IntegerQ[n ] && IntegersQ[2*m, 2*n, 2*p]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 1.60 (sec) , antiderivative size = 787, normalized size of antiderivative = 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 )}\, F\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}\, \left (2 a x -x^{2}+1\right ) \sqrt {-\left (a -x \right ) \left (x^{2}+1\right ) \left (a^{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}\, \Pi \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}\, \Pi \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}\, \Pi \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}\, \Pi \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 | \(\text {Expression too large to display}\) | \(1463\) |
int((-a+x-(a^2+1)^(1/2))/(-a+x+(a^2+1)^(1/2))/((-a+x)*(x^2+1))^(1/2),x,met hod=_RETURNVERBOSE)
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)*(2*a*x-x^2+1)*(-(a-x)*(x^2+1)*(a^2+1))^(1/2)/(-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))*Ellipti cPi(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)))
Time = 0.30 (sec) , antiderivative size = 546, normalized size of antiderivative = 8.27 \[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=\left [\frac {1}{4} \, \sqrt {-2 \, a - 2 \, \sqrt {a^{2} + 1}} \log \left (-\frac {8 \, a x^{7} + x^{8} + 4 \, {\left (2 \, a^{2} + 15\right )} x^{6} - 8 \, {\left (4 \, a^{3} + 15 \, a\right )} x^{5} + 2 \, {\left (8 \, a^{4} + 80 \, a^{2} + 67\right )} x^{4} + 64 \, a^{4} - 8 \, {\left (20 \, a^{3} + 37 \, a\right )} x^{3} + 4 \, {\left (16 \, a^{4} + 74 \, a^{2} + 15\right )} x^{2} + 48 \, a^{2} - 4 \, {\left (a x^{6} + 2 \, {\left (2 \, a^{2} + 3\right )} x^{5} - {\left (4 \, a^{3} - a\right )} x^{4} - 8 \, a^{3} - {\left (4 \, a^{3} + 29 \, a\right )} x^{2} + 20 \, x^{3} + 2 \, {\left (10 \, a^{2} + 3\right )} x - {\left (4 \, a x^{5} + x^{6} - {\left (4 \, a^{2} - 15\right )} x^{4} - 16 \, a x^{3} + {\left (4 \, a^{2} + 15\right )} x^{2} + 8 \, a^{2} - 20 \, a x + 1\right )} \sqrt {a^{2} + 1} - 5 \, a\right )} \sqrt {-a x^{2} + x^{3} - a + x} \sqrt {-2 \, a - 2 \, \sqrt {a^{2} + 1}} - 8 \, {\left (24 \, a^{3} + 13 \, a\right )} x + 16 \, {\left (a x^{6} - x^{7} + 15 \, a x^{4} - 7 \, x^{5} - {\left (12 \, a^{2} + 7\right )} x^{3} + 4 \, a^{3} + {\left (4 \, a^{3} + 15 \, a\right )} x^{2} - {\left (12 \, a^{2} + 1\right )} x + a\right )} \sqrt {a^{2} + 1} + 1}{8 \, a x^{7} - x^{8} - 4 \, {\left (6 \, a^{2} - 1\right )} x^{6} + 8 \, {\left (4 \, a^{3} - 3 \, a\right )} x^{5} - 2 \, {\left (8 \, a^{4} - 24 \, a^{2} + 3\right )} x^{4} - 8 \, {\left (4 \, a^{3} - 3 \, a\right )} x^{3} - 4 \, {\left (6 \, a^{2} - 1\right )} x^{2} - 8 \, a x - 1}\right ), -\frac {1}{2} \, \sqrt {2 \, a + 2 \, \sqrt {a^{2} + 1}} \arctan \left (-\frac {\sqrt {-a x^{2} + x^{3} - a + x} {\left (2 \, a^{2} - 2 \, a x - x^{2} - 2 \, \sqrt {a^{2} + 1} {\left (a - x\right )} - 1\right )} \sqrt {2 \, a + 2 \, \sqrt {a^{2} + 1}}}{4 \, {\left (a x^{2} - x^{3} + a - x\right )}}\right )\right ] \]
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")
[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)*s qrt(-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*sqrt(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)*s qrt(2*a + 2*sqrt(a^2 + 1))/(a*x^2 - x^3 + a - x))]
Timed out. \[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=\text {Timed out} \]
\[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=\int { \frac {a - x + \sqrt {a^{2} + 1}}{\sqrt {-{\left (x^{2} + 1\right )} {\left (a - x\right )}} {\left (a - x - \sqrt {a^{2} + 1}\right )}} \,d x } \]
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")
\[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=\int { \frac {a - x + \sqrt {a^{2} + 1}}{\sqrt {-{\left (x^{2} + 1\right )} {\left (a - x\right )}} {\left (a - x - \sqrt {a^{2} + 1}\right )}} \,d x } \]
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")
Timed out. \[ \int \frac {-a-\sqrt {1+a^2}+x}{\left (-a+\sqrt {1+a^2}+x\right ) \sqrt {(-a+x) \left (1+x^2\right )}} \, dx=\int -\frac {a-x+\sqrt {a^2+1}}{\sqrt {-\left (x^2+1\right )\,\left (a-x\right )}\,\left (x-a+\sqrt {a^2+1}\right )} \,d x \]