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

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 = 2.37246, 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 \[ \frac{4 \sqrt{a^2+1} \sqrt{x^2+1} \sqrt{\frac{a-x}{a+i}} \Pi \left (\frac{2}{1-i \left (a-\sqrt{a^2+1}\right )};\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\left (1-i \left (a-\sqrt{a^2+1}\right )\right ) \sqrt{\left (x^2+1\right ) (-(a-x))}}+\frac{2 i \sqrt{x^2+1} \sqrt{\frac{a-x}{a+i}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2}{1-i a}\right )}{\sqrt{\left (x^2+1\right ) (-(a-x))}} \]

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)
)])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [C]  time = 0.56329, size = 213, normalized size = 3.23 \[ \frac{2 \sqrt{\frac{a-x}{a+i}} \left (2 i \sqrt{a^2+1} \sqrt{1-i x} \sqrt{x^2+1} \Pi \left (\frac{2 i}{a-\sqrt{a^2+1}+i};\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2 i}{a+i}\right )-\left (\sqrt{a^2+1}-a-i\right ) \sqrt{1+i x} (x+i) F\left (\sin ^{-1}\left (\frac{\sqrt{1-i x}}{\sqrt{2}}\right )|\frac{2 i}{a+i}\right )\right )}{\left (-\sqrt{a^2+1}+a+i\right ) \sqrt{1-i x} \sqrt{\left (x^2+1\right ) (x-a)}} \]

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)*Ellip
ticF[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 = 0.168, size = 1275, normalized size = 19.3 \[ \text{result too large to display} \]

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)*(-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))*Elliptic
Pi(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))*Elliptic
Pi(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))+(-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)*Ell
ipticPi(((-a+x)/(-a-I))^(1/2),(a+I)/(a^2+1)^(1/2),((a+I)/(a-I))^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a - x + sqrt(a^2 + 1))/(sqrt(-(x^2 + 1)*(a - x))*(a - x - sqrt(a^2 + 1))),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 = 0.616919, size = 1, normalized size = 0.02 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*sqrt(-2*a - 2*sqrt(a^2 + 1))*log((256*a^6 + (32*a^6 + 48*a^4 + 18*a^2 + 1)*
x^4 + 320*a^4 - 12*(16*a^5 + 20*a^3 + 5*a)*x^3 + 2*(128*a^6 + 184*a^4 + 64*a^2 +
3)*x^2 + 82*a^2 + 4*(32*a^5 + 36*a^3 + (16*a^5 + 20*a^3 + 5*a)*x^2 - 2*(8*a^4 +
8*a^2 + 1)*x + (32*a^4 + (16*a^4 + 12*a^2 + 1)*x^2 + 20*a^2 - 8*(2*a^3 + a)*x +
1)*sqrt(a^2 + 1) + 7*a)*sqrt(-a*x^2 + x^3 - a + x)*sqrt(-2*a - 2*sqrt(a^2 + 1))
- 4*(64*a^5 + 76*a^3 + 17*a)*x + 2*(128*a^5 + (16*a^5 + 16*a^3 + 3*a)*x^4 - 6*(
16*a^4 + 12*a^2 + 1)*x^3 + 96*a^3 + 4*(32*a^5 + 30*a^3 + 5*a)*x^2 - 2*(64*a^4 +
44*a^2 + 3)*x + 9*a)*sqrt(a^2 + 1) + 1)/((32*a^6 + 48*a^4 + 18*a^2 + 1)*x^4 + 4*
(16*a^5 + 20*a^3 + 5*a)*x^3 + 6*(8*a^4 + 8*a^2 + 1)*x^2 + 2*a^2 + 4*(4*a^3 + 3*a
)*x + 2*((16*a^5 + 16*a^3 + 3*a)*x^4 + 2*(16*a^4 + 12*a^2 + 1)*x^3 + 12*(2*a^3 +
a)*x^2 + 2*(4*a^2 + 1)*x + a)*sqrt(a^2 + 1) + 1)), 1/2*sqrt(2*a + 2*sqrt(a^2 +
1))*arctan(1/2*(8*a^3 + (4*a^3 + 3*a)*x^2 - 2*(2*a^2 + 1)*x + ((4*a^2 + 1)*x^2 +
8*a^2 - 4*a*x + 1)*sqrt(a^2 + 1) + 5*a)/(sqrt(-a*x^2 + x^3 - a + x)*(2*a^2 + 2*
sqrt(a^2 + 1)*a + 1)*sqrt(2*a + 2*sqrt(a^2 + 1))))]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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