Optimal. Leaf size=56 \[ \frac{\left (x-\sqrt{a+x^2}\right )^{b+1}}{2 (b+1)}-\frac{a \left (x-\sqrt{a+x^2}\right )^{b-1}}{2 (1-b)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0458865, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (x-\sqrt{a+x^2}\right )^{b+1}}{2 (b+1)}-\frac{a \left (x-\sqrt{a+x^2}\right )^{b-1}}{2 (1-b)} \]
Antiderivative was successfully verified.
[In] Int[(x - Sqrt[a + x^2])^b,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x - \sqrt{a + x^{2}}\right )^{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x-(x**2+a)**(1/2))**b,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.03978, size = 39, normalized size = 0.7 \[ \frac{\left (x-\sqrt{a+x^2}\right )^b \left (b \left (-\sqrt{a+x^2}\right )-x\right )}{b^2-1} \]
Antiderivative was successfully verified.
[In] Integrate[(x - Sqrt[a + x^2])^b,x]
[Out]
_______________________________________________________________________________________
Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int \left ( x-\sqrt{{x}^{2}+a} \right ) ^{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x-(x^2+a)^(1/2))^b,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x - \sqrt{x^{2} + a}\right )}^{b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^b,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.229939, size = 45, normalized size = 0.8 \[ -\frac{{\left (\sqrt{x^{2} + a} b + x\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{b}}{b^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^b,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x - \sqrt{a + x^{2}}\right )^{b}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x-(x**2+a)**(1/2))**b,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x - \sqrt{x^{2} + a}\right )}^{b}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^b,x, algorithm="giac")
[Out]