Optimal. Leaf size=56 \[ \frac{\left (x-\sqrt{a+x^2}\right )^{n+1}}{2 (n+1)}-\frac{a \left (x-\sqrt{a+x^2}\right )^{n-1}}{2 (1-n)} \]
[Out]
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Rubi [A] time = 0.0453643, 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 \[ \frac{\left (x-\sqrt{a+x^2}\right )^{n+1}}{2 (n+1)}-\frac{a \left (x-\sqrt{a+x^2}\right )^{n-1}}{2 (1-n)} \]
Antiderivative was successfully verified.
[In] Int[(x - Sqrt[a + x^2])^n,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x - \sqrt{a + x^{2}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x-(x**2+a)**(1/2))**n,x)
[Out]
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Mathematica [A] time = 0.0235853, size = 39, normalized size = 0.7 \[ \frac{\left (x-\sqrt{a+x^2}\right )^n \left (n \left (-\sqrt{a+x^2}\right )-x\right )}{n^2-1} \]
Antiderivative was successfully verified.
[In] Integrate[(x - Sqrt[a + x^2])^n,x]
[Out]
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Maple [F] time = 0.024, size = 0, normalized size = 0. \[ \int \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x-(x^2+a)^(1/2))^n,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.299274, size = 45, normalized size = 0.8 \[ -\frac{{\left (\sqrt{x^{2} + a} n + x\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x - \sqrt{a + x^{2}}\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x-(x**2+a)**(1/2))**n,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - sqrt(x^2 + a))^n,x, algorithm="giac")
[Out]