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

Optimal. Leaf size=20 \[ -\frac{\left (x-\sqrt{a+x^2}\right )^n}{n} \]

[Out]

-((x - Sqrt[a + x^2])^n/n)

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Rubi [A]  time = 0.094739, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{\left (x-\sqrt{a+x^2}\right )^n}{n} \]

Antiderivative was successfully verified.

[In]  Int[(x - Sqrt[a + x^2])^n/Sqrt[a + x^2],x]

[Out]

-((x - Sqrt[a + x^2])^n/n)

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Rubi in Sympy [A]  time = 9.91696, size = 14, normalized size = 0.7 \[ - \frac{\left (x - \sqrt{a + x^{2}}\right )^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x-(x**2+a)**(1/2))**n/(x**2+a)**(1/2),x)

[Out]

-(x - sqrt(a + x**2))**n/n

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Mathematica [A]  time = 0.0308736, size = 20, normalized size = 1. \[ -\frac{\left (x-\sqrt{a+x^2}\right )^n}{n} \]

Antiderivative was successfully verified.

[In]  Integrate[(x - Sqrt[a + x^2])^n/Sqrt[a + x^2],x]

[Out]

-((x - Sqrt[a + x^2])^n/n)

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{1 \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}{\frac{1}{\sqrt{{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x-(x^2+a)^(1/2))^n/(x^2+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.738753, size = 24, normalized size = 1.2 \[ -\frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - sqrt(x^2 + a))^n/sqrt(x^2 + a),x, algorithm="maxima")

[Out]

-(x - sqrt(x^2 + a))^n/n

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Fricas [A]  time = 0.312866, size = 24, normalized size = 1.2 \[ -\frac{{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x - sqrt(x^2 + a))^n/n

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Sympy [A]  time = 10.3882, size = 36, normalized size = 1.8 \[ \begin{cases} - \frac{\left (x - \sqrt{a + x^{2}}\right )^{n}}{n} & \text{for}\: n \neq 0 \\\begin{cases} \operatorname{asinh}{\left (x \sqrt{\frac{1}{a}} \right )} & \text{for}\: a > 0 \\\operatorname{acosh}{\left (x \sqrt{- \frac{1}{a}} \right )} & \text{for}\: a < 0 \end{cases} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x-(x**2+a)**(1/2))**n/(x**2+a)**(1/2),x)

[Out]

Piecewise((-(x - sqrt(a + x**2))**n/n, Ne(n, 0)), (Piecewise((asinh(x*sqrt(1/a))
, a > 0), (acosh(x*sqrt(-1/a)), a < 0)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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