3.836 \(\int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx\)

Optimal. Leaf size=26 \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcCosh[x]/2

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Rubi [A]  time = 0.0032451, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {90, 52} \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\frac{1}{2} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[-1 + x]*Sqrt[1 + x]),x]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcCosh[x]/2

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{-1+x} \sqrt{1+x}} \, dx &=\frac{1}{2} \sqrt{-1+x} x \sqrt{1+x}+\frac{1}{2} \int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx\\ &=\frac{1}{2} \sqrt{-1+x} x \sqrt{1+x}+\frac{1}{2} \cosh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0098209, size = 36, normalized size = 1.38 \[ \frac{1}{2} \sqrt{x-1} \sqrt{x+1} x+\tanh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[-1 + x]*Sqrt[1 + x]),x]

[Out]

(Sqrt[-1 + x]*x*Sqrt[1 + x])/2 + ArcTanh[Sqrt[-1 + x]/Sqrt[1 + x]]

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Maple [B]  time = 0.012, size = 40, normalized size = 1.5 \begin{align*}{\frac{1}{2}\sqrt{-1+x}\sqrt{1+x} \left ( x\sqrt{{x}^{2}-1}+\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(-1+x)^(1/2)*(1+x)^(1/2)*(x*(x^2-1)^(1/2)+ln(x+(x^2-1)^(1/2)))/(x^2-1)^(1/2)

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Maxima [A]  time = 1.35675, size = 36, normalized size = 1.38 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} - 1} x + \frac{1}{2} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 - 1)*x + 1/2*log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 1.54204, size = 95, normalized size = 3.65 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x - \frac{1}{2} \, \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(x + 1)*sqrt(x - 1)*x - 1/2*log(sqrt(x + 1)*sqrt(x - 1) - x)

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Sympy [C]  time = 5.42951, size = 87, normalized size = 3.35 \begin{align*} \frac{{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(-1+x)**(1/2)/(1+x)**(1/2),x)

[Out]

meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4, 0, 0), ()), x**(-2))/(4*pi**(3/2)) - I*mei
jerg(((-3/2, -5/4, -1, -3/4, -1/2, 1), ()), ((-5/4, -3/4), (-3/2, -1, -1, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**
(3/2))

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Giac [A]  time = 1.94229, size = 42, normalized size = 1.62 \begin{align*} \frac{1}{2} \, \sqrt{x + 1} \sqrt{x - 1} x - \log \left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(-1+x)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(x + 1)*sqrt(x - 1)*x - log(abs(-sqrt(x + 1) + sqrt(x - 1)))