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

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

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Rubi [A]  time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{2} \sqrt {x-1} \sqrt {x+1} x+\frac {1}{2} \cosh ^{-1}(x) \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.38 \begin {gather*} \frac {1}{2} \sqrt {x-1} \sqrt {x+1} x+\tanh ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {x+1}}\right ) \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.07, size = 62, normalized size = 2.38 \begin {gather*} \frac {\frac {(x-1)^{3/2}}{(x+1)^{3/2}}+\frac {\sqrt {x-1}}{\sqrt {x+1}}}{\left (\frac {x-1}{x+1}-1\right )^2}+\tanh ^{-1}\left (\frac {\sqrt {x-1}}{\sqrt {x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-1 + x)^(3/2)/(1 + x)^(3/2) + Sqrt[-1 + x]/Sqrt[1 + x])/(-1 + (-1 + x)/(1 + x))^2 + ArcTanh[Sqrt[-1 + x]/Sqr
t[1 + x]]

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fricas [A]  time = 1.36, size = 32, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x - \frac {1}{2} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \end {gather*}

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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giac [A]  time = 1.18, size = 30, normalized size = 1.15 \begin {gather*} \frac {1}{2} \, \sqrt {x + 1} \sqrt {x - 1} x - \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \end {gather*}

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(sqrt(x + 1) - sqrt(x - 1))

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maple [B]  time = 0.02, size = 40, normalized size = 1.54 \begin {gather*} \frac {\sqrt {x -1}\, \sqrt {x +1}\, \left (\sqrt {x^{2}-1}\, x +\ln \left (x +\sqrt {x^{2}-1}\right )\right )}{2 \sqrt {x^{2}-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x-1)^(1/2)*(x+1)^(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 = 0.60, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{2} \, \sqrt {x^{2} - 1} x + \frac {1}{2} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \end {gather*}

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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mupad [B]  time = 7.51, size = 194, normalized size = 7.46 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )-\frac {\frac {14\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {14\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {x+1}-1\right )}^5}+\frac {2\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {x+1}-1\right )}^7}+\frac {2\,\left (\sqrt {x-1}-\mathrm {i}\right )}{\sqrt {x+1}-1}}{1+\frac {6\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh(((x - 1)^(1/2) - 1i)/((x + 1)^(1/2) - 1)) - ((14*((x - 1)^(1/2) - 1i)^3)/((x + 1)^(1/2) - 1)^3 + (14*(
(x - 1)^(1/2) - 1i)^5)/((x + 1)^(1/2) - 1)^5 + (2*((x - 1)^(1/2) - 1i)^7)/((x + 1)^(1/2) - 1)^7 + (2*((x - 1)^
(1/2) - 1i))/((x + 1)^(1/2) - 1))/((6*((x - 1)^(1/2) - 1i)^4)/((x + 1)^(1/2) - 1)^4 - (4*((x - 1)^(1/2) - 1i)^
2)/((x + 1)^(1/2) - 1)^2 - (4*((x - 1)^(1/2) - 1i)^6)/((x + 1)^(1/2) - 1)^6 + ((x - 1)^(1/2) - 1i)^8/((x + 1)^
(1/2) - 1)^8 + 1)

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sympy [C]  time = 24.12, size = 87, normalized size = 3.35 \begin {gather*} \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 {gather*}

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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