3.848 \(\int \frac{1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{1}{\sqrt{x-1} \sqrt{x+1}}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

[Out]

-(1/(Sqrt[-1 + x]*Sqrt[1 + x])) - ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

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Rubi [A]  time = 0.0051011, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {104, 92, 203} \[ -\frac{1}{\sqrt{x-1} \sqrt{x+1}}-\tan ^{-1}\left (\sqrt{x-1} \sqrt{x+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((-1 + x)^(3/2)*x*(1 + x)^(3/2)),x]

[Out]

-(1/(Sqrt[-1 + x]*Sqrt[1 + x])) - ArcTan[Sqrt[-1 + x]*Sqrt[1 + x]]

Rule 104

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

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(-1+x)^{3/2} x (1+x)^{3/2}} \, dx &=-\frac{1}{\sqrt{-1+x} \sqrt{1+x}}-\int \frac{1}{\sqrt{-1+x} x \sqrt{1+x}} \, dx\\ &=-\frac{1}{\sqrt{-1+x} \sqrt{1+x}}-\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+x} \sqrt{1+x}\right )\\ &=-\frac{1}{\sqrt{-1+x} \sqrt{1+x}}-\tan ^{-1}\left (\sqrt{-1+x} \sqrt{1+x}\right )\\ \end{align*}

Mathematica [C]  time = 0.010061, size = 31, normalized size = 0.89 \[ -\frac{\, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};1-x^2\right )}{\sqrt{x-1} \sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((-1 + x)^(3/2)*x*(1 + x)^(3/2)),x]

[Out]

-(Hypergeometric2F1[-1/2, 1, 1/2, 1 - x^2]/(Sqrt[-1 + x]*Sqrt[1 + x]))

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Maple [A]  time = 0.017, size = 51, normalized size = 1.5 \begin{align*}{ \left ( \arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ){x}^{2}-\arctan \left ({\frac{1}{\sqrt{{x}^{2}-1}}} \right ) -\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}{\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{-1+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-1+x)^(3/2)/x/(1+x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.74001, size = 20, normalized size = 0.57 \begin{align*} -\frac{1}{\sqrt{x^{2} - 1}} + \arcsin \left (\frac{1}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(3/2)/x/(1+x)^(3/2),x, algorithm="maxima")

[Out]

-1/sqrt(x^2 - 1) + arcsin(1/abs(x))

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Fricas [A]  time = 1.53882, size = 119, normalized size = 3.4 \begin{align*} -\frac{2 \,{\left (x^{2} - 1\right )} \arctan \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) + \sqrt{x + 1} \sqrt{x - 1}}{x^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(3/2)/x/(1+x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 13.8836, size = 58, normalized size = 1.66 \begin{align*} - \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & 1, 2, \frac{5}{2} \\\frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, 1 & \\\frac{3}{4}, \frac{5}{4} & 0, \frac{1}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)**(3/2)/x/(1+x)**(3/2),x)

[Out]

-meijerg(((5/4, 7/4, 1), (1, 2, 5/2)), ((5/4, 3/2, 7/4, 2, 5/2), (0,)), x**(-2))/(2*pi**(3/2)) - I*meijerg(((0
, 1/2, 3/4, 1, 5/4, 1), ()), ((3/4, 5/4), (0, 1/2, 3/2, 0)), exp_polar(2*I*pi)/x**2)/(2*pi**(3/2))

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Giac [B]  time = 1.41745, size = 73, normalized size = 2.09 \begin{align*} -\frac{\sqrt{x + 1}}{2 \, \sqrt{x - 1}} + \frac{2}{{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2} + 2} + 2 \, \arctan \left (\frac{1}{2} \,{\left (\sqrt{x + 1} - \sqrt{x - 1}\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-1+x)^(3/2)/x/(1+x)^(3/2),x, algorithm="giac")

[Out]

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