3.448 \(\int \frac{(-\sqrt{1-x}-\sqrt{1+x}) (\sqrt{1-x}+\sqrt{1+x})}{x} \, dx\)

Optimal. Leaf size=32 \[ -2 \sqrt{1-x^2}+2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \log (x) \]

[Out]

-2*Sqrt[1 - x^2] + 2*ArcTanh[Sqrt[1 - x^2]] - 2*Log[x]

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Rubi [A]  time = 0.195786, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6688, 6742, 266, 50, 63, 206} \[ -2 \sqrt{1-x^2}+2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]))/x,x]

[Out]

-2*Sqrt[1 - x^2] + 2*ArcTanh[Sqrt[1 - x^2]] - 2*Log[x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0295521, size = 32, normalized size = 1. \[ -2 \sqrt{1-x^2}+2 \tanh ^{-1}\left (\sqrt{1-x^2}\right )-2 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*Sqrt[1 - x^2] + 2*ArcTanh[Sqrt[1 - x^2]] - 2*Log[x]

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Maple [A]  time = 0.002, size = 51, normalized size = 1.6 \begin{align*} -2\,\ln \left ( x \right ) -2\,{\frac{\sqrt{1-x}\sqrt{1+x} \left ( \sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) \right ) }{\sqrt{-{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46965, size = 55, normalized size = 1.72 \begin{align*} -2 \, \sqrt{-x^{2} + 1} - 2 \, \log \left (x\right ) + 2 \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.919679, size = 111, normalized size = 3.47 \begin{align*} -2 \, \sqrt{x + 1} \sqrt{-x + 1} - 2 \, \log \left (x\right ) - 2 \, \log \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{2}{x}\, dx - \int \frac{2 \sqrt{1 - x} \sqrt{x + 1}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(2/x, x) - Integral(2*sqrt(1 - x)*sqrt(x + 1)/x, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError