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

Optimal. Leaf size=28 \[ \sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.320949, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 7, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.18, Rules used = {2103, 6688, 14, 266, 50, 63, 206} \[ \sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2103

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

Rule 6688

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[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{\sqrt{1-x}+\sqrt{1+x}}{-\sqrt{1-x}+\sqrt{1+x}} \, dx &=\frac{1}{2} \int \frac{\sqrt{1-x} \left (\sqrt{1-x}+\sqrt{1+x}\right )}{x} \, dx+\frac{1}{2} \int \frac{\sqrt{1+x} \left (\sqrt{1-x}+\sqrt{1+x}\right )}{x} \, dx\\ &=\frac{1}{2} \int \frac{1-x+\sqrt{1-x^2}}{x} \, dx+\frac{1}{2} \int \frac{1+x+\sqrt{1-x^2}}{x} \, dx\\ &=\frac{1}{2} \int \left (-1+\frac{1}{x}+\frac{\sqrt{1-x^2}}{x}\right ) \, dx+\frac{1}{2} \int \left (1+\frac{1}{x}+\frac{\sqrt{1-x^2}}{x}\right ) \, dx\\ &=\log (x)+2 \left (\frac{1}{2} \int \frac{\sqrt{1-x^2}}{x} \, dx\right )\\ &=\log (x)+2 \left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x} \, dx,x,x^2\right )\right )\\ &=\log (x)+2 \left (\frac{\sqrt{1-x^2}}{2}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,x^2\right )\right )\\ &=\log (x)+2 \left (\frac{\sqrt{1-x^2}}{2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-x^2}\right )\right )\\ &=2 \left (\frac{\sqrt{1-x^2}}{2}-\frac{1}{2} \tanh ^{-1}\left (\sqrt{1-x^2}\right )\right )+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.158765, size = 48, normalized size = 1.71 \[ \sqrt{1-x^2}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )+\log (x)+2 \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )+\sin ^{-1}(x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 48, normalized size = 1.7 \begin{align*} \ln \left ( x \right ) +{\sqrt{1-x}\sqrt{1+x} \left ( \sqrt{-{x}^{2}+1}-{\it Artanh} \left ({\frac{1}{\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(((1-x)^(1/2)+(1+x)^(1/2))/(-(1-x)^(1/2)+(1+x)^(1/2)),x)

[Out]

ln(x)+(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)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + 1} + \sqrt{-x + 1}}{\sqrt{x + 1} - \sqrt{-x + 1}}\,{d x} \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, algorithm="maxima")

[Out]

integrate((sqrt(x + 1) + sqrt(-x + 1))/(sqrt(x + 1) - sqrt(-x + 1)), x)

________________________________________________________________________________________

Fricas [A]  time = 0.923263, size = 101, normalized size = 3.61 \begin{align*} \sqrt{x + 1} \sqrt{-x + 1} + \log \left (x\right ) + \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, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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, algorithm="giac")

[Out]

Exception raised: NotImplementedError