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

Optimal. Leaf size=45 \[ -\log \left (\sqrt{1-x^2}+1\right )-\frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}+1}+\frac{1}{2} \sin ^{-1}(x)^2 \]

[Out]

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

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Rubi [A]  time = 0.12133, antiderivative size = 51, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {6742, 277, 216, 4791, 4627, 266, 63, 206, 4693, 29, 4641} \[ \frac{\sqrt{1-x^2} \sin ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt{1-x^2}\right )-\log (x)+\frac{1}{2} \sin ^{-1}(x)^2-\frac{\sin ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[x]/(1 + Sqrt[1 - x^2]),x]

[Out]

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

Rule 6742

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 216

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

Rule 4791

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(Px_.)*((f_) + (g_.)*((d_) + (e_.)*(x_)^2)^(p_))^(m_.), x_Symbol]
:> With[{u = ExpandIntegrand[Px*(f + g*(d + e*x^2)^p)^m*(a + b*ArcSin[c*x])^n, x]}, Int[u, x] /; SumQ[u]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && PolynomialQ[Px, x] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && IntegersQ[m,
n]

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 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])

Rule 4693

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 1)), x] + (-Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m + 1
)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x] + Dist[(c^2*Sqrt[d + e*x^2])/(f^2*
(m + 1)*Sqrt[1 - c^2*x^2]), Int[((f*x)^(m + 2)*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x]) /; FreeQ[{a,
b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rubi steps

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

Mathematica [A]  time = 0.0415444, size = 44, normalized size = 0.98 \[ -\log \left (\sqrt{1-x^2}+1\right )+\frac{\left (\sqrt{1-x^2}-1\right ) \sin ^{-1}(x)}{x}+\frac{1}{2} \sin ^{-1}(x)^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int{\arcsin \left ( x \right ) \left ( 1+\sqrt{-{x}^{2}+1} \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (x\right )}{\sqrt{-x^{2} + 1} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(x)/(sqrt(-x^2 + 1) + 1), x)

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Fricas [A]  time = 2.80343, size = 182, normalized size = 4.04 \begin{align*} \frac{x \arcsin \left (x\right )^{2} - 2 \, x \log \left (x\right ) - x \log \left (\sqrt{-x^{2} + 1} + 1\right ) + x \log \left (\sqrt{-x^{2} + 1} - 1\right ) + 2 \, \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - 2 \, \arcsin \left (x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}{\left (x \right )}}{\sqrt{1 - x^{2}} + 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14176, size = 77, normalized size = 1.71 \begin{align*} \frac{1}{2} \, \arcsin \left (x\right )^{2} - \frac{x \arcsin \left (x\right )}{\sqrt{-x^{2} + 1} + 1} - 2 \, \log \left (2\right ) + \log \left (2 \, \sqrt{-x^{2} + 1} + 2\right ) - 2 \, \log \left (\sqrt{-x^{2} + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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