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

Optimal. Leaf size=3 \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

[Out]

ArcTan[ArcSin[x]]

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Rubi [A]  time = 0.0469675, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6696, 203} \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[ArcSin[x]]

Rule 6696

Int[(u_.)*((a_.) + (b_.)*(y_)^(n_))^(p_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Dist[q, Subst[In
t[(a + b*x^n)^p, x], x, y], x] /;  !FalseQ[q]] /; FreeQ[{a, b, n, p}, x]

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

Mathematica [A]  time = 0.05518, size = 3, normalized size = 1. \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[ArcSin[x]]

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Maple [A]  time = 0.005, size = 4, normalized size = 1.3 \begin{align*} \arctan \left ( \arcsin \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(arcsin(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.15427, size = 26, normalized size = 8.67 \begin{align*} \arctan \left (\arcsin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(arcsin(x))

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Sympy [A]  time = 0.41762, size = 3, normalized size = 1. \begin{align*} \operatorname{atan}{\left (\operatorname{asin}{\left (x \right )} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(asin(x))

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Giac [A]  time = 1.06363, size = 4, normalized size = 1.33 \begin{align*} \arctan \left (\arcsin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(arcsin(x))