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 = 0.05, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 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])

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]

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*}

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Mathematica [A]  time = 0.06, size = 3, normalized size = 1.00 \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[ArcSin[x]]

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fricas [A]  time = 0.46, size = 3, normalized size = 1.00 \[ \arctan \left (\arcsin \relax (x)\right ) \]

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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giac [A]  time = 1.13, size = 3, normalized size = 1.00 \[ \arctan \left (\arcsin \relax (x)\right ) \]

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))

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maple [A]  time = 0.01, size = 4, normalized size = 1.33 \[ \arctan \left (\arcsin \relax (x )\right ) \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} + 1} {\left (\arcsin \relax (x)^{2} + 1\right )}}\,{d x} \]

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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mupad [B]  time = 3.10, size = 43, normalized size = 14.33 \[ \frac {\ln \left (\frac {-1+\mathrm {asin}\relax (x)\,1{}\mathrm {i}}{\sqrt {1-x^2}}\right )\,1{}\mathrm {i}}{2}-\frac {\ln \left (\frac {1+\mathrm {asin}\relax (x)\,1{}\mathrm {i}}{\sqrt {1-x^2}}\right )\,1{}\mathrm {i}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((asin(x)*1i - 1)/(1 - x^2)^(1/2))*1i)/2 - (log((asin(x)*1i + 1)/(1 - x^2)^(1/2))*1i)/2

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sympy [A]  time = 0.41, size = 3, normalized size = 1.00 \[ \operatorname {atan}{\left (\operatorname {asin}{\relax (x )} \right )} \]

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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