3.415 \(\int \sin ^{-1}(1+x^2)^2 \, dx\)

Optimal. Leaf size=40 \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]

[Out]

-8*x + (4*Sqrt[-2*x^2 - x^4]*ArcSin[1 + x^2])/x + x*ArcSin[1 + x^2]^2

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Rubi [A]  time = 0.0062437, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4814, 8} \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[1 + x^2]^2,x]

[Out]

-8*x + (4*Sqrt[-2*x^2 - x^4]*ArcSin[1 + x^2])/x + x*ArcSin[1 + x^2]^2

Rule 4814

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)^2]*(b_.))^(n_), x_Symbol] :> Simp[x*(a + b*ArcSin[c + d*x^2])^n, x] + (-
Dist[4*b^2*n*(n - 1), Int[(a + b*ArcSin[c + d*x^2])^(n - 2), x], x] + Simp[(2*b*n*Sqrt[-2*c*d*x^2 - d^2*x^4]*(
a + b*ArcSin[c + d*x^2])^(n - 1))/(d*x), x]) /; FreeQ[{a, b, c, d}, x] && EqQ[c^2, 1] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \sin ^{-1}\left (1+x^2\right )^2 \, dx &=\frac{4 \sqrt{-2 x^2-x^4} \sin ^{-1}\left (1+x^2\right )}{x}+x \sin ^{-1}\left (1+x^2\right )^2-8 \int 1 \, dx\\ &=-8 x+\frac{4 \sqrt{-2 x^2-x^4} \sin ^{-1}\left (1+x^2\right )}{x}+x \sin ^{-1}\left (1+x^2\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0132387, size = 40, normalized size = 1. \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[1 + x^2]^2,x]

[Out]

-8*x + (4*Sqrt[-2*x^2 - x^4]*ArcSin[1 + x^2])/x + x*ArcSin[1 + x^2]^2

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \begin{align*} \int \left ( \arcsin \left ({x}^{2}+1 \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.26228, size = 84, normalized size = 2.1 \begin{align*} x \arctan \left (\frac{\sqrt{-x^{4} - 2 \, x^{2}}{\left (x^{2} + 1\right )}}{x^{4} + 2 \, x^{2}}\right )^{2} - 8 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arctan(sqrt(-x^4 - 2*x^2)*(x^2 + 1)/(x^4 + 2*x^2))^2 - 8*x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(x**2 + 1)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsin(x^2 + 1)^2, x)