3.5.15 \(\int \text {ArcSin}(1+x^2)^2 \, dx\) [415]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4898, 8} \begin {gather*} x \text {ArcSin}\left (x^2+1\right )^2+\frac {4 \sqrt {-x^4-2 x^2} \text {ArcSin}\left (x^2+1\right )}{x}-8 x \end {gather*}

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 8

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

Rule 4898

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]

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 = 0.01, size = 40, normalized size = 1.00 \begin {gather*} -8 x+\frac {4 \sqrt {-2 x^2-x^4} \text {ArcSin}\left (1+x^2\right )}{x}+x \text {ArcSin}\left (1+x^2\right )^2 \end {gather*}

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

________________________________________________________________________________________

Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \arcsin \left (x^{2}+1\right )^{2}\, dx\]

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)

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found sqrt((-_SAGE_VAR_x^2)-2)

________________________________________________________________________________________

Fricas [A]
time = 3.18, size = 39, normalized size = 0.98 \begin {gather*} 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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asin}^{2}{\left (x^{2} + 1 \right )}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {asin}\left (x^2+1\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(x^2 + 1)^2,x)

[Out]

int(asin(x^2 + 1)^2, x)

________________________________________________________________________________________