3.5.10 \(\int (a-b \text {ArcSin}(1-d x^2))^2 \, dx\) [410]

Optimal. Leaf size=67 \[ -8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2 \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4898, 8} \begin {gather*} \frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2-8 b^2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a - b*ArcSin[1 - d*x^2])^2,x]

[Out]

-8*b^2*x + (4*b*Sqrt[2*d*x^2 - d^2*x^4]*(a - b*ArcSin[1 - d*x^2]))/(d*x) + x*(a - b*ArcSin[1 - d*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 \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2 \, dx &=\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2-\left (8 b^2\right ) \int 1 \, dx\\ &=-8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 67, normalized size = 1.00 \begin {gather*} -8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}{d x}+x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a - b*ArcSin[1 - d*x^2])^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x^2-1))^2,x)

[Out]

int((a+b*arcsin(d*x^2-1))^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x^2-1))^2,x, algorithm="maxima")

[Out]

2*(x*arcsin(d*x^2 - 1) - 2*(d^(3/2)*x^2 - 2*sqrt(d))/(sqrt(-d*x^2 + 2)*d))*a*b + (x*arctan2(d*x^2 - 1, sqrt(-d
*x^2 + 2)*sqrt(d)*x)^2 + 4*sqrt(d)*integrate(sqrt(-d*x^2 + 2)*x*arctan2(d*x^2 - 1, sqrt(-d*x^2 + 2)*sqrt(d)*x)
/(d*x^2 - 2), x))*b^2 + a^2*x

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Fricas [A]
time = 2.41, size = 91, normalized size = 1.36 \begin {gather*} \frac {b^{2} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 2 \, a b d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{2} - 8 \, b^{2}\right )} d x^{2} + 4 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} {\left (b^{2} \arcsin \left (d x^{2} - 1\right ) + a b\right )}}{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x^2-1))^2,x, algorithm="fricas")

[Out]

(b^2*d*x^2*arcsin(d*x^2 - 1)^2 + 2*a*b*d*x^2*arcsin(d*x^2 - 1) + (a^2 - 8*b^2)*d*x^2 + 4*sqrt(-d^2*x^4 + 2*d*x
^2)*(b^2*arcsin(d*x^2 - 1) + a*b))/(d*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x**2-1))**2,x)

[Out]

Integral((a + b*asin(d*x**2 - 1))**2, x)

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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((a+b*arcsin(d*x^2-1))^2,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x^2 - 1) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(d*x^2 - 1))^2,x)

[Out]

int((a + b*asin(d*x^2 - 1))^2, x)

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