\(\int (a-b \arcsin (1-d x^2))^2 \, dx\) [410]

Optimal result

Integrand size = 16, antiderivative size = 67 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=-8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \]

[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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4898, 8} \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2-8 b^2 x \]

[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*} \text {integral}& = \frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \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 \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=-8 b^2 x+\frac {4 b \sqrt {2 d x^2-d^2 x^4} \left (a-b \arcsin \left (1-d x^2\right )\right )}{d x}+x \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \]

[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

Maple [F]

\[\int {\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.36 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\frac {b^{2} d x^{2} \arcsin \left (d x^{2} - 1\right )^{2} + 2 \, a b d x^{2} \arcsin \left (d x^{2} - 1\right ) + {\left (a^{2} - 8 \, b^{2}\right )} d x^{2} + 4 \, \sqrt {-d^{2} x^{4} + 2 \, d x^{2}} {\left (b^{2} \arcsin \left (d x^{2} - 1\right ) + a b\right )}}{d x} \]

[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)

Sympy [F]

\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int \left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{2}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int { {\left (b \arcsin \left (d x^{2} - 1\right ) + a\right )}^{2} \,d x } \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (61) = 122\).

Time = 0.44 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.39 \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=2 \, {\left (x \arcsin \left (d x^{2} - 1\right ) - \frac {2 \, \sqrt {2} \mathrm {sgn}\left (x\right )}{\sqrt {d}} + \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} a b + {\left (x \arcsin \left (d x^{2} - 1\right )^{2} + \frac {2 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 4 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} + \frac {4 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (d x^{2} - 1\right ) + \frac {2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{2} + a^{2} x \]

[In]

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

[Out]

2*(x*arcsin(d*x^2 - 1) - 2*sqrt(2)*sgn(x)/sqrt(d) + 2*sqrt(-d^2*x^2 + 2*d)/(d*sgn(x)))*a*b + (x*arcsin(d*x^2 -
1)^2 + 2*(sqrt(2)*pi*sqrt(d)*abs(d) - 4*sqrt(2)*d^(3/2))*sgn(x)/(d*abs(d)) + 4*(sqrt(-d^2*x^2 + 2*d)*arcsin(d
*x^2 - 1) + 2*(sqrt(2)*sqrt(d) - sqrt(d^2*x^2))*d/abs(d))/(d*sgn(x)))*b^2 + a^2*x

Mupad [F(-1)]

Timed out. \[ \int \left (a-b \arcsin \left (1-d x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^2 \,d x \]

[In]

int((a + b*asin(d*x^2 - 1))^2,x)

[Out]

int((a + b*asin(d*x^2 - 1))^2, x)