\(\int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx\) [331]

Optimal result

Integrand size = 33, antiderivative size = 13 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arcsin (a+b x)} \]

[Out]

-1/b/arcsin(b*x+a)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4891, 4737} \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arcsin (a+b x)} \]

[In]

Int[1/(Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2),x]

[Out]

-(1/(b*ArcSin[a + b*x]))

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4891

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Di
st[1/d, Subst[Int[(-C/d^2 + (C/d^2)*x^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A, B
, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \arcsin (x)^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {1}{b \arcsin (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arcsin (a+b x)} \]

[In]

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

[Out]

-(1/(b*ArcSin[a + b*x]))

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
default	\(-\frac {1}{b \arcsin \left (b x +a \right )}\)	\(14\)

[In]

int(1/arcsin(b*x+a)^2/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/b/arcsin(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arcsin \left (b x + a\right )} \]

[In]

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

[Out]

-1/(b*arcsin(b*x + a))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).

Time = 0.46 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=\begin {cases} - \frac {1}{b \operatorname {asin}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}^{2}{\left (a \right )}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-1/(b*asin(a + b*x)), Ne(b, 0)), (x/(sqrt(1 - a**2)*asin(a)**2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).

Time = 0.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )} \]

[In]

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

[Out]

-1/(b*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)))

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b \arcsin \left (b x + a\right )} \]

[In]

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

[Out]

-1/(b*arcsin(b*x + a))

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \arcsin (a+b x)^2} \, dx=-\frac {1}{b\,\mathrm {asin}\left (a+b\,x\right )} \]

[In]

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

[Out]

-1/(b*asin(a + b*x))