3.329 \(\int \frac {\sin ^{-1}(a+b x)}{\sqrt {1-a^2-2 a b x-b^2 x^2}} \, dx\)

Optimal. Leaf size=15 \[ \frac {\sin ^{-1}(a+b x)^2}{2 b} \]

[Out]

1/2*arcsin(b*x+a)^2/b

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Rubi [A]  time = 0.04, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {4807, 4641} \[ \frac {\sin ^{-1}(a+b x)^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a + b*x]^2/(2*b)

Rule 4641

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

Rule 4807

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*x^2)/d^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*} \int \frac {\sin ^{-1}(a+b x)}{\sqrt {1-a^2-2 a b x-b^2 x^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sin ^{-1}(a+b x)^2}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 15, normalized size = 1.00 \[ \frac {\sin ^{-1}(a+b x)^2}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a + b*x]^2/(2*b)

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fricas [A]  time = 0.45, size = 13, normalized size = 0.87 \[ \frac {\arcsin \left (b x + a\right )^{2}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(b*x + a)^2/b

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giac [A]  time = 0.35, size = 13, normalized size = 0.87 \[ \frac {\arcsin \left (b x + a\right )^{2}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(b*x + a)^2/b

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maple [A]  time = 0.10, size = 14, normalized size = 0.93 \[ \frac {\arcsin \left (b x +a \right )^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsin(b*x+a)^2/b

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maxima [B]  time = 0.42, size = 83, normalized size = 5.53 \[ -\frac {\arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b} - \frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.26, size = 13, normalized size = 0.87 \[ \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asin(a + b*x)^2/(2*b)

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sympy [A]  time = 0.65, size = 24, normalized size = 1.60 \[ \begin {cases} \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x \operatorname {asin}{\relax (a )}}{\sqrt {1 - a^{2}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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