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

Optimal. Leaf size=111 \[ -\frac{(a+b x) \sqrt{1-(a+b x)^2}}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{\sin ^{-1}(a+b x)}{4 b} \]

[Out]

-((a + b*x)*Sqrt[1 - (a + b*x)^2])/(4*b) + ArcSin[a + b*x]/(4*b) - ((a + b*x)^2*ArcSin[a + b*x])/(2*b) + ((a +
b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b) + ArcSin[a + b*x]^3/(6*b)

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Rubi [A]  time = 0.12721, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4807, 4647, 4641, 4627, 321, 216} \[ -\frac{(a+b x) \sqrt{1-(a+b x)^2}}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{\sin ^{-1}(a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)*Sqrt[1 - (a + b*x)^2])/(4*b) + ArcSin[a + b*x]/(4*b) - ((a + b*x)^2*ArcSin[a + b*x])/(2*b) + ((a +
b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*b) + ArcSin[a + b*x]^3/(6*b)

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]

Rule 4647

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

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 4627

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{1-x^2} \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b}-\frac{\operatorname{Subst}\left (\int x \sin ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{(a+b x) \sqrt{1-(a+b x)^2}}{4 b}-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{4 b}\\ &=-\frac{(a+b x) \sqrt{1-(a+b x)^2}}{4 b}+\frac{\sin ^{-1}(a+b x)}{4 b}-\frac{(a+b x)^2 \sin ^{-1}(a+b x)}{2 b}+\frac{(a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b}+\frac{\sin ^{-1}(a+b x)^3}{6 b}\\ \end{align*}

Mathematica [A]  time = 0.0988498, size = 116, normalized size = 1.05 \[ \frac{-3 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1}+6 (a+b x) \sqrt{-a^2-2 a b x-b^2 x^2+1} \sin ^{-1}(a+b x)^2-3 \left (2 a^2+4 a b x+2 b^2 x^2-1\right ) \sin ^{-1}(a+b x)+2 \sin ^{-1}(a+b x)^3}{12 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(a + b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2] - 3*(-1 + 2*a^2 + 4*a*b*x + 2*b^2*x^2)*ArcSin[a + b*x] + 6*(a
+ b*x)*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]*ArcSin[a + b*x]^2 + 2*ArcSin[a + b*x]^3)/(12*b)

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Maple [A]  time = 0.059, size = 179, normalized size = 1.6 \begin{align*}{\frac{1}{12\,b} \left ( 6\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb-6\,\arcsin \left ( bx+a \right ){x}^{2}{b}^{2}+6\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a-12\,\arcsin \left ( bx+a \right ) xab+2\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}-6\,\arcsin \left ( bx+a \right ){a}^{2}-3\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb-3\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a+3\,\arcsin \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*arcsin(b*x+a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b-6*arcsin(b*x+a)*x^2*b^2+6*arcsin(b*x+a)^2*(-b^2*x^2
-2*a*b*x-a^2+1)^(1/2)*a-12*arcsin(b*x+a)*x*a*b+2*arcsin(b*x+a)^3-6*arcsin(b*x+a)*a^2-3*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*x*b-3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a+3*arcsin(b*x+a))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.38593, size = 223, normalized size = 2.01 \begin{align*} \frac{2 \, \arcsin \left (b x + a\right )^{3} - 3 \,{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} - 1\right )} \arcsin \left (b x + a\right ) + 3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (2 \,{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2} - b x - a\right )}}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(2*arcsin(b*x + a)^3 - 3*(2*b^2*x^2 + 4*a*b*x + 2*a^2 - 1)*arcsin(b*x + a) + 3*sqrt(-b^2*x^2 - 2*a*b*x -
a^2 + 1)*(2*(b*x + a)*arcsin(b*x + a)^2 - b*x - a))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27094, size = 169, normalized size = 1.52 \begin{align*} \frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )} \arcsin \left (b x + a\right )^{2}}{2 \, b} + \frac{\arcsin \left (b x + a\right )^{3}}{6 \, b} - \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )}{2 \, b} - \frac{\sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{4 \, b} - \frac{\arcsin \left (b x + a\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)^2/b + 1/6*arcsin(b*x + a)^3/b - 1/2*(b^2*x^2
+ 2*a*b*x + a^2 - 1)*arcsin(b*x + a)/b - 1/4*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/b - 1/4*arcsin(b*x +
a)/b