3.4.14 \(\int \sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)^2 \, dx\) [314]

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

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 = {4891, 4741, 4737, 4723, 327, 222} \begin {gather*} \frac {\text {ArcSin}(a+b x)^3}{6 b}+\frac {(a+b x) \sqrt {1-(a+b x)^2} \text {ArcSin}(a+b x)^2}{2 b}-\frac {(a+b x)^2 \text {ArcSin}(a+b x)}{2 b}+\frac {\text {ArcSin}(a+b x)}{4 b}-\frac {(a+b x) \sqrt {1-(a+b x)^2}}{4 b} \end {gather*}

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]

-1/4*((a + b*x)*Sqrt[1 - (a + b*x)^2])/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 222

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]

Rule 327

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 4723

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 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 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 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*} \int \sqrt {1-a^2-2 a b x-b^2 x^2} \sin ^{-1}(a+b x)^2 \, dx &=\frac {\text {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 {\text {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\sqrt {1-x^2}} \, dx,x,a+b x\right )}{2 b}-\frac {\text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 0.07, size = 116, normalized size = 1.05 \begin {gather*} \frac {-3 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2}-3 \left (-1+2 a^2+4 a b x+2 b^2 x^2\right ) \text {ArcSin}(a+b x)+6 (a+b x) \sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)^2+2 \text {ArcSin}(a+b x)^3}{12 b} \end {gather*}

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 = 0.42, size = 179, normalized size = 1.61

method	result	size
default	\(\frac {6 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -6 \arcsin \left (b x +a \right ) b^{2} x^{2}+6 \arcsin \left (b x +a \right )^{2} \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a -12 \arcsin \left (b x +a \right ) a b x +2 \arcsin \left (b x +a \right )^{3}-6 \arcsin \left (b x +a \right ) a^{2}-3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, b x -3 \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, a +3 \arcsin \left (b x +a \right )}{12 b}\)	\(179\)

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,method=_RETURNVERBOSE)

[Out]

1/12*(6*arcsin(b*x+a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*b*x-6*arcsin(b*x+a)*b^2*x^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)*a*b*x+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)*b*x-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]
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(arcsin(b*x+a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 7.71, size = 91, normalized size = 0.82 \begin {gather*} \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 {gather*}

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

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 = 0.46, size = 125, normalized size = 1.13 \begin {gather*} \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 {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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