3.4.28 \(\int \frac {\text {ArcSin}(a+b x)^2}{\sqrt {1-a^2-2 a b x-b^2 x^2}} \, dx\) [328]

Optimal. Leaf size=15 \[ \frac {\text {ArcSin}(a+b x)^3}{3 b} \]

[Out]

1/3*arcsin(b*x+a)^3/b

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Rubi [A]
time = 0.05, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {4891, 4737} \begin {gather*} \frac {\text {ArcSin}(a+b x)^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a + b*x]^3/(3*b)

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

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Mathematica [A]
time = 0.02, size = 15, normalized size = 1.00 \begin {gather*} \frac {\text {ArcSin}(a+b x)^3}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a + b*x]^3/(3*b)

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Maple [A]
time = 0.37, size = 14, normalized size = 0.93

method	result	size
default	\(\frac {\arcsin \left (b x +a \right )^{3}}{3 b}\)	\(14\)

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]

1/3*arcsin(b*x+a)^3/b

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (13) = 26\).
time = 0.50, size = 130, normalized size = 8.67 \begin {gather*} -\frac {\arcsin \left (b x + a\right )^{2} \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 (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 )^{2}}{b} - \frac {\arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{3}}{3 \, 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="maxima")

[Out]

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

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Fricas [A]
time = 2.14, size = 13, normalized size = 0.87 \begin {gather*} \frac {\arcsin \left (b x + a\right )^{3}}{3 \, 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]

1/3*arcsin(b*x + a)^3/b

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
time = 0.42, size = 26, normalized size = 1.73 \begin {gather*} \begin {cases} \frac {\operatorname {asin}^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\\frac {x \operatorname {asin}^{2}{\left (a \right )}}{\sqrt {1 - a^{2}}} & \text {otherwise} \end {cases} \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]

Piecewise((asin(a + b*x)**3/(3*b), Ne(b, 0)), (x*asin(a)**2/sqrt(1 - a**2), True))

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Giac [A]
time = 0.45, size = 13, normalized size = 0.87 \begin {gather*} \frac {\arcsin \left (b x + a\right )^{3}}{3 \, 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]

1/3*arcsin(b*x + a)^3/b

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Mupad [B]
time = 0.28, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {asin}\left (a+b\,x\right )}^3}{3\,b} \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]

asin(a + b*x)^3/(3*b)

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