3.4.32 \(\int \frac {1}{\sqrt {1-a^2-2 a b x-b^2 x^2} \text {ArcSin}(a+b x)^3} \, dx\) [332]

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

[Out]

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

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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 {1}{2 b \text {ArcSin}(a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(b*ArcSin[a + b*x]^2)

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

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b \text {ArcSin}(a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(b*ArcSin[a + b*x]^2)

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

method	result	size
default	\(-\frac {1}{2 b \arcsin \left (b x +a \right )^{2}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arcsin(b*x+a)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
time = 16.59, size = 33, normalized size = 2.20 \begin {gather*} -\frac {1}{2 \, b \arctan \left (b x + a, \sqrt {b x + a + 1} \sqrt {-b x - a + 1}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(b*x+a)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2/(b*arctan2(b*x + a, sqrt(b*x + a + 1)*sqrt(-b*x - a + 1))^2)

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Fricas [A]
time = 1.62, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{2 \, b \arcsin \left (b x + a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(b*x+a)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x, algorithm="fricas")

[Out]

-1/2/(b*arcsin(b*x + a)^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (14) = 28\).
time = 0.75, size = 29, normalized size = 1.93 \begin {gather*} \begin {cases} - \frac {1}{2 b \operatorname {asin}^{2}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {1 - a^{2}} \operatorname {asin}^{3}{\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asin(b*x+a)**3/(-b**2*x**2-2*a*b*x-a**2+1)**(1/2),x)

[Out]

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

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Giac [A]
time = 0.48, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{2 \, b \arcsin \left (b x + a\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(b*x+a)^3/(-b^2*x^2-2*a*b*x-a^2+1)^(1/2),x, algorithm="giac")

[Out]

-1/2/(b*arcsin(b*x + a)^2)

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Mupad [B]
time = 0.26, size = 13, normalized size = 0.87 \begin {gather*} -\frac {1}{2\,b\,{\mathrm {asin}\left (a+b\,x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(asin(a + b*x)^3*(1 - b^2*x^2 - 2*a*b*x - a^2)^(1/2)),x)

[Out]

-1/(2*b*asin(a + b*x)^2)

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