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

Optimal. Leaf size=46 \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}} \]

[Out]

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

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Rubi [A]  time = 0.0843548, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4807, 4643, 4641} \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{c-c (a+b x)^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4643

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

Rubi steps

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

Mathematica [A]  time = 0.0414277, size = 54, normalized size = 1.17 \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{-c \left (a^2+2 a b x+b^2 x^2-1\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.035, size = 80, normalized size = 1.7 \begin{align*} -{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}{2\,b \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}-1 \right ) c}\sqrt{-c \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}-1 \right ) }\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b^{2} c x^{2} - 2 \, a b c x -{\left (a^{2} - 1\right )} c} \arcsin \left (b x + a\right )}{b^{2} c x^{2} + 2 \, a b c x +{\left (a^{2} - 1\right )} c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (b x + a\right )}{\sqrt{-b^{2} c x^{2} - 2 \, a b c x -{\left (a^{2} - 1\right )} c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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