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

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 = 0.158231, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {247, 217, 203, 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[c - c*(a + b*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 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 217

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 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{c-c (a+b 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 = 0.100684, size = 46, normalized size = 1. \[ \frac{\sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{2 b \sqrt{-c \left ((a+b x)^2-1\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, 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)/(c-c*(b*x+a)^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)/(c-c*(b*x+a)^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)/(c-c*(b*x+a)^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)/(c-c*(b*x+a)**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{-{\left (b x + a\right )}^{2} c + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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