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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 46, normalized size of antiderivative = 1.00, 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 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]

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 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]

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 = 0.05, 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))])

________________________________________________________________________________________

fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b^{2} c x^{2} - 2 \, a b c x - {\left (a^{2} - 1\right )} c} \arcsin \left (b x + a\right )}{b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} - 1\right )} c}, x\right ) \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (b x + a\right )}{\sqrt {-b^{2} c x^{2} - 2 \, a b c x - {\left (a^{2} - 1\right )} c}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 80, normalized size = 1.74 \[ -\frac {\sqrt {-c \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}\, \sqrt {-b^{2} x^{2}-2 a b x -a^{2}+1}\, \arcsin \left (b x +a \right )^{2}}{2 b \left (b^{2} x^{2}+2 a b x +a^{2}-1\right ) c} \]

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

________________________________________________________________________________________

maxima [B]  time = 0.43, size = 200, normalized size = 4.35 \[ \frac {\sqrt {c} \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )^{2}}{2 \, \sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} - 1\right )} b^{2} c^{2}}} - \frac {\arcsin \left (b x + a\right ) \arcsin \left (-\frac {b^{2} c x + a b c}{\sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} - 1\right )} b^{2} c^{2}}}\right )}{b \sqrt {c}} - \frac {\arcsin \left (-\frac {b^{2} c x + a b c}{\sqrt {a^{2} b^{2} c^{2} - {\left (a^{2} - 1\right )} b^{2} c^{2}}}\right ) \arcsin \left (-\frac {b^{2} x + a b}{\sqrt {a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}}}\right )}{b \sqrt {c}} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asin}\left (a+b\,x\right )}{\sqrt {-c\,b^2\,x^2-2\,a\,c\,b\,x-c\,\left (a^2-1\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}{\left (a + b x \right )}}{\sqrt {- c \left (a + b x - 1\right ) \left (a + b x + 1\right )}}\, dx \]

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)

________________________________________________________________________________________