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

Optimal. Leaf size=18 \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {4641} \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*c*(a + b*ArcSin[c*x])))

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \[ -\frac {1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1/(b*c*(a + b*ArcSin[c*x])))

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 18, normalized size = 1.00 \[ -\frac {1}{b^{2} c \arcsin \left (c x\right ) + a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*c*arcsin(c*x) + a*b*c)

________________________________________________________________________________________

giac [A]  time = 0.52, size = 18, normalized size = 1.00 \[ -\frac {1}{b^{2} c \arcsin \left (c x\right ) + a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*c*arcsin(c*x) + a*b*c)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 19, normalized size = 1.06 \[ -\frac {1}{b c \left (a +b \arcsin \left (c x \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 18, normalized size = 1.00 \[ -\frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )} b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*arcsin(c*x) + a)*b*c)

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 18, normalized size = 1.00 \[ -\frac {1}{c\,\mathrm {asin}\left (c\,x\right )\,b^2+a\,c\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*c*asin(c*x) + a*b*c)

________________________________________________________________________________________

sympy [A]  time = 2.11, size = 53, normalized size = 2.94 \[ \begin {cases} \frac {x}{a^{2}} & \text {for}\: b = 0 \wedge c = 0 \\\frac {\begin {cases} - \frac {i \operatorname {acosh}{\left (c x \right )}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {\operatorname {asin}{\left (c x \right )}}{c} & \text {otherwise} \end {cases}}{a^{2}} & \text {for}\: b = 0 \\\frac {x}{a^{2}} & \text {for}\: c = 0 \\- \frac {1}{a b c + b^{2} c \operatorname {asin}{\left (c x \right )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x/a**2, Eq(b, 0) & Eq(c, 0)), (Piecewise((-I*acosh(c*x)/c, Abs(c**2*x**2) > 1), (asin(c*x)/c, True)
)/a**2, Eq(b, 0)), (x/a**2, Eq(c, 0)), (-1/(a*b*c + b**2*c*asin(c*x)), True))

________________________________________________________________________________________