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])))

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Rubi [A]  time = 0.0437099, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, 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 = 0.007966, size = 18, normalized size = 1. \[ -\frac{1}{b c \left (a+b \sin ^{-1}(c x)\right )} \]

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])))

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Maple [A]  time = 0.006, size = 19, normalized size = 1.1 \begin{align*} -{\frac{1}{bc \left ( a+b\arcsin \left ( cx \right ) \right ) }} \end{align*}

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))

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Maxima [A]  time = 1.49348, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b \arcsin \left (c x\right ) + a\right )} b c} \end{align*}

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)

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Fricas [A]  time = 2.26342, size = 43, normalized size = 2.39 \begin{align*} -\frac{1}{b^{2} c \arcsin \left (c x\right ) + a b c} \end{align*}

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)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError

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Giac [A]  time = 1.1342, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b \arcsin \left (c x\right ) + a\right )} b c} \end{align*}

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*arcsin(c*x) + a)*b*c)