3.420 \(\int \frac{1}{(1-c^2 x^2)^{3/2} (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{2 c \text{Unintegrable}\left (\frac{x}{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac{1}{b c \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.107705, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{1}{b c \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}+\frac{(2 c) \int \frac{x}{\left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b}\\ \end{align*}

Mathematica [A]  time = 2.56931, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[1/((1 - c^2*x^2)^(3/2)*(a + b*ArcSin[c*x])^2), x]

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Maple [A]  time = 0.194, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x)

[Out]

int(1/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x)

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-c^{2} x^{2} + 1}}{a^{2} c^{4} x^{4} - 2 \, a^{2} c^{2} x^{2} +{\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \arcsin \left (c x\right )^{2} + a^{2} + 2 \,{\left (a b c^{4} x^{4} - 2 \, a b c^{2} x^{2} + a b\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a^2*c^4*x^4 - 2*a^2*c^2*x^2 + (b^2*c^4*x^4 - 2*b^2*c^2*x^2 + b^2)*arcsin(c*x)^2 +
a^2 + 2*(a*b*c^4*x^4 - 2*a*b*c^2*x^2 + a*b)*arcsin(c*x)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c**2*x**2+1)**(3/2)/(a+b*asin(c*x))**2,x)

[Out]

Integral(1/((-(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((-c^2*x^2 + 1)^(3/2)*(b*arcsin(c*x) + a)^2), x)