3.377 \(\int \frac{1}{(c-a^2 c x^2) \sin ^{-1}(a x)^2} \, dx\)

Optimal. Leaf size=58 \[ \frac{a \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)},x\right )}{c}-\frac{1}{a c \sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0956909, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.76265, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right ) \sin ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}c{x}^{2}+c \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(-a^2*c*x^2+c)/arcsin(a*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/(-a^2*c*x^2+c)/arcsin(a*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{1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-1/((a^2*c*x^2 - c)*arcsin(a*x)^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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