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

Optimal. Leaf size=59 \[ \frac{3 a \text{Unintegrable}\left (\frac{x}{\left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)},x\right )}{c^2}-\frac{1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)} \]

[Out]

-(1/(a*c^2*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])) + (3*a*Unintegrable[x/((1 - a^2*x^2)^(5/2)*ArcSin[a*x]), x])/c^2

________________________________________________________________________________________

Rubi [A]  time = 0.0974797, 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 = {} \[ \int \frac{1}{\left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(1/(a*c^2*(1 - a^2*x^2)^(3/2)*ArcSin[a*x])) + (3*a*Defer[Int][x/((1 - a^2*x^2)^(5/2)*ArcSin[a*x]), x])/c^2

Rubi steps

\begin{align*} \int \frac{1}{\left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^2} \, dx &=-\frac{1}{a c^2 \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}+\frac{(3 a) \int \frac{x}{\left (1-a^2 x^2\right )^{5/2} \sin ^{-1}(a x)} \, dx}{c^2}\\ \end{align*}

Mathematica [A]  time = 14.5328, size = 0, normalized size = 0. \[ \int \frac{1}{\left (c-a^2 c x^2\right )^2 \sin ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.289, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}c{x}^{2}+c \right ) ^{2} \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)^2/arcsin(a*x)^2,x)

[Out]

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

________________________________________________________________________________________

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)^2/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{a^{4} x^{4} \operatorname{asin}^{2}{\left (a x \right )} - 2 a^{2} x^{2} \operatorname{asin}^{2}{\left (a x \right )} + \operatorname{asin}^{2}{\left (a x \right )}}\, dx}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(a**4*x**4*asin(a*x)**2 - 2*a**2*x**2*asin(a*x)**2 + asin(a*x)**2), x)/c**2

________________________________________________________________________________________

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

[Out]

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