\(\int \frac {1}{(c-a^2 c x^2) \arcsin (a x)^2} \, dx\) [377]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=-\frac {1}{a c \sqrt {1-a^2 x^2} \arcsin (a x)}+\frac {a \text {Int}\left (\frac {x}{\left (1-a^2 x^2\right )^{3/2} \arcsin (a x)},x\right )}{c} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 4.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right ) \arcsin \left (a x \right )^{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=- \frac {\int \frac {1}{a^{2} x^{2} \operatorname {asin}^{2}{\left (a x \right )} - \operatorname {asin}^{2}{\left (a x \right )}}\, dx}{c} \]

[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

Maxima [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 153, normalized size of antiderivative = 7.65 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]

[In]

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

[Out]

((a^4*c*x^2 - a^2*c)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x/((a^4
*c*x^4 - 2*a^2*c*x^2 + c)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + sqrt(a*x + 1)*sqrt(-a*x + 1))/((a^
3*c*x^2 - a*c)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)))

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int { -\frac {1}{{\left (a^{2} c x^{2} - c\right )} \arcsin \left (a x\right )^{2}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\left (c-a^2 c x^2\right ) \arcsin (a x)^2} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^2\,\left (c-a^2\,c\,x^2\right )} \,d x \]

[In]

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

[Out]

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