\(\int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx\) [167]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\text {Int}\left (\frac {1}{x^2 (a+b \arcsin (c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

Int[1/(x^2*(a + b*ArcSin[c*x])^2),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 34.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.97 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 182, normalized size of antiderivative = 13.00 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-((b^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)*integrate((c^2*x^2 - 2)*sqrt(c*x + 1)*sqr
t(-c*x + 1)/(a*b*c^3*x^5 - a*b*c*x^3 + (b^2*c^3*x^5 - b^2*c*x^3)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))),
x) + sqrt(c*x + 1)*sqrt(-c*x + 1))/(b^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)

Giac [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int(1/(x^2*(a + b*asin(c*x))^2),x)

[Out]

int(1/(x^2*(a + b*asin(c*x))^2), x)