\(\int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx\) [172]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\text {Int}\left (\frac {1}{x^2 (a+b \arcsin (c x))^3},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (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 = {} \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx \]

[In]

Int[1/(x^2*(a + b*ArcSin[c*x])^3),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 15.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

\[\int \frac {1}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.79 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int \frac {1}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 283, normalized size of antiderivative = 20.21 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(a*c^2*x^2 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (b*c^2*x^2 - 2*b)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1)) + 2*(b^4*c^2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*x^3*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1)) + a^2*b^2*c^2*x^3)*integrate(1/2*(c^2*x^2 - 6)/(b^3*c^2*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt
(-c*x + 1)) + a*b^2*c^2*x^4), x) - 2*a)/(b^4*c^2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^
2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2*x^3)

Giac [N/A]

Not integrable

Time = 1.39 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int { \frac {1}{{\left (b \arcsin \left (c x\right ) + a\right )}^{3} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 (a+b \arcsin (c x))^3} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]

[In]

int(1/(x^2*(a + b*asin(c*x))^3),x)

[Out]

int(1/(x^2*(a + b*asin(c*x))^3), x)