\(\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx\) [415]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=-\frac {1}{b c x^2 (a+b \arcsin (c x))}-\frac {2 \text {Int}\left (\frac {1}{x^3 (a+b \arcsin (c x))},x\right )}{b c} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.07 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^{2} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.29 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

-(2*(b^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)*integrate(1/(b^2*c*x^3*arctan2(c*x, sqr
t(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^3), 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 = 0.79 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]

[In]

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

[Out]

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