\(\int \frac {(1-c^2 x^2)^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx\) [395]

   Optimal result
   Rubi [N/A]
   Mathematica [N/A]
   Maple [N/A] (verified)
   Fricas [N/A]
   Sympy [N/A]
   Maxima [N/A]
   Giac [N/A]
   Mupad [N/A]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.16 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(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 = 2.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}} \,d x } \]

[In]

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

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((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 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \arcsin (c x))^2} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

[In]

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

[Out]

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