\(\int \frac {1}{x (1-c^2 x^2)^{3/2} (a+b \arcsin (c x))^2} \, dx\) [421]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 40.72 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.86 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

integral(sqrt(-c^2*x^2 + 1)/(a^2*c^4*x^5 - 2*a^2*c^2*x^3 + a^2*x + (b^2*c^4*x^5 - 2*b^2*c^2*x^3 + b^2*x)*arcsi
n(c*x)^2 + 2*(a*b*c^4*x^5 - 2*a*b*c^2*x^3 + a*b*x)*arcsin(c*x)), x)

Sympy [N/A]

Not integrable

Time = 6.62 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 209, normalized size of antiderivative = 7.46 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

((a*b*c^3*x^3 - a*b*c*x + (b^2*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*integrate((3*c^2
*x^2 - 1)/(a*b*c^5*x^6 - 2*a*b*c^3*x^4 + a*b*c*x^2 + (b^2*c^5*x^6 - 2*b^2*c^3*x^4 + b^2*c*x^2)*arctan2(c*x, sq
rt(c*x + 1)*sqrt(-c*x + 1))), x) + 1)/(a*b*c^3*x^3 - a*b*c*x + (b^2*c^3*x^3 - b^2*c*x)*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1)))

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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