\(\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx\) [414]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

Sympy [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int \frac {1}{x \sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2} \, dx=\int { \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \arcsin \left (c x\right ) + a\right )}^{2} x} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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