3.415 \(\int \frac{1}{x^2 \sqrt{1-c^2 x^2} (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{2 \text{Unintegrable}\left (\frac{1}{x^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac{1}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )} \]

[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 [A]  time = 0.1493, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac{1}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )}-\frac{2 \int \frac{1}{x^3 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end{align*}

Mathematica [A]  time = 1.28753, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.093, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\frac{2 \,{\left (b^{2} c x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{2}\right )} \int \frac{1}{{\left (b \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a\right )} x^{3}}\,{d x}}{b c} + 1}{b^{2} c x^{2} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} x^{2} + 1}}{a^{2} c^{2} x^{4} - a^{2} x^{2} +{\left (b^{2} c^{2} x^{4} - b^{2} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} x^{4} - a b x^{2}\right )} \arcsin \left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-c^{2} x^{2} + 1}{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)