3.388 \(\int \frac{\sqrt{1-c^2 x^2}}{x^4 (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x]

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Rubi [A]  time = 0.120872, 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 = {} \[ \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2),x]

[Out]

Defer[Int][Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=\int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 3.44463, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2),x]

[Out]

Integrate[Sqrt[1 - c^2*x^2]/(x^4*(a + b*ArcSin[c*x])^2), x]

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Maple [A]  time = 5.385, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{2} x^{2} + 2 \,{\left (b^{2} c x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{4}\right )} \int \frac{c^{2} x^{2} - 2}{b^{2} c x^{5} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{5}}\,{d x} - 1}{b^{2} c x^{4} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right ) + a b c x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{4} \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((-c**2*x**2+1)**(1/2)/x**4/(a+b*asin(c*x))**2,x)

[Out]

Integral(sqrt(-(c*x - 1)*(c*x + 1))/(x**4*(a + b*asin(c*x))**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(1/2)/x^4/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

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