3.422 \(\int \frac{1}{x^2 (1-c^2 x^2)^{3/2} (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.132117, 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 = {} \[ \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 31.4241, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.642, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}} \left ( -{c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (a b c^{3} x^{4} - a b c x^{2} +{\left (b^{2} c^{3} x^{4} - b^{2} c x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )\right )} \int \frac{2 \, c^{2} x^{2} - 1}{a b c^{5} x^{7} - 2 \, a b c^{3} x^{5} + a b c x^{3} +{\left (b^{2} c^{5} x^{7} - 2 \, b^{2} c^{3} x^{5} + b^{2} c x^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}\,{d x} + 1}{a b c^{3} x^{4} - a b c x^{2} +{\left (b^{2} c^{3} x^{4} - b^{2} c x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*b*c^3*x^4 - a*b*c*x^2 + (b^2*c^3*x^4 - b^2*c*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*integrate(2*
(2*c^2*x^2 - 1)/(a*b*c^5*x^7 - 2*a*b*c^3*x^5 + a*b*c*x^3 + (b^2*c^5*x^7 - 2*b^2*c^3*x^5 + b^2*c*x^3)*arctan2(c
*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) + 1)/(a*b*c^3*x^4 - a*b*c*x^2 + (b^2*c^3*x^4 - b^2*c*x^2)*arctan2(c*x,
sqrt(c*x + 1)*sqrt(-c*x + 1)))

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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}}{a^{2} c^{4} x^{6} - 2 \, a^{2} c^{2} x^{4} + a^{2} x^{2} +{\left (b^{2} c^{4} x^{6} - 2 \, b^{2} c^{2} x^{4} + b^{2} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{4} x^{6} - 2 \, 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/(-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^6 - 2*a^2*c^2*x^4 + a^2*x^2 + (b^2*c^4*x^6 - 2*b^2*c^2*x^4 + b^2*x^2)*a
rcsin(c*x)^2 + 2*(a*b*c^4*x^6 - 2*a*b*c^2*x^4 + a*b*x^2)*arcsin(c*x)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}}{\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/(-c^2*x^2+1)^(3/2)/(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

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