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

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sqrt {1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 3.99, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x^4 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b^{2} x^{4} \arcsin \left (c x\right )^{2} + 2 \, a b x^{4} \arcsin \left (c x\right ) + a^{2} x^{4}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2} + 1}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}\,{d x} \]

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)

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maple [A]  time = 6.53, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x^{4} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{2} x^{2} + 2 \, {\left (b^{2} c x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{4}\right )} \int \frac {c^{2} x^{2} - 2}{b^{2} c x^{5} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{5}}\,{d x} - 1}{b^{2} c x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{4}} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

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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