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

Optimal. Leaf size=100 \[ -\frac{2 \text{Unintegrable}\left (\frac{1-c^2 x^2}{x^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac{2 c \text{Unintegrable}\left (\frac{1-c^2 x^2}{x \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac{\left (1-c^2 x^2\right )^2}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.24623, 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 = {} \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.34198, size = 0, normalized size = 0. \[ \int \frac{\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.341, 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((-c^2*x^2+1)^(3/2)/x^2/(a+b*arcsin(c*x))^2,x)

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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