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

Optimal. Leaf size=102 \[ -\frac {2 c \text {Int}\left (\frac {1-c^2 x^2}{x \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b}-\frac {2 \text {Int}\left (\frac {1-c^2 x^2}{x^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac {\left (1-c^2 x^2\right )^2}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.25, 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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 4.62, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{b^{2} x^{2} \arcsin \left (c x\right )^{2} + 2 \, a b x^{2} \arcsin \left (c x\right ) + a^{2} x^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{2}}\,{d x} \]

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)

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maple [A]  time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(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 = 0.00, size = 0, normalized size = 0.00 \[ -\frac {c^{4} x^{4} - 2 \, c^{2} x^{2} - 2 \, {\left (b^{2} c x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{2}\right )} \int \frac {c^{4} x^{4} - 1}{b^{2} c x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{3}}\,{d x} + 1}{b^{2} c x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x^{2}} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

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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