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

Optimal. Leaf size=47 \[ -\frac {2 \text {Int}\left (\frac {1}{x^3 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac {1}{b c x^2 \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.15, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.39, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1}}{a^{2} c^{2} x^{4} - a^{2} x^{2} + {\left (b^{2} c^{2} x^{4} - b^{2} x^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} x^{4} - a b x^{2}\right )} \arcsin \left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(b^2*c*x^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2)*integrate(1/(b^2*c*x^3*arctan2(c*x, sqr
t(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^3), x) + 1)/(b^2*c*x^2*arctan2(c*x, sqrt(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 = -0.02 \[ \int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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