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

Optimal. Leaf size=17 \[ \text {Int}\left (\frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )^3},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^2*(a + b*ArcSin[c*x])^3),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 19.12, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} x^{2} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} x^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b x^{2} \arcsin \left (c x\right ) + a^{3} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \left (a +b \arcsin \left (c x \right )\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a c^{2} x^{2} + \sqrt {c x + 1} \sqrt {-c x + 1} b c x + {\left (b c^{2} x^{2} - 2 \, b\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (b^{4} c^{2} x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c^{2} x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c^{2} x^{3}\right )} \int \frac {c^{2} x^{2} - 6}{b^{3} c^{2} x^{4} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b^{2} c^{2} x^{4}}\,{d x} - 2 \, a}{2 \, {\left (b^{4} c^{2} x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, a b^{3} c^{2} x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a^{2} b^{2} c^{2} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*c^2*x^2 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (b*c^2*x^2 - 2*b)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1)) + 2*(b^4*c^2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^2*x^3*arctan2(c*x, sqrt(c*x +
1)*sqrt(-c*x + 1)) + a^2*b^2*c^2*x^3)*integrate(1/2*(c^2*x^2 - 6)/(b^3*c^2*x^4*arctan2(c*x, sqrt(c*x + 1)*sqrt
(-c*x + 1)) + a*b^2*c^2*x^4), x) - 2*a)/(b^4*c^2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b^3*c^
2*x^3*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a^2*b^2*c^2*x^3)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{x^2\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*asin(c*x))^3),x)

[Out]

int(1/(x^2*(a + b*asin(c*x))^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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