3.503 \(\int \frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}},x\right ) \]

[Out]

Unintegrable(arcsin(a*x)^n/x^2/(-a^2*x^2+1)^(1/2),x)

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][ArcSin[a*x]^n/(x^2*Sqrt[1 - a^2*x^2]), x]

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx &=\int \frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ \end {align*}

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Mathematica [A]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{-1}(a x)^n}{x^2 \sqrt {1-a^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{n}}{a^{2} x^{4} - x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x \right )^{n}}{x^{2} \sqrt {-a^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^n}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^n/(x^2*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(asin(a*x)^n/(x^2*(1 - a^2*x^2)^(1/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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