3.6.2 \(\int \frac {\text {ArcSin}(a x)^n}{x \sqrt {1-a^2 x^2}} \, dx\) [502]

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

[Out]

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

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Rubi [A]
time = 0.07, 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 = {} \begin {gather*} \int \frac {\text {ArcSin}(a x)^n}{x \sqrt {1-a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcSin}(a x)^n}{x \sqrt {1-a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (a x \right )^{n}}{x \sqrt {-a^{2} x^{2}+1}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^n/x/(-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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^n/x/(-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^3 - x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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