3.6.1 \(\int \frac {\text {ArcSin}(a x)^n}{\sqrt {1-a^2 x^2}} \, dx\) [501]

Optimal. Leaf size=17 \[ \frac {\text {ArcSin}(a x)^{1+n}}{a (1+n)} \]

[Out]

arcsin(a*x)^(1+n)/a/(1+n)

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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4737} \begin {gather*} \frac {\text {ArcSin}(a x)^{n+1}}{a (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]^(1 + n)/(a*(1 + n))

Rule 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\text {ArcSin}(a x)^{1+n}}{a (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]^(1 + n)/(a*(1 + n))

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Maple [A]
time = 0.11, size = 18, normalized size = 1.06

method	result	size
derivativedivides	\(\frac {\arcsin \left (a x \right )^{1+n}}{a \left (1+n \right )}\)	\(18\)
default	\(\frac {\arcsin \left (a x \right )^{1+n}}{a \left (1+n \right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)^(1+n)/a/(1+n)

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Maxima [A]
time = 0.48, size = 17, normalized size = 1.00 \begin {gather*} \frac {\arcsin \left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)^(n + 1)/(a*(n + 1))

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Fricas [A]
time = 2.21, size = 18, normalized size = 1.06 \begin {gather*} \frac {\arcsin \left (a x\right )^{n} \arcsin \left (a x\right )}{a n + a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)^n*arcsin(a*x)/(a*n + a)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (12) = 24\).
time = 0.36, size = 34, normalized size = 2.00 \begin {gather*} \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge n = -1 \\0^{n} x & \text {for}\: a = 0 \\\frac {\log {\left (\operatorname {asin}{\left (a x \right )} \right )}}{a} & \text {for}\: n = -1 \\\frac {\operatorname {asin}{\left (a x \right )} \operatorname {asin}^{n}{\left (a x \right )}}{a n + a} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(n, -1)), (0**n*x, Eq(a, 0)), (log(asin(a*x))/a, Eq(n, -1)), (asin(a*x)*asin(a*
x)**n/(a*n + a), True))

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Giac [A]
time = 0.41, size = 17, normalized size = 1.00 \begin {gather*} \frac {\arcsin \left (a x\right )^{n + 1}}{a {\left (n + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(a*x)^(n + 1)/(a*(n + 1))

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Mupad [B]
time = 0.31, size = 33, normalized size = 1.94 \begin {gather*} \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {asin}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }n=-1\\ \frac {{\mathrm {asin}\left (a\,x\right )}^{n+1}}{a\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(n == -1, log(asin(a*x))/a, n ~= -1, asin(a*x)^(n + 1)/(a*(n + 1)))

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