3.2.7 \(\int \frac {\text {ArcSin}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [107]

Optimal. Leaf size=28 \[ -\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{x}+a \log (x) \]

[Out]

a*ln(x)-arcsin(a*x)*(-a^2*x^2+1)^(1/2)/x

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Rubi [A]
time = 0.04, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4771, 29} \begin {gather*} a \log (x)-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 4771

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d
+ e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{x}+a \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - a^2*x^2]*ArcSin[a*x])/x) + a*Log[x]

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Maple [A]
time = 0.07, size = 32, normalized size = 1.14

method	result	size
default	\(-\frac {-\ln \left (a x \right ) a x +\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-ln(a*x)*a*x+arcsin(a*x)*(-a^2*x^2+1)^(1/2))/x

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Maxima [A]
time = 0.48, size = 26, normalized size = 0.93 \begin {gather*} a \log \left (x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*log(x) - sqrt(-a^2*x^2 + 1)*arcsin(a*x)/x

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Fricas [A]
time = 1.91, size = 28, normalized size = 1.00 \begin {gather*} \frac {a x \log \left (x\right ) - \sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*x*log(x) - sqrt(-a^2*x^2 + 1)*arcsin(a*x))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}{\left (a x \right )}}{x^{2} \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)/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (26) = 52\).
time = 0.40, size = 67, normalized size = 2.39 \begin {gather*} \frac {1}{2} \, {\left (\frac {a^{4} x}{{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{x {\left | a \right |}}\right )} \arcsin \left (a x\right ) + a \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a^4*x/((sqrt(-a^2*x^2 + 1)*abs(a) + a)*abs(a)) - (sqrt(-a^2*x^2 + 1)*abs(a) + a)/(x*abs(a)))*arcsin(a*x)
+ a*log(abs(x))

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

[Out]

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

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