3.3.70 \(\int \frac {\text {ArcSin}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [270]

Optimal. Leaf size=76 \[ -i a \text {ArcSin}(a x)^2-\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{x}+2 a \text {ArcSin}(a x) \log \left (1-e^{2 i \text {ArcSin}(a x)}\right )-i a \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(a x)}\right ) \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4771, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{x}-i a \text {Li}_2\left (e^{2 i \text {ArcSin}(a x)}\right )-i a \text {ArcSin}(a x)^2+2 a \text {ArcSin}(a x) \log \left (1-e^{2 i \text {ArcSin}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

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)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+(2 a) \int \frac {\sin ^{-1}(a x)}{x} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+(2 a) \text {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}-(4 i a) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-(2 a) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+(i a) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-i a \sin ^{-1}(a x)^2-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{x}+2 a \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-i a \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 72, normalized size = 0.95 \begin {gather*} \text {ArcSin}(a x) \left (-\frac {\left (i a x+\sqrt {1-a^2 x^2}\right ) \text {ArcSin}(a x)}{x}+2 a \log \left (1-e^{2 i \text {ArcSin}(a x)}\right )\right )-i a \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(a x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]*(-(((I*a*x + Sqrt[1 - a^2*x^2])*ArcSin[a*x])/x) + 2*a*Log[1 - E^((2*I)*ArcSin[a*x])]) - I*a*PolyLo
g[2, E^((2*I)*ArcSin[a*x])]

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Maple [A]
time = 0.20, size = 141, normalized size = 1.86

method	result	size
default	\(\frac {\left (i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{2}}{x}-2 i a \left (i \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+i \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2}+\polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(I*a*x-(-a^2*x^2+1)^(1/2))*arcsin(a*x)^2/x-2*I*a*(I*arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+I*arcsin(a*x)*l
n(1-I*a*x-(-a^2*x^2+1)^(1/2))+arcsin(a*x)^2+polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+polylog(2,I*a*x+(-a^2*x^2+1)^
(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2 - 2*a*x*integrate(arctan2(a*x, sqr
t(a*x + 1)*sqrt(-a*x + 1))/x, x))/x

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Fricas [F]
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)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [F]
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)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{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)^2/(x^2*(1 - a^2*x^2)^(1/2)),x)

[Out]

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

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