3.39 \(\int \frac {\sin ^{-1}(a x)^4}{x^2} \, dx\)

Optimal. Leaf size=156 \[ 12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]

[Out]

-arcsin(a*x)^4/x-8*a*arcsin(a*x)^3*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+12*I*a*arcsin(a*x)^2*polylog(2,-I*a*x-(-a
^2*x^2+1)^(1/2))-12*I*a*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))-24*a*arcsin(a*x)*polylog(3,-I*a*x-(-
a^2*x^2+1)^(1/2))+24*a*arcsin(a*x)*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))-24*I*a*polylog(4,-I*a*x-(-a^2*x^2+1)^(1
/2))+24*I*a*polylog(4,I*a*x+(-a^2*x^2+1)^(1/2))

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Rubi [A]  time = 0.19, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4627, 4709, 4183, 2531, 6609, 2282, 6589} \[ 12 i a \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {PolyLog}\left (3,-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {PolyLog}\left (3,e^{i \sin ^{-1}(a x)}\right )-24 i a \text {PolyLog}\left (4,-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {PolyLog}\left (4,e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^4/x^2,x]

[Out]

-(ArcSin[a*x]^4/x) - 8*a*ArcSin[a*x]^3*ArcTanh[E^(I*ArcSin[a*x])] + (12*I)*a*ArcSin[a*x]^2*PolyLog[2, -E^(I*Ar
cSin[a*x])] - (12*I)*a*ArcSin[a*x]^2*PolyLog[2, E^(I*ArcSin[a*x])] - 24*a*ArcSin[a*x]*PolyLog[3, -E^(I*ArcSin[
a*x])] + 24*a*ArcSin[a*x]*PolyLog[3, E^(I*ArcSin[a*x])] - (24*I)*a*PolyLog[4, -E^(I*ArcSin[a*x])] + (24*I)*a*P
olyLog[4, E^(I*ArcSin[a*x])]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 4183

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

Rule 4627

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcSi
n[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4709

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[1/(c^(m
+ 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2
*d + e, 0] && GtQ[d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6609

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

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\sin ^{-1}(a x)^4}{x}+(4 a) \int \frac {\sin ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sin ^{-1}(a x)^4}{x}+(4 a) \operatorname {Subst}\left (\int x^3 \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )-(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(12 a) \operatorname {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-(24 i a) \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )+(24 i a) \operatorname {Subst}\left (\int x \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )+(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )-(24 a) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-(24 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )+(24 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )\\ &=-\frac {\sin ^{-1}(a x)^4}{x}-8 a \sin ^{-1}(a x)^3 \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )+12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )-12 i a \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )-24 a \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )+24 a \sin ^{-1}(a x) \text {Li}_3\left (e^{i \sin ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (e^{i \sin ^{-1}(a x)}\right )\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 198, normalized size = 1.27 \[ a \left (12 i \sin ^{-1}(a x)^2 \text {Li}_2\left (e^{-i \sin ^{-1}(a x)}\right )+12 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+24 \sin ^{-1}(a x) \text {Li}_3\left (e^{-i \sin ^{-1}(a x)}\right )-24 \sin ^{-1}(a x) \text {Li}_3\left (-e^{i \sin ^{-1}(a x)}\right )-24 i \text {Li}_4\left (e^{-i \sin ^{-1}(a x)}\right )-24 i \text {Li}_4\left (-e^{i \sin ^{-1}(a x)}\right )-\frac {\sin ^{-1}(a x)^4}{a x}+i \sin ^{-1}(a x)^4+4 \sin ^{-1}(a x)^3 \log \left (1-e^{-i \sin ^{-1}(a x)}\right )-4 \sin ^{-1}(a x)^3 \log \left (1+e^{i \sin ^{-1}(a x)}\right )-\frac {i \pi ^4}{2}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSin[a*x]^4/x^2,x]

[Out]

a*((-1/2*I)*Pi^4 + I*ArcSin[a*x]^4 - ArcSin[a*x]^4/(a*x) + 4*ArcSin[a*x]^3*Log[1 - E^((-I)*ArcSin[a*x])] - 4*A
rcSin[a*x]^3*Log[1 + E^(I*ArcSin[a*x])] + (12*I)*ArcSin[a*x]^2*PolyLog[2, E^((-I)*ArcSin[a*x])] + (12*I)*ArcSi
n[a*x]^2*PolyLog[2, -E^(I*ArcSin[a*x])] + 24*ArcSin[a*x]*PolyLog[3, E^((-I)*ArcSin[a*x])] - 24*ArcSin[a*x]*Pol
yLog[3, -E^(I*ArcSin[a*x])] - (24*I)*PolyLog[4, E^((-I)*ArcSin[a*x])] - (24*I)*PolyLog[4, -E^(I*ArcSin[a*x])])

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{4}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^2,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^4/x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{4}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^2,x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^4/x^2, x)

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maple [A]  time = 0.11, size = 241, normalized size = 1.54 \[ -\frac {\arcsin \left (a x \right )^{4}}{x}-4 a \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+4 a \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-24 a \arcsin \left (a x \right ) \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 a \arcsin \left (a x \right ) \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 i a \arcsin \left (a x \right )^{2} \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-12 i a \arcsin \left (a x \right )^{2} \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 i a \polylog \left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i a \polylog \left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^4/x^2,x)

[Out]

-arcsin(a*x)^4/x-4*a*arcsin(a*x)^3*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+4*a*arcsin(a*x)^3*ln(1-I*a*x-(-a^2*x^2+1)^(1
/2))-24*a*arcsin(a*x)*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))+24*a*arcsin(a*x)*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2)
)+12*I*a*arcsin(a*x)^2*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-12*I*a*arcsin(a*x)^2*polylog(2,I*a*x+(-a^2*x^2+1)^
(1/2))-24*I*a*polylog(4,-I*a*x-(-a^2*x^2+1)^(1/2))+24*I*a*polylog(4,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 \[ -\frac {\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{4} + 4 \, a x \int \frac {\sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{\sqrt {a x + 1} {\left (a x - 1\right )} x}\,{d x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^4/x^2,x, algorithm="maxima")

[Out]

-(arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4 + 4*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqr
t(a*x + 1)*sqrt(-a*x + 1))^3/(a^2*x^3 - x), x))/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^4/x^2,x)

[Out]

int(asin(a*x)^4/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**4/x**2,x)

[Out]

Integral(asin(a*x)**4/x**2, x)

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