\(\int \frac {\arcsin (a x)^4}{x^2} \, dx\) [39]

Optimal result

Integrand size = 10, antiderivative size = 156 \[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=-\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-24 a \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+24 a \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \]

[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))

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4723, 4803, 4268, 2611, 6744, 2320, 6724} \[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-24 a \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+24 a \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right )-\frac {\arcsin (a x)^4}{x} \]

[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 2320

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 2611

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 4268

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 4723

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 4803

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

Rule 6724

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 6744

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*} \text {integral}& = -\frac {\arcsin (a x)^4}{x}+(4 a) \int \frac {\arcsin (a x)^3}{x \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {\arcsin (a x)^4}{x}+(4 a) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )-(12 a) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+(12 a) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-(24 i a) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )+(24 i a) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-24 a \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+24 a \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )+(24 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arcsin (a x)\right )-(24 a) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-24 a \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+24 a \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-(24 i a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arcsin (a x)}\right )+(24 i a) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arcsin (a x)}\right ) \\ & = -\frac {\arcsin (a x)^4}{x}-8 a \arcsin (a x)^3 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-12 i a \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-24 a \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+24 a \arcsin (a x) \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )-24 i a \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )+24 i a \operatorname {PolyLog}\left (4,e^{i \arcsin (a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.27 \[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=a \left (-\frac {i \pi ^4}{2}+i \arcsin (a x)^4-\frac {\arcsin (a x)^4}{a x}+4 \arcsin (a x)^3 \log \left (1-e^{-i \arcsin (a x)}\right )-4 \arcsin (a x)^3 \log \left (1+e^{i \arcsin (a x)}\right )+12 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arcsin (a x)}\right )+12 i \arcsin (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+24 \arcsin (a x) \operatorname {PolyLog}\left (3,e^{-i \arcsin (a x)}\right )-24 \arcsin (a x) \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )-24 i \operatorname {PolyLog}\left (4,e^{-i \arcsin (a x)}\right )-24 i \operatorname {PolyLog}\left (4,-e^{i \arcsin (a x)}\right )\right ) \]

[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])])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.53

method	result	size
derivativedivides	\(a \left (-\frac {\arcsin \left (a x \right )^{4}}{a x}+4 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+24 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-12 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(238\)
default	\(a \left (-\frac {\arcsin \left (a x \right )^{4}}{a x}+4 \arcsin \left (a x \right )^{3} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-4 \arcsin \left (a x \right )^{3} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+24 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 \arcsin \left (a x \right ) \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-12 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+12 i \arcsin \left (a x \right )^{2} \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+24 i \operatorname {polylog}\left (4, i a x +\sqrt {-a^{2} x^{2}+1}\right )-24 i \operatorname {polylog}\left (4, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(238\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{2}} \,d x } \]

[In]

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

[Out]

integral(arcsin(a*x)^4/x^2, x)

Sympy [F]

\[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=\int \frac {\operatorname {asin}^{4}{\left (a x \right )}}{x^{2}}\, dx \]

[In]

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

[Out]

Integral(asin(a*x)**4/x**2, x)

Maxima [F]

\[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{2}} \,d x } \]

[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

Giac [F]

\[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{4}}{x^{2}} \,d x } \]

[In]

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

[Out]

integrate(arcsin(a*x)^4/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^4}{x^2} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^4}{x^2} \,d x \]

[In]

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

[Out]

int(asin(a*x)^4/x^2, x)