\(\int \frac {\arcsin (a x)^2}{x} \, dx\) [17]

Optimal result

Integrand size = 10, antiderivative size = 71 \[ \int \frac {\arcsin (a x)^2}{x} \, dx=-\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4721, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {\arcsin (a x)^2}{x} \, dx=-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right )-\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right ) \]

[In]

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

[Out]

(-1/3*I)*ArcSin[a*x]^3 + ArcSin[a*x]^2*Log[1 - E^((2*I)*ArcSin[a*x])] - I*ArcSin[a*x]*PolyLog[2, E^((2*I)*ArcS
in[a*x])] + PolyLog[3, E^((2*I)*ArcSin[a*x])]/2

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 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 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 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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^2 \cot (x) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{3} i \arcsin (a x)^3-2 i \text {Subst}\left (\int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-2 \text {Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+i \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i \arcsin (a x)}\right ) \\ & = -\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{2 i \arcsin (a x)}\right )-i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 i \arcsin (a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\arcsin (a x)^2}{x} \, dx=\frac {1}{3} i \arcsin (a x)^3+\arcsin (a x)^2 \log \left (1-e^{-2 i \arcsin (a x)}\right )+i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{-2 i \arcsin (a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \arcsin (a x)}\right ) \]

[In]

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

[Out]

(I/3)*ArcSin[a*x]^3 + ArcSin[a*x]^2*Log[1 - E^((-2*I)*ArcSin[a*x])] + I*ArcSin[a*x]*PolyLog[2, E^((-2*I)*ArcSi
n[a*x])] + PolyLog[3, E^((-2*I)*ArcSin[a*x])]/2

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.38

method	result	size
derivativedivides	\(-\frac {i \arcsin \left (a x \right )^{3}}{3}+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\)	\(169\)
default	\(-\frac {i \arcsin \left (a x \right )^{3}}{3}+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\)	\(169\)

[In]

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

[Out]

-1/3*I*arcsin(a*x)^3+arcsin(a*x)^2*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*I*arcsin(a*x)*polylog(2,I*a*x+(-a^2*x^2+1)
^(1/2))+2*polylog(3,I*a*x+(-a^2*x^2+1)^(1/2))+arcsin(a*x)^2*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))-2*I*arcsin(a*x)*pol
ylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))+2*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))

Fricas [F]

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

[In]

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

[Out]

integral(arcsin(a*x)^2/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(asin(a*x)**2/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(arcsin(a*x)^2/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(arcsin(a*x)^2/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(asin(a*x)^2/x, x)