Integrand size = 24, antiderivative size = 92 \[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+2 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]
-2*arcsin(a*x)^2*arctanh(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*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))+2*polylog(3,I*a*x+(-a^2*x^2 +1)^(1/2))
Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\arcsin (a x)^2 \log \left (1-e^{i \arcsin (a x)}\right )-\arcsin (a x)^2 \log \left (1+e^{i \arcsin (a x)}\right )+2 i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )+2 \operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right ) \]
ArcSin[a*x]^2*Log[1 - E^(I*ArcSin[a*x])] - ArcSin[a*x]^2*Log[1 + E^(I*ArcS in[a*x])] + (2*I)*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - (2*I)*ArcSi n[a*x]*PolyLog[2, E^(I*ArcSin[a*x])] - 2*PolyLog[3, -E^(I*ArcSin[a*x])] + 2*PolyLog[3, E^(I*ArcSin[a*x])]
Time = 0.45 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5218, 3042, 4671, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle \int \frac {\arcsin (a x)^2}{a x}d\arcsin (a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \arcsin (a x)^2 \csc (\arcsin (a x))d\arcsin (a x)\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -2 \int \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+2 \int \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle 2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )d\arcsin (a x)\right )-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle 2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\int e^{-i \arcsin (a x)} \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}\right )-2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -2 \arcsin (a x)^2 \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arcsin (a x)}\right )\right )-2 \left (i \arcsin (a x) \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arcsin (a x)}\right )\right )\) |
-2*ArcSin[a*x]^2*ArcTanh[E^(I*ArcSin[a*x])] + 2*(I*ArcSin[a*x]*PolyLog[2, -E^(I*ArcSin[a*x])] - PolyLog[3, -E^(I*ArcSin[a*x])]) - 2*(I*ArcSin[a*x]*P olyLog[2, E^(I*ArcSin[a*x])] - PolyLog[3, E^(I*ArcSin[a*x])])
3.3.69.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
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] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(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] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Time = 0.14 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right )\) | \(161\) |
-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)*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))
\[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
\[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x\,\sqrt {1-a^2\,x^2}} \,d x \]