Integrand size = 10, antiderivative size = 66 \[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=-\frac {\arcsin (a x)^2}{x}-4 a \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+2 i a \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i a \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]
-arcsin(a*x)^2/x-4*a*arcsin(a*x)*arctanh(I*a*x+(-a^2*x^2+1)^(1/2))+2*I*a*p olylog(2,-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))
Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.32 \[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=a \left (-\arcsin (a x) \left (\frac {\arcsin (a x)}{a x}-2 \log \left (1-e^{i \arcsin (a x)}\right )+2 \log \left (1+e^{i \arcsin (a x)}\right )\right )+2 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-2 i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right ) \]
a*(-(ArcSin[a*x]*(ArcSin[a*x]/(a*x) - 2*Log[1 - E^(I*ArcSin[a*x])] + 2*Log [1 + E^(I*ArcSin[a*x])])) + (2*I)*PolyLog[2, -E^(I*ArcSin[a*x])] - (2*I)*P olyLog[2, E^(I*ArcSin[a*x])])
Time = 0.38 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5138, 5218, 3042, 4671, 2715, 2838}
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.
\(\displaystyle \int \frac {\arcsin (a x)^2}{x^2} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle 2 a \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^2}{x}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle 2 a \int \frac {\arcsin (a x)}{a x}d\arcsin (a x)-\frac {\arcsin (a x)^2}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 2 a \int \arcsin (a x) \csc (\arcsin (a x))d\arcsin (a x)-\frac {\arcsin (a x)^2}{x}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {\arcsin (a x)^2}{x}+2 a \left (-\int \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+\int \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arcsin (a x)^2}{x}+2 a \left (i \int e^{-i \arcsin (a x)} \log \left (1-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-i \int e^{-i \arcsin (a x)} \log \left (1+e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arcsin (a x)^2}{x}+2 a \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )\) |
-(ArcSin[a*x]^2/x) + 2*a*(-2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + I*Po lyLog[2, -E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])])
3.1.18.3.1 Defintions of rubi rules used
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[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]
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]
Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.82
method | result | size |
derivativedivides | \(a \left (-\frac {\arcsin \left (a x \right )^{2}}{a x}+2 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(120\) |
default | \(a \left (-\frac {\arcsin \left (a x \right )^{2}}{a x}+2 \arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 \arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \operatorname {dilog}\left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\) | \(120\) |
a*(-arcsin(a*x)^2/a/x+2*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-2*arcsi n(a*x)*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+2*I*dilog(1+I*a*x+(-a^2*x^2+1)^(1/2) )-2*I*dilog(1-I*a*x-(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{2}} \,d x } \]
-(2*a*x*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)* sqrt(-a*x + 1))/(a^2*x^3 - x), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)/x
\[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^2} \,d x \]