Integrand size = 10, antiderivative size = 116 \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x^2}-\frac {\arcsin (a x)^2}{3 x^3}-\frac {2}{3} a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]
-1/3*a^2/x-1/3*arcsin(a*x)^2/x^3-2/3*a^3*arcsin(a*x)*arctanh(I*a*x+(-a^2*x ^2+1)^(1/2))+1/3*I*a^3*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))-1/3*I*a^3*poly log(2,I*a*x+(-a^2*x^2+1)^(1/2))-1/3*a*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/x^2
Time = 0.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=-\frac {a^2 x^2+a x \sqrt {1-a^2 x^2} \arcsin (a x)+\arcsin (a x)^2-a^3 x^3 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )+a^3 x^3 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )}{3 x^3} \]
-1/3*(a^2*x^2 + a*x*Sqrt[1 - a^2*x^2]*ArcSin[a*x] + ArcSin[a*x]^2 - a^3*x^ 3*ArcSin[a*x]*Log[1 - E^(I*ArcSin[a*x])] + a^3*x^3*ArcSin[a*x]*Log[1 + E^( I*ArcSin[a*x])] - I*a^3*x^3*PolyLog[2, -E^(I*ArcSin[a*x])] + I*a^3*x^3*Pol yLog[2, E^(I*ArcSin[a*x])])/x^3
Time = 0.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5138, 5204, 15, 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.
\(\displaystyle \int \frac {\arcsin (a x)^2}{x^4} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle \frac {2}{3} a \int \frac {\arcsin (a x)}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\arcsin (a x)^2}{3 x^3}\) |
\(\Big \downarrow \) 5204 |
\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}}dx+\frac {1}{2} a \int \frac {1}{x^2}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}\right )-\frac {\arcsin (a x)^2}{3 x^3}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)}{x \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )-\frac {\arcsin (a x)^2}{3 x^3}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a^2 \int \frac {\arcsin (a x)}{a x}d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )-\frac {\arcsin (a x)^2}{3 x^3}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2}{3} a \left (\frac {1}{2} a^2 \int \arcsin (a x) \csc (\arcsin (a x))d\arcsin (a x)-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )-\frac {\arcsin (a x)^2}{3 x^3}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {\arcsin (a x)^2}{3 x^3}+\frac {2}{3} a \left (\frac {1}{2} a^2 \left (-\int \log \left (1-e^{i \arcsin (a x)}\right )d\arcsin (a x)+\int \log \left (1+e^{i \arcsin (a x)}\right )d\arcsin (a x)-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {\arcsin (a x)^2}{3 x^3}+\frac {2}{3} a \left (\frac {1}{2} a^2 \left (i \int e^{-i \arcsin (a x)} \log \left (1-e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-i \int e^{-i \arcsin (a x)} \log \left (1+e^{i \arcsin (a x)}\right )de^{i \arcsin (a x)}-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {\arcsin (a x)^2}{3 x^3}+\frac {2}{3} a \left (\frac {1}{2} a^2 \left (-2 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )\right )-\frac {\sqrt {1-a^2 x^2} \arcsin (a x)}{2 x^2}-\frac {a}{2 x}\right )\) |
-1/3*ArcSin[a*x]^2/x^3 + (2*a*(-1/2*a/x - (Sqrt[1 - a^2*x^2]*ArcSin[a*x])/ (2*x^2) + (a^2*(-2*ArcSin[a*x]*ArcTanh[E^(I*ArcSin[a*x])] + I*PolyLog[2, - E^(I*ArcSin[a*x])] - I*PolyLog[2, E^(I*ArcSin[a*x])]))/2))/3
3.1.20.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] + (Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] - Simp[b* c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*( 1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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 = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(a^{3} \left (-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}\right )\) | \(149\) |
default | \(a^{3} \left (-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}\right )\) | \(149\) |
a^3*(-1/3*(arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x+arcsin(a*x)^2+a^2*x^2)/a^3/x ^3+1/3*arcsin(a*x)*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))-1/3*I*polylog(2,I*a*x+(- a^2*x^2+1)^(1/2))-1/3*arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))+1/3*I*pol ylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))
\[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=\int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{4}} \,d x } \]
-1/3*(6*a*x^3*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt (a*x + 1)*sqrt(-a*x + 1))/(a^2*x^5 - x^3), x) + arctan2(a*x, sqrt(a*x + 1) *sqrt(-a*x + 1))^2)/x^3
\[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=\int { \frac {\arcsin \left (a x\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x^4} \,d x \]