\(\int \frac {\arcsin (a x)^2}{x^4} \, dx\) [20]

Optimal result

Integrand size = 10, antiderivative size = 116 \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x^2}-\frac {\arcsin (a x)^2}{3 x^3}-\frac {2}{3} a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right ) \]

[Out]

-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*polylog(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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4723, 4789, 4803, 4268, 2317, 2438, 30} \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=-\frac {2}{3} a^3 \arcsin (a x) \text {arctanh}\left (e^{i \arcsin (a x)}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )-\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)}{3 x^2}-\frac {a^2}{3 x}-\frac {\arcsin (a x)^2}{3 x^3} \]

[In]

Int[ArcSin[a*x]^2/x^4,x]

[Out]

-1/3*a^2/x - (a*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(3*x^2) - ArcSin[a*x]^2/(3*x^3) - (2*a^3*ArcSin[a*x]*ArcTanh[E^
(I*ArcSin[a*x])])/3 + (I/3)*a^3*PolyLog[2, -E^(I*ArcSin[a*x])] - (I/3)*a^3*PolyLog[2, E^(I*ArcSin[a*x])]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

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 4789

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] + (Dist[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] - Dist[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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[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]

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

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {\arcsin (a x)^2}{x^4} \, dx=-\frac {a^2 x^2+a x \sqrt {1-a^2 x^2} \arcsin (a x)+\arcsin (a x)^2-a^3 x^3 \arcsin (a x) \log \left (1-e^{i \arcsin (a x)}\right )+a^3 x^3 \arcsin (a x) \log \left (1+e^{i \arcsin (a x)}\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (a x)}\right )+i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (a x)}\right )}{3 x^3} \]

[In]

Integrate[ArcSin[a*x]^2/x^4,x]

[Out]

-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*P
olyLog[2, E^(I*ArcSin[a*x])])/x^3

Maple [A] (verified)

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	\(a^{3} \left (-\frac {\arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x +\arcsin \left (a x \right )^{2}+a^{2} x^{2}}{3 a^{3} x^{3}}+\frac {\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}-\frac {\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )}{3}+\frac {i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )}{3}\right )\)	\(149\)

[In]

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

[Out]

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*p
olylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral(arcsin(a*x)^2/x^4, x)

Sympy [F]

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

[In]

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

[Out]

Integral(asin(a*x)**2/x**4, x)

Maxima [F]

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

[In]

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

[Out]

-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

Giac [F]

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

[In]

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

[Out]

integrate(arcsin(a*x)^2/x^4, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(asin(a*x)^2/x^4, x)