\(\int \frac {\arcsin (a x)^3}{x^5} \, dx\) [31]

Optimal result

Integrand size = 10, antiderivative size = 169 \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \]

[Out]

-1/4*a^2*arcsin(a*x)/x^2-1/2*I*a^4*arcsin(a*x)^2-1/4*arcsin(a*x)^3/x^4+a^4*arcsin(a*x)*ln(1-(I*a*x+(-a^2*x^2+1
)^(1/2))^2)-1/2*I*a^4*polylog(2,(I*a*x+(-a^2*x^2+1)^(1/2))^2)-1/4*a^3*(-a^2*x^2+1)^(1/2)/x-1/4*a*arcsin(a*x)^2
*(-a^2*x^2+1)^(1/2)/x^3-1/2*a^3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)/x

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {4723, 4789, 4771, 4721, 3798, 2221, 2317, 2438, 270} \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i a^4 \arcsin (a x)^2+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {\arcsin (a x)^3}{4 x^4} \]

[In]

Int[ArcSin[a*x]^3/x^5,x]

[Out]

-1/4*(a^3*Sqrt[1 - a^2*x^2])/x - (a^2*ArcSin[a*x])/(4*x^2) - (I/2)*a^4*ArcSin[a*x]^2 - (a*Sqrt[1 - a^2*x^2]*Ar
cSin[a*x]^2)/(4*x^3) - (a^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(2*x) - ArcSin[a*x]^3/(4*x^4) + a^4*ArcSin[a*x]*L
og[1 - E^((2*I)*ArcSin[a*x])] - (I/2)*a^4*PolyLog[2, E^((2*I)*ArcSin[a*x])]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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 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 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 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 4771

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[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, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\arcsin (a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\arcsin (a x)^2}{x^4 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {\arcsin (a x)^3}{4 x^4}+\frac {1}{2} a^2 \int \frac {\arcsin (a x)}{x^3} \, dx+\frac {1}{2} a^3 \int \frac {\arcsin (a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+a^4 \int \frac {\arcsin (a x)}{x} \, dx \\ & = -\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \text {Subst}(\int x \cot (x) \, dx,x,\arcsin (a x)) \\ & = -\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}-\left (2 i a^4\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-a^4 \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arcsin (a x)\right ) \\ & = -\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )+\frac {1}{2} \left (i a^4\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arcsin (a x)}\right ) \\ & = -\frac {a^3 \sqrt {1-a^2 x^2}}{4 x}-\frac {a^2 \arcsin (a x)}{4 x^2}-\frac {1}{2} i a^4 \arcsin (a x)^2-\frac {a \sqrt {1-a^2 x^2} \arcsin (a x)^2}{4 x^3}-\frac {a^3 \sqrt {1-a^2 x^2} \arcsin (a x)^2}{2 x}-\frac {\arcsin (a x)^3}{4 x^4}+a^4 \arcsin (a x) \log \left (1-e^{2 i \arcsin (a x)}\right )-\frac {1}{2} i a^4 \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.69 \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\frac {1}{4} \left (-\frac {\arcsin (a x)^3}{x^4}+a^4 \left (-\frac {\sqrt {1-a^2 x^2} \left (1+\left (2+\frac {1}{a^2 x^2}\right ) \arcsin (a x)^2\right )}{a x}-\arcsin (a x) \left (\frac {1}{a^2 x^2}+2 i \arcsin (a x)-4 \log \left (1-e^{2 i \arcsin (a x)}\right )\right )-2 i \operatorname {PolyLog}\left (2,e^{2 i \arcsin (a x)}\right )\right )\right ) \]

[In]

Integrate[ArcSin[a*x]^3/x^5,x]

[Out]

(-(ArcSin[a*x]^3/x^4) + a^4*(-((Sqrt[1 - a^2*x^2]*(1 + (2 + 1/(a^2*x^2))*ArcSin[a*x]^2))/(a*x)) - ArcSin[a*x]*
(1/(a^2*x^2) + (2*I)*ArcSin[a*x] - 4*Log[1 - E^((2*I)*ArcSin[a*x])]) - (2*I)*PolyLog[2, E^((2*I)*ArcSin[a*x])]
))/4

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(a^{4} \left (-\frac {-2 i \arcsin \left (a x \right )^{2} a^{4} x^{4}+2 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}+\arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )^{3}+a^{2} x^{2} \arcsin \left (a x \right )}{4 a^{4} x^{4}}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right )^{2}-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(231\)
default	\(a^{4} \left (-\frac {-2 i \arcsin \left (a x \right )^{2} a^{4} x^{4}+2 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-i a^{4} x^{4}+\arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x +a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )^{3}+a^{2} x^{2} \arcsin \left (a x \right )}{4 a^{4} x^{4}}+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \arcsin \left (a x \right )^{2}-i \operatorname {polylog}\left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )\right )\)	\(231\)

[In]

int(arcsin(a*x)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

a^4*(-1/4*(-2*I*arcsin(a*x)^2*a^4*x^4+2*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*x^3-I*a^4*x^4+arcsin(a*x)^2*(-a^2
*x^2+1)^(1/2)*a*x+a^3*x^3*(-a^2*x^2+1)^(1/2)+arcsin(a*x)^3+a^2*x^2*arcsin(a*x))/a^4/x^4+arcsin(a*x)*ln(1-I*a*x
-(-a^2*x^2+1)^(1/2))+arcsin(a*x)*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))-I*arcsin(a*x)^2-I*polylog(2,I*a*x+(-a^2*x^2+1)
^(1/2))-I*polylog(2,-I*a*x-(-a^2*x^2+1)^(1/2)))

Fricas [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{5}} \,d x } \]

[In]

integrate(arcsin(a*x)^3/x^5,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^3/x^5, x)

Sympy [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{5}}\, dx \]

[In]

integrate(asin(a*x)**3/x**5,x)

[Out]

Integral(asin(a*x)**3/x**5, x)

Maxima [F]

\[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int { \frac {\arcsin \left (a x\right )^{3}}{x^{5}} \,d x } \]

[In]

integrate(arcsin(a*x)^3/x^5,x, algorithm="maxima")

[Out]

-1/4*(12*a*x^4*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2/(a^2*x^
6 - x^4), x) + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)/x^4

Giac [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(arcsin(a*x)^3/x^5,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {\arcsin (a x)^3}{x^5} \, dx=\int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^5} \,d x \]

[In]

int(asin(a*x)^3/x^5,x)

[Out]

int(asin(a*x)^3/x^5, x)