\(\int \frac {1}{\arcsin (a x)^4} \, dx\) [71]

Optimal result

Integrand size = 6, antiderivative size = 78 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{6 a} \]

[Out]

1/6*x/arcsin(a*x)^2+1/6*Si(arcsin(a*x))/a-1/3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3+1/6*(-a^2*x^2+1)^(1/2)/a/arcs
in(a*x)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4717, 4807, 4809, 3380} \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}-\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {\text {Si}(\arcsin (a x))}{6 a}+\frac {x}{6 \arcsin (a x)^2} \]

[In]

Int[ArcSin[a*x]^(-4),x]

[Out]

-1/3*Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x]^3) + x/(6*ArcSin[a*x]^2) + Sqrt[1 - a^2*x^2]/(6*a*ArcSin[a*x]) + SinInte
gral[ArcSin[a*x]]/(6*a)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4717

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 - c^2*x^2]*((a + b*ArcSin[c*x])^(n + 1)/
(b*c*(n + 1))), x] + Dist[c/(b*(n + 1)), Int[x*((a + b*ArcSin[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4807

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] - Dist[f*(m/(b
*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /;
FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*c
^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x],
x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2, 0] && IGtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {1}{3} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}-\frac {1}{6} \int \frac {1}{\arcsin (a x)^2} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {1}{6} a \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx \\ & = -\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{6 a} \\ & = -\frac {\sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {x}{6 \arcsin (a x)^2}+\frac {\sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{6 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\frac {-2 \sqrt {1-a^2 x^2}+a x \arcsin (a x)+\sqrt {1-a^2 x^2} \arcsin (a x)^2+\arcsin (a x)^3 \text {Si}(\arcsin (a x))}{6 a \arcsin (a x)^3} \]

[In]

Integrate[ArcSin[a*x]^(-4),x]

[Out]

(-2*Sqrt[1 - a^2*x^2] + a*x*ArcSin[a*x] + Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + ArcSin[a*x]^3*SinIntegral[ArcSin[a
*x]])/(6*a*ArcSin[a*x]^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arcsin \left (a x \right )^{3}}+\frac {a x}{6 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{6}}{a}\)	\(63\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{3 \arcsin \left (a x \right )^{3}}+\frac {a x}{6 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{6 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{6}}{a}\)	\(63\)

[In]

int(1/arcsin(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/3/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/6*a*x/arcsin(a*x)^2+1/6/arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/6*Si(arc
sin(a*x)))

Fricas [F]

\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^4,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^(-4), x)

Sympy [F]

\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int \frac {1}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(1/asin(a*x)**4,x)

[Out]

Integral(asin(a*x)**(-4), x)

Maxima [F]

\[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int { \frac {1}{\arcsin \left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/arcsin(a*x)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{6 \, a} + \frac {x}{6 \, \arcsin \left (a x\right )^{2}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{6 \, a \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a \arcsin \left (a x\right )^{3}} \]

[In]

integrate(1/arcsin(a*x)^4,x, algorithm="giac")

[Out]

1/6*sin_integral(arcsin(a*x))/a + 1/6*x/arcsin(a*x)^2 + 1/6*sqrt(-a^2*x^2 + 1)/(a*arcsin(a*x)) - 1/3*sqrt(-a^2
*x^2 + 1)/(a*arcsin(a*x)^3)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arcsin (a x)^4} \, dx=\int \frac {1}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]

[In]

int(1/asin(a*x)^4,x)

[Out]

int(1/asin(a*x)^4, x)