\(\int \frac {x^2}{\arcsin (a x)^4} \, dx\) [69]

Optimal result

Integrand size = 10, antiderivative size = 141 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{24 a^3}-\frac {9 \text {Si}(3 \arcsin (a x))}{8 a^3} \]

[Out]

-1/3*x/a^2/arcsin(a*x)^2+1/2*x^3/arcsin(a*x)^2+1/24*Si(arcsin(a*x))/a^3-9/8*Si(3*arcsin(a*x))/a^3-1/3*x^2*(-a^
2*x^2+1)^(1/2)/a/arcsin(a*x)^3-1/3*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+3/2*x^2*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4729, 4807, 4727, 3380, 4717, 4809} \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\frac {\text {Si}(\arcsin (a x))}{24 a^3}-\frac {9 \text {Si}(3 \arcsin (a x))}{8 a^3}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}-\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {x^3}{2 \arcsin (a x)^2} \]

[In]

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

[Out]

-1/3*(x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) - x/(3*a^2*ArcSin[a*x]^2) + x^3/(2*ArcSin[a*x]^2) - Sqrt[1 - a^
2*x^2]/(3*a^3*ArcSin[a*x]) + (3*x^2*Sqrt[1 - a^2*x^2])/(2*a*ArcSin[a*x]) + SinIntegral[ArcSin[a*x]]/(24*a^3) -
(9*SinIntegral[3*ArcSin[a*x]])/(8*a^3)

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 4727

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 - c^2*x^2]*((a + b*ArcSin
[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[
-a/b + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x]
&& IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 4729

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

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 {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {2 \int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx}{3 a}-a \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {3}{2} \int \frac {x^2}{\arcsin (a x)^2} \, dx+\frac {\int \frac {1}{\arcsin (a x)^2} \, dx}{3 a^2} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}-\frac {3 \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{2 a^3}-\frac {\int \frac {x}{\sqrt {1-a^2 x^2} \arcsin (a x)} \, dx}{3 a} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}-\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{3 a^3}+\frac {3 \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^3}-\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{8 a^3} \\ & = -\frac {x^2 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x}{3 a^2 \arcsin (a x)^2}+\frac {x^3}{2 \arcsin (a x)^2}-\frac {\sqrt {1-a^2 x^2}}{3 a^3 \arcsin (a x)}+\frac {3 x^2 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{24 a^3}-\frac {9 \text {Si}(3 \arcsin (a x))}{8 a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\frac {-\frac {8 a^2 x^2 \sqrt {1-a^2 x^2}}{\arcsin (a x)^3}+\frac {4 a x \left (-2+3 a^2 x^2\right )}{\arcsin (a x)^2}+\frac {4 \sqrt {1-a^2 x^2} \left (-2+9 a^2 x^2\right )}{\arcsin (a x)}-80 \text {Si}(\arcsin (a x))-27 (-3 \text {Si}(\arcsin (a x))+\text {Si}(3 \arcsin (a x)))}{24 a^3} \]

[In]

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

[Out]

((-8*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcSin[a*x]^3 + (4*a*x*(-2 + 3*a^2*x^2))/ArcSin[a*x]^2 + (4*Sqrt[1 - a^2*x^2]*
(-2 + 9*a^2*x^2))/ArcSin[a*x] - 80*SinIntegral[ArcSin[a*x]] - 27*(-3*SinIntegral[ArcSin[a*x]] + SinIntegral[3*
ArcSin[a*x]]))/(24*a^3)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arcsin \left (a x \right )^{3}}+\frac {a x}{24 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{24}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{12 \arcsin \left (a x \right )^{3}}+\frac {a x}{24 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{24}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\sin \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )^{2}}-\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {9 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{8}}{a^{3}}\)	\(117\)

[In]

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

[Out]

1/a^3*(-1/12/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/24*a*x/arcsin(a*x)^2+1/24/arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/24*
Si(arcsin(a*x))+1/12/arcsin(a*x)^3*cos(3*arcsin(a*x))-1/8/arcsin(a*x)^2*sin(3*arcsin(a*x))-3/8/arcsin(a*x)*cos
(3*arcsin(a*x))-9/8*Si(3*arcsin(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^2/arcsin(a*x)^4, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/asin(a*x)**4, x)

Maxima [F]

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

[In]

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

[Out]

-1/6*(6*a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3*integrate(1/6*(27*a^2*x^3 - 20*x)*sqrt(a*x + 1)*sqrt(
-a*x + 1)/((a^3*x^2 - a)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + (2*a^2*x^2 - (9*a^2*x^2 - 2)*arctan
2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (3*a^3*x^3 - 2*a*x)*arctan2(a*x, sqrt(a
*x + 1)*sqrt(-a*x + 1)))/(a^3*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\arcsin (a x)^4} \, dx=\frac {{\left (a^{2} x^{2} - 1\right )} x}{2 \, a^{2} \arcsin \left (a x\right )^{2}} - \frac {9 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{8 \, a^{3}} + \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{24 \, a^{3}} - \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 \, a^{3} \arcsin \left (a x\right )} + \frac {x}{6 \, a^{2} \arcsin \left (a x\right )^{2}} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{3} \arcsin \left (a x\right )} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3} \arcsin \left (a x\right )^{3}} \]

[In]

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

[Out]

1/2*(a^2*x^2 - 1)*x/(a^2*arcsin(a*x)^2) - 9/8*sin_integral(3*arcsin(a*x))/a^3 + 1/24*sin_integral(arcsin(a*x))
/a^3 - 3/2*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)) + 1/6*x/(a^2*arcsin(a*x)^2) + 7/6*sqrt(-a^2*x^2 + 1)/(a^3*ar
csin(a*x)) + 1/3*(-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^2 + 1)/(a^3*arcsin(a*x)^3)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/asin(a*x)^4, x)