\(\int \frac {x^3}{\arcsin (a x)^4} \, dx\) [68]

Optimal result

Integrand size = 10, antiderivative size = 144 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{3 a^4}+\frac {4 \operatorname {CosIntegral}(4 \arcsin (a x))}{3 a^4} \]

[Out]

-1/2*x^2/a^2/arcsin(a*x)^2+2/3*x^4/arcsin(a*x)^2-1/3*Ci(2*arcsin(a*x))/a^4+4/3*Ci(4*arcsin(a*x))/a^4-1/3*x^3*(
-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-x*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+8/3*x^3*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x
)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4729, 4807, 4727, 3383} \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{3 a^4}+\frac {4 \operatorname {CosIntegral}(4 \arcsin (a x))}{3 a^4}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}-\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {2 x^4}{3 \arcsin (a x)^2} \]

[In]

Int[x^3/ArcSin[a*x]^4,x]

[Out]

-1/3*(x^3*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) - x^2/(2*a^2*ArcSin[a*x]^2) + (2*x^4)/(3*ArcSin[a*x]^2) - (x*Sq
rt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (8*x^3*Sqrt[1 - a^2*x^2])/(3*a*ArcSin[a*x]) - CosIntegral[2*ArcSin[a*x]]/
(3*a^4) + (4*CosIntegral[4*ArcSin[a*x]])/(3*a^4)

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx}{a}-\frac {1}{3} (4 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {8}{3} \int \frac {x^3}{\arcsin (a x)^2} \, dx+\frac {\int \frac {x}{\arcsin (a x)^2} \, dx}{a^2} \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{a^4}-\frac {8 \text {Subst}\left (\int \left (\frac {\cos (2 x)}{2 x}-\frac {\cos (4 x)}{2 x}\right ) \, dx,x,\arcsin (a x)\right )}{3 a^4} \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}+\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{a^4}-\frac {4 \text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{3 a^4}+\frac {4 \text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arcsin (a x)\right )}{3 a^4} \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {x^2}{2 a^2 \arcsin (a x)^2}+\frac {2 x^4}{3 \arcsin (a x)^2}-\frac {x \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {8 x^3 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)}-\frac {\operatorname {CosIntegral}(2 \arcsin (a x))}{3 a^4}+\frac {4 \operatorname {CosIntegral}(4 \arcsin (a x))}{3 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.74 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\frac {\frac {a x \left (-2 a^2 x^2 \sqrt {1-a^2 x^2}+a x \left (-3+4 a^2 x^2\right ) \arcsin (a x)+2 \sqrt {1-a^2 x^2} \left (-3+8 a^2 x^2\right ) \arcsin (a x)^2\right )}{\arcsin (a x)^3}-2 \operatorname {CosIntegral}(2 \arcsin (a x))+8 \operatorname {CosIntegral}(4 \arcsin (a x))}{6 a^4} \]

[In]

Integrate[x^3/ArcSin[a*x]^4,x]

[Out]

((a*x*(-2*a^2*x^2*Sqrt[1 - a^2*x^2] + a*x*(-3 + 4*a^2*x^2)*ArcSin[a*x] + 2*Sqrt[1 - a^2*x^2]*(-3 + 8*a^2*x^2)*
ArcSin[a*x]^2))/ArcSin[a*x]^3 - 2*CosIntegral[2*ArcSin[a*x]] + 8*CosIntegral[4*ArcSin[a*x]])/(6*a^4)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\)	\(114\)
default	\(\frac {-\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{3}}-\frac {\cos \left (2 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}+\frac {\sin \left (2 \arcsin \left (a x \right )\right )}{6 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{3}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{24 \arcsin \left (a x \right )^{3}}+\frac {\cos \left (4 \arcsin \left (a x \right )\right )}{12 \arcsin \left (a x \right )^{2}}-\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{3 \arcsin \left (a x \right )}+\frac {4 \,\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{3}}{a^{4}}\)	\(114\)

[In]

int(x^3/arcsin(a*x)^4,x,method=_RETURNVERBOSE)

[Out]

1/a^4*(-1/12/arcsin(a*x)^3*sin(2*arcsin(a*x))-1/12/arcsin(a*x)^2*cos(2*arcsin(a*x))+1/6/arcsin(a*x)*sin(2*arcs
in(a*x))-1/3*Ci(2*arcsin(a*x))+1/24/arcsin(a*x)^3*sin(4*arcsin(a*x))+1/12/arcsin(a*x)^2*cos(4*arcsin(a*x))-1/3
/arcsin(a*x)*sin(4*arcsin(a*x))+4/3*Ci(4*arcsin(a*x)))

Fricas [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \]

[In]

integrate(x^3/arcsin(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^3/arcsin(a*x)^4, x)

Sympy [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^{3}}{\operatorname {asin}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(x**3/asin(a*x)**4,x)

[Out]

Integral(x**3/asin(a*x)**4, x)

Maxima [F]

\[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int { \frac {x^{3}}{\arcsin \left (a x\right )^{4}} \,d x } \]

[In]

integrate(x^3/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(1/3*(32*a^4*x^4 - 30*a^2*x^2 + 3)*sqrt(a*x
+ 1)*sqrt(-a*x + 1)/((a^5*x^2 - a^3)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))), x) + 2*(a^2*x^3 - (8*a^2*x^3
- 3*x)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (4*a^3*x^4 - 3*a*x^2)*arc
tan2(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 = 174, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=-\frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )} + \frac {4 \, \operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} - \frac {\operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{3 \, a^{4}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {2 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{3 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{3 \, a^{3} \arcsin \left (a x\right )^{3}} + \frac {5 \, {\left (a^{2} x^{2} - 1\right )}}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac {1}{6 \, a^{4} \arcsin \left (a x\right )^{2}} \]

[In]

integrate(x^3/arcsin(a*x)^4,x, algorithm="giac")

[Out]

-8/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)) + 5/3*sqrt(-a^2*x^2 + 1)*x/(a^3*arcsin(a*x)) + 4/3*cos_integral(
4*arcsin(a*x))/a^4 - 1/3*cos_integral(2*arcsin(a*x))/a^4 + 1/3*(-a^2*x^2 + 1)^(3/2)*x/(a^3*arcsin(a*x)^3) + 2/
3*(a^2*x^2 - 1)^2/(a^4*arcsin(a*x)^2) - 1/3*sqrt(-a^2*x^2 + 1)*x/(a^3*arcsin(a*x)^3) + 5/6*(a^2*x^2 - 1)/(a^4*
arcsin(a*x)^2) + 1/6/(a^4*arcsin(a*x)^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\arcsin (a x)^4} \, dx=\int \frac {x^3}{{\mathrm {asin}\left (a\,x\right )}^4} \,d x \]

[In]

int(x^3/asin(a*x)^4,x)

[Out]

int(x^3/asin(a*x)^4, x)