\(\int \frac {x^4}{\arcsin (a x)^4} \, dx\) [67]

Optimal result

Integrand size = 10, antiderivative size = 158 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{48 a^5}-\frac {27 \text {Si}(3 \arcsin (a x))}{32 a^5}+\frac {125 \text {Si}(5 \arcsin (a x))}{96 a^5} \]

[Out]

-2/3*x^3/a^2/arcsin(a*x)^2+5/6*x^5/arcsin(a*x)^2+1/48*Si(arcsin(a*x))/a^5-27/32*Si(3*arcsin(a*x))/a^5+125/96*S
i(5*arcsin(a*x))/a^5-1/3*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^3-2*x^2*(-a^2*x^2+1)^(1/2)/a^3/arcsin(a*x)+25/6*
x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4729, 4807, 4727, 3380} \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=\frac {\text {Si}(\arcsin (a x))}{48 a^5}-\frac {27 \text {Si}(3 \arcsin (a x))}{32 a^5}+\frac {125 \text {Si}(5 \arcsin (a x))}{96 a^5}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}-\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {5 x^5}{6 \arcsin (a x)^2} \]

[In]

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

[Out]

-1/3*(x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^3) - (2*x^3)/(3*a^2*ArcSin[a*x]^2) + (5*x^5)/(6*ArcSin[a*x]^2) - (
2*x^2*Sqrt[1 - a^2*x^2])/(a^3*ArcSin[a*x]) + (25*x^4*Sqrt[1 - a^2*x^2])/(6*a*ArcSin[a*x]) + SinIntegral[ArcSin
[a*x]]/(48*a^5) - (27*SinIntegral[3*ArcSin[a*x]])/(32*a^5) + (125*SinIntegral[5*ArcSin[a*x]])/(96*a^5)

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 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^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}+\frac {4 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx}{3 a}-\frac {1}{3} (5 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arcsin (a x)^3} \, dx \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {25}{6} \int \frac {x^4}{\arcsin (a x)^2} \, dx+\frac {2 \int \frac {x^2}{\arcsin (a x)^2} \, dx}{a^2} \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {2 \text {Subst}\left (\int \left (-\frac {\sin (x)}{4 x}+\frac {3 \sin (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^5}-\frac {25 \text {Subst}\left (\int \left (-\frac {\sin (x)}{8 x}+\frac {9 \sin (3 x)}{16 x}-\frac {5 \sin (5 x)}{16 x}\right ) \, dx,x,\arcsin (a x)\right )}{6 a^5} \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {25 x^4 \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 )}{2 a^5}+\frac {25 \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{48 a^5}+\frac {125 \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\arcsin (a x)\right )}{96 a^5}+\frac {3 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^5}-\frac {75 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{32 a^5} \\ & = -\frac {x^4 \sqrt {1-a^2 x^2}}{3 a \arcsin (a x)^3}-\frac {2 x^3}{3 a^2 \arcsin (a x)^2}+\frac {5 x^5}{6 \arcsin (a x)^2}-\frac {2 x^2 \sqrt {1-a^2 x^2}}{a^3 \arcsin (a x)}+\frac {25 x^4 \sqrt {1-a^2 x^2}}{6 a \arcsin (a x)}+\frac {\text {Si}(\arcsin (a x))}{48 a^5}-\frac {27 \text {Si}(3 \arcsin (a x))}{32 a^5}+\frac {125 \text {Si}(5 \arcsin (a x))}{96 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=\frac {-32 a^4 x^4 \sqrt {1-a^2 x^2}-64 a^3 x^3 \arcsin (a x)+80 a^5 x^5 \arcsin (a x)-192 a^2 x^2 \sqrt {1-a^2 x^2} \arcsin (a x)^2+400 a^4 x^4 \sqrt {1-a^2 x^2} \arcsin (a x)^2+2 \arcsin (a x)^3 \text {Si}(\arcsin (a x))-81 \arcsin (a x)^3 \text {Si}(3 \arcsin (a x))+125 \arcsin (a x)^3 \text {Si}(5 \arcsin (a x))}{96 a^5 \arcsin (a x)^3} \]

[In]

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

[Out]

(-32*a^4*x^4*Sqrt[1 - a^2*x^2] - 64*a^3*x^3*ArcSin[a*x] + 80*a^5*x^5*ArcSin[a*x] - 192*a^2*x^2*Sqrt[1 - a^2*x^
2]*ArcSin[a*x]^2 + 400*a^4*x^4*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2 + 2*ArcSin[a*x]^3*SinIntegral[ArcSin[a*x]] - 81
*ArcSin[a*x]^3*SinIntegral[3*ArcSin[a*x]] + 125*ArcSin[a*x]^3*SinIntegral[5*ArcSin[a*x]])/(96*a^5*ArcSin[a*x]^
3)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )^{3}}+\frac {a x}{48 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{48}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )^{3}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{48 \arcsin \left (a x \right )^{3}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )^{2}}+\frac {25 \cos \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )}+\frac {125 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{24 \arcsin \left (a x \right )^{3}}+\frac {a x}{48 \arcsin \left (a x \right )^{2}}+\frac {\sqrt {-a^{2} x^{2}+1}}{48 \arcsin \left (a x \right )}+\frac {\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{48}+\frac {\cos \left (3 \arcsin \left (a x \right )\right )}{16 \arcsin \left (a x \right )^{3}}-\frac {3 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{48 \arcsin \left (a x \right )^{3}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )^{2}}+\frac {25 \cos \left (5 \arcsin \left (a x \right )\right )}{96 \arcsin \left (a x \right )}+\frac {125 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{96}}{a^{5}}\)	\(171\)

[In]

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

[Out]

1/a^5*(-1/24/arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)+1/48*a*x/arcsin(a*x)^2+1/48/arcsin(a*x)*(-a^2*x^2+1)^(1/2)+1/48*
Si(arcsin(a*x))+1/16/arcsin(a*x)^3*cos(3*arcsin(a*x))-3/32/arcsin(a*x)^2*sin(3*arcsin(a*x))-9/32/arcsin(a*x)*c
os(3*arcsin(a*x))-27/32*Si(3*arcsin(a*x))-1/48/arcsin(a*x)^3*cos(5*arcsin(a*x))+5/96/arcsin(a*x)^2*sin(5*arcsi
n(a*x))+25/96/arcsin(a*x)*cos(5*arcsin(a*x))+125/96*Si(5*arcsin(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^4/arcsin(a*x)^4, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**4/asin(a*x)**4, x)

Maxima [F]

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

[In]

integrate(x^4/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*(125*a^4*x^5 - 136*a^2*x^3 + 24*x)*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^4 - (25*a
^2*x^4 - 12*x^2)*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)*sqrt(a*x + 1)*sqrt(-a*x + 1) - (5*a^3*x^5 - 4*a
*x^3)*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 = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arcsin (a x)^4} \, dx=\frac {5 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} + \frac {25 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac {{\left (a^{2} x^{2} - 1\right )} x}{a^{4} \arcsin \left (a x\right )^{2}} + \frac {125 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{96 \, a^{5}} - \frac {27 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} + \frac {\operatorname {Si}\left (\arcsin \left (a x\right )\right )}{48 \, a^{5}} - \frac {19 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )} + \frac {x}{6 \, a^{4} \arcsin \left (a x\right )^{2}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} + \frac {13 \, \sqrt {-a^{2} x^{2} + 1}}{6 \, a^{5} \arcsin \left (a x\right )} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{3 \, a^{5} \arcsin \left (a x\right )^{3}} \]

[In]

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

[Out]

5/6*(a^2*x^2 - 1)^2*x/(a^4*arcsin(a*x)^2) + 25/6*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)) + (a^2*x
^2 - 1)*x/(a^4*arcsin(a*x)^2) + 125/96*sin_integral(5*arcsin(a*x))/a^5 - 27/32*sin_integral(3*arcsin(a*x))/a^5
+ 1/48*sin_integral(arcsin(a*x))/a^5 - 19/3*(-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)) + 1/6*x/(a^4*arcsin(a*x)^2
) - 1/3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^3) + 13/6*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)) + 2
/3*(-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)^3) - 1/3*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^3)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^4/asin(a*x)^4, x)