\(\int \frac {x^4}{\arcsin (a x)^3} \, dx\) [60]

Optimal result

Integrand size = 10, antiderivative size = 98 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}-\frac {2 x^3}{a^2 \arcsin (a x)}+\frac {5 x^5}{2 \arcsin (a x)}-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{16 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arcsin (a x))}{32 a^5}-\frac {25 \operatorname {CosIntegral}(5 \arcsin (a x))}{32 a^5} \]

[Out]

-2*x^3/a^2/arcsin(a*x)+5/2*x^5/arcsin(a*x)-1/16*Ci(arcsin(a*x))/a^5+27/32*Ci(3*arcsin(a*x))/a^5-25/32*Ci(5*arc
sin(a*x))/a^5-1/2*x^4*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)^2

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4729, 4807, 4731, 4491, 3383} \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=-\frac {\operatorname {CosIntegral}(\arcsin (a x))}{16 a^5}+\frac {27 \operatorname {CosIntegral}(3 \arcsin (a x))}{32 a^5}-\frac {25 \operatorname {CosIntegral}(5 \arcsin (a x))}{32 a^5}-\frac {2 x^3}{a^2 \arcsin (a x)}-\frac {x^4 \sqrt {1-a^2 x^2}}{2 a \arcsin (a x)^2}+\frac {5 x^5}{2 \arcsin (a x)} \]

[In]

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

[Out]

-1/2*(x^4*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x]^2) - (2*x^3)/(a^2*ArcSin[a*x]) + (5*x^5)/(2*ArcSin[a*x]) - CosInte
gral[ArcSin[a*x]]/(16*a^5) + (27*CosIntegral[3*ArcSin[a*x]])/(32*a^5) - (25*CosIntegral[5*ArcSin[a*x]])/(32*a^
5)

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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

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 4731

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=-\frac {16 a^4 x^4 \sqrt {1-a^2 x^2}+64 a^3 x^3 \arcsin (a x)-80 a^5 x^5 \arcsin (a x)+2 \arcsin (a x)^2 \operatorname {CosIntegral}(\arcsin (a x))-27 \arcsin (a x)^2 \operatorname {CosIntegral}(3 \arcsin (a x))+25 \arcsin (a x)^2 \operatorname {CosIntegral}(5 \arcsin (a x))}{32 a^5 \arcsin (a x)^2} \]

[In]

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

[Out]

-1/32*(16*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] + 2*ArcSin[a*x]^2*CosInt
egral[ArcSin[a*x]] - 27*ArcSin[a*x]^2*CosIntegral[3*ArcSin[a*x]] + 25*ArcSin[a*x]^2*CosIntegral[5*ArcSin[a*x]]
)/(a^5*ArcSin[a*x]^2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{16 \arcsin \left (a x \right )^{2}}+\frac {a x}{16 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{16}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{32}}{a^{5}}\)	\(121\)
default	\(\frac {-\frac {\sqrt {-a^{2} x^{2}+1}}{16 \arcsin \left (a x \right )^{2}}+\frac {a x}{16 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (\arcsin \left (a x \right )\right )}{16}+\frac {3 \cos \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}-\frac {9 \sin \left (3 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Ci}\left (3 \arcsin \left (a x \right )\right )}{32}-\frac {\cos \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )^{2}}+\frac {5 \sin \left (5 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Ci}\left (5 \arcsin \left (a x \right )\right )}{32}}{a^{5}}\)	\(121\)

[In]

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

[Out]

1/a^5*(-1/16/arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)+1/16*a*x/arcsin(a*x)-1/16*Ci(arcsin(a*x))+3/32/arcsin(a*x)^2*cos
(3*arcsin(a*x))-9/32/arcsin(a*x)*sin(3*arcsin(a*x))+27/32*Ci(3*arcsin(a*x))-1/32/arcsin(a*x)^2*cos(5*arcsin(a*
x))+5/32/arcsin(a*x)*sin(5*arcsin(a*x))-25/32*Ci(5*arcsin(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^4/arcsin(a*x)^3, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**4/asin(a*x)**3, x)

Maxima [F]

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

[In]

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

[Out]

-1/2*(sqrt(a*x + 1)*sqrt(-a*x + 1)*a*x^4 + arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2*integrate((25*a^2*x^4
- 12*x^2)/arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)), x) - (5*a^2*x^5 - 4*x^3)*arctan2(a*x, sqrt(a*x + 1)*sqrt
(-a*x + 1)))/(a^2*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.73 \[ \int \frac {x^4}{\arcsin (a x)^3} \, dx=\frac {5 \, {\left (a^{2} x^{2} - 1\right )}^{2} x}{2 \, a^{4} \arcsin \left (a x\right )} + \frac {3 \, {\left (a^{2} x^{2} - 1\right )} x}{a^{4} \arcsin \left (a x\right )} + \frac {x}{2 \, a^{4} \arcsin \left (a x\right )} - \frac {25 \, \operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} + \frac {27 \, \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{32 \, a^{5}} - \frac {\operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{16 \, a^{5}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{5} \arcsin \left (a x\right )^{2}} + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} \arcsin \left (a x\right )^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{2 \, a^{5} \arcsin \left (a x\right )^{2}} \]

[In]

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

[Out]

5/2*(a^2*x^2 - 1)^2*x/(a^4*arcsin(a*x)) + 3*(a^2*x^2 - 1)*x/(a^4*arcsin(a*x)) + 1/2*x/(a^4*arcsin(a*x)) - 25/3
2*cos_integral(5*arcsin(a*x))/a^5 + 27/32*cos_integral(3*arcsin(a*x))/a^5 - 1/16*cos_integral(arcsin(a*x))/a^5
- 1/2*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^2) + (-a^2*x^2 + 1)^(3/2)/(a^5*arcsin(a*x)^2) - 1/2
*sqrt(-a^2*x^2 + 1)/(a^5*arcsin(a*x)^2)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^4/asin(a*x)^3, x)