\(\int \frac {x^5}{\arcsin (a x)^2} \, dx\) [52]

Optimal result

Integrand size = 10, antiderivative size = 71 \[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=-\frac {x^5 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {5 \operatorname {CosIntegral}(2 \arcsin (a x))}{16 a^6}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^6}+\frac {3 \operatorname {CosIntegral}(6 \arcsin (a x))}{16 a^6} \]

[Out]

5/16*Ci(2*arcsin(a*x))/a^6-1/2*Ci(4*arcsin(a*x))/a^6+3/16*Ci(6*arcsin(a*x))/a^6-x^5*(-a^2*x^2+1)^(1/2)/a/arcsi
n(a*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4727, 3383} \[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=\frac {5 \operatorname {CosIntegral}(2 \arcsin (a x))}{16 a^6}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^6}+\frac {3 \operatorname {CosIntegral}(6 \arcsin (a x))}{16 a^6}-\frac {x^5 \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]

[In]

Int[x^5/ArcSin[a*x]^2,x]

[Out]

-((x^5*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + (5*CosIntegral[2*ArcSin[a*x]])/(16*a^6) - CosIntegral[4*ArcSin[a*
x]]/(2*a^6) + (3*CosIntegral[6*ArcSin[a*x]])/(16*a^6)

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^5 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \left (\frac {5 \cos (2 x)}{16 x}-\frac {\cos (4 x)}{2 x}+\frac {3 \cos (6 x)}{16 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^6} \\ & = -\frac {x^5 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {3 \text {Subst}\left (\int \frac {\cos (6 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^6}+\frac {5 \text {Subst}\left (\int \frac {\cos (2 x)}{x} \, dx,x,\arcsin (a x)\right )}{16 a^6}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{x} \, dx,x,\arcsin (a x)\right )}{2 a^6} \\ & = -\frac {x^5 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {5 \operatorname {CosIntegral}(2 \arcsin (a x))}{16 a^6}-\frac {\operatorname {CosIntegral}(4 \arcsin (a x))}{2 a^6}+\frac {3 \operatorname {CosIntegral}(6 \arcsin (a x))}{16 a^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10 \[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=-\frac {-10 \arcsin (a x) \operatorname {CosIntegral}(2 \arcsin (a x))+16 \arcsin (a x) \operatorname {CosIntegral}(4 \arcsin (a x))-6 \arcsin (a x) \operatorname {CosIntegral}(6 \arcsin (a x))+5 \sin (2 \arcsin (a x))-4 \sin (4 \arcsin (a x))+\sin (6 \arcsin (a x))}{32 a^6 \arcsin (a x)} \]

[In]

Integrate[x^5/ArcSin[a*x]^2,x]

[Out]

-1/32*(-10*ArcSin[a*x]*CosIntegral[2*ArcSin[a*x]] + 16*ArcSin[a*x]*CosIntegral[4*ArcSin[a*x]] - 6*ArcSin[a*x]*
CosIntegral[6*ArcSin[a*x]] + 5*Sin[2*ArcSin[a*x]] - 4*Sin[4*ArcSin[a*x]] + Sin[6*ArcSin[a*x]])/(a^6*ArcSin[a*x
])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {-\frac {5 \sin \left (2 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {5 \,\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{2}-\frac {\sin \left (6 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {3 \,\operatorname {Ci}\left (6 \arcsin \left (a x \right )\right )}{16}}{a^{6}}\)	\(78\)
default	\(\frac {-\frac {5 \sin \left (2 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {5 \,\operatorname {Ci}\left (2 \arcsin \left (a x \right )\right )}{16}+\frac {\sin \left (4 \arcsin \left (a x \right )\right )}{8 \arcsin \left (a x \right )}-\frac {\operatorname {Ci}\left (4 \arcsin \left (a x \right )\right )}{2}-\frac {\sin \left (6 \arcsin \left (a x \right )\right )}{32 \arcsin \left (a x \right )}+\frac {3 \,\operatorname {Ci}\left (6 \arcsin \left (a x \right )\right )}{16}}{a^{6}}\)	\(78\)

[In]

int(x^5/arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^6*(-5/32/arcsin(a*x)*sin(2*arcsin(a*x))+5/16*Ci(2*arcsin(a*x))+1/8/arcsin(a*x)*sin(4*arcsin(a*x))-1/2*Ci(4
*arcsin(a*x))-1/32/arcsin(a*x)*sin(6*arcsin(a*x))+3/16*Ci(6*arcsin(a*x)))

Fricas [F]

\[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=\int { \frac {x^{5}}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^5/arcsin(a*x)^2, x)

Sympy [F]

\[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=\int \frac {x^{5}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**5/asin(a*x)**2,x)

[Out]

Integral(x**5/asin(a*x)**2, x)

Maxima [F]

\[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=\int { \frac {x^{5}}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

-(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^5 - a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate((6*a^2*x^6 - 5*x^4
)*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))/(a*arctan2(a*x,
sqrt(a*x + 1)*sqrt(-a*x + 1)))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.69 \[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=-\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{5} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{5} \arcsin \left (a x\right )} + \frac {3 \, \operatorname {Ci}\left (6 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} - \frac {\operatorname {Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{6}} + \frac {5 \, \operatorname {Ci}\left (2 \, \arcsin \left (a x\right )\right )}{16 \, a^{6}} \]

[In]

integrate(x^5/arcsin(a*x)^2,x, algorithm="giac")

[Out]

-(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x/(a^5*arcsin(a*x)) + 2*(-a^2*x^2 + 1)^(3/2)*x/(a^5*arcsin(a*x)) - sqrt(-a
^2*x^2 + 1)*x/(a^5*arcsin(a*x)) + 3/16*cos_integral(6*arcsin(a*x))/a^6 - 1/2*cos_integral(4*arcsin(a*x))/a^6 +
5/16*cos_integral(2*arcsin(a*x))/a^6

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5}{\arcsin (a x)^2} \, dx=\int \frac {x^5}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^5/asin(a*x)^2,x)

[Out]

int(x^5/asin(a*x)^2, x)