\(\int \frac {x^6}{\arcsin (a x)^2} \, dx\) [51]

Optimal result

Integrand size = 10, antiderivative size = 83 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {5 \text {Si}(\arcsin (a x))}{64 a^7}+\frac {27 \text {Si}(3 \arcsin (a x))}{64 a^7}-\frac {25 \text {Si}(5 \arcsin (a x))}{64 a^7}+\frac {7 \text {Si}(7 \arcsin (a x))}{64 a^7} \]

[Out]

-5/64*Si(arcsin(a*x))/a^7+27/64*Si(3*arcsin(a*x))/a^7-25/64*Si(5*arcsin(a*x))/a^7+7/64*Si(7*arcsin(a*x))/a^7-x
^6*(-a^2*x^2+1)^(1/2)/a/arcsin(a*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4727, 3380} \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {5 \text {Si}(\arcsin (a x))}{64 a^7}+\frac {27 \text {Si}(3 \arcsin (a x))}{64 a^7}-\frac {25 \text {Si}(5 \arcsin (a x))}{64 a^7}+\frac {7 \text {Si}(7 \arcsin (a x))}{64 a^7}-\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)} \]

[In]

Int[x^6/ArcSin[a*x]^2,x]

[Out]

-((x^6*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - (5*SinIntegral[ArcSin[a*x]])/(64*a^7) + (27*SinIntegral[3*ArcSin[
a*x]])/(64*a^7) - (25*SinIntegral[5*ArcSin[a*x]])/(64*a^7) + (7*SinIntegral[7*ArcSin[a*x]])/(64*a^7)

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {5 \sin (x)}{64 x}+\frac {27 \sin (3 x)}{64 x}-\frac {25 \sin (5 x)}{64 x}+\frac {7 \sin (7 x)}{64 x}\right ) \, dx,x,\arcsin (a x)\right )}{a^7} \\ & = -\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{64 a^7}+\frac {7 \text {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\arcsin (a x)\right )}{64 a^7}-\frac {25 \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\arcsin (a x)\right )}{64 a^7}+\frac {27 \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{64 a^7} \\ & = -\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {5 \text {Si}(\arcsin (a x))}{64 a^7}+\frac {27 \text {Si}(3 \arcsin (a x))}{64 a^7}-\frac {25 \text {Si}(5 \arcsin (a x))}{64 a^7}+\frac {7 \text {Si}(7 \arcsin (a x))}{64 a^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \arcsin (a x) \text {Si}(\arcsin (a x))-27 \arcsin (a x) \text {Si}(3 \arcsin (a x))+25 \arcsin (a x) \text {Si}(5 \arcsin (a x))-7 \arcsin (a x) \text {Si}(7 \arcsin (a x))}{64 a^7 \arcsin (a x)} \]

[In]

Integrate[x^6/ArcSin[a*x]^2,x]

[Out]

-1/64*(64*a^6*x^6*Sqrt[1 - a^2*x^2] + 5*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] - 27*ArcSin[a*x]*SinIntegral[3*Ar
cSin[a*x]] + 25*ArcSin[a*x]*SinIntegral[5*ArcSin[a*x]] - 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]])/(a^7*ArcSin
[a*x])

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {-\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arcsin \left (a x \right )}-\frac {5 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{64}+\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{64}-\frac {5 \cos \left (5 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{64}+\frac {\cos \left (7 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {7 \,\operatorname {Si}\left (7 \arcsin \left (a x \right )\right )}{64}}{a^{7}}\)	\(105\)
default	\(\frac {-\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arcsin \left (a x \right )}-\frac {5 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{64}+\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{64}-\frac {5 \cos \left (5 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{64}+\frac {\cos \left (7 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {7 \,\operatorname {Si}\left (7 \arcsin \left (a x \right )\right )}{64}}{a^{7}}\)	\(105\)

[In]

int(x^6/arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/a^7*(-5/64/arcsin(a*x)*(-a^2*x^2+1)^(1/2)-5/64*Si(arcsin(a*x))+9/64/arcsin(a*x)*cos(3*arcsin(a*x))+27/64*Si(
3*arcsin(a*x))-5/64/arcsin(a*x)*cos(5*arcsin(a*x))-25/64*Si(5*arcsin(a*x))+1/64/arcsin(a*x)*cos(7*arcsin(a*x))
+7/64*Si(7*arcsin(a*x)))

Fricas [F]

\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^6/arcsin(a*x)^2, x)

Sympy [F]

\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int \frac {x^{6}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]

[In]

integrate(x**6/asin(a*x)**2,x)

[Out]

Integral(x**6/asin(a*x)**2, x)

Maxima [F]

\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

-(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^6 - a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate((7*a^2*x^7 - 6*x^5
)*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (73) = 146\).

Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.94 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} + \frac {7 \, \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {27 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{7} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} \]

[In]

integrate(x^6/arcsin(a*x)^2,x, algorithm="giac")

[Out]

-(a^2*x^2 - 1)^3*sqrt(-a^2*x^2 + 1)/(a^7*arcsin(a*x)) - 3*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)/(a^7*arcsin(a*x))
+ 7/64*sin_integral(7*arcsin(a*x))/a^7 - 25/64*sin_integral(5*arcsin(a*x))/a^7 + 27/64*sin_integral(3*arcsin(
a*x))/a^7 - 5/64*sin_integral(arcsin(a*x))/a^7 + 3*(-a^2*x^2 + 1)^(3/2)/(a^7*arcsin(a*x)) - sqrt(-a^2*x^2 + 1)
/(a^7*arcsin(a*x))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int \frac {x^6}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

[In]

int(x^6/asin(a*x)^2,x)

[Out]

int(x^6/asin(a*x)^2, x)