Integrand size = 10, antiderivative size = 83 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}-\frac {5 \text {Si}(\arcsin (a x))}{64 a^7}+\frac {27 \text {Si}(3 \arcsin (a x))}{64 a^7}-\frac {25 \text {Si}(5 \arcsin (a x))}{64 a^7}+\frac {7 \text {Si}(7 \arcsin (a x))}{64 a^7} \]
-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)
Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {64 a^6 x^6 \sqrt {1-a^2 x^2}+5 \arcsin (a x) \text {Si}(\arcsin (a x))-27 \arcsin (a x) \text {Si}(3 \arcsin (a x))+25 \arcsin (a x) \text {Si}(5 \arcsin (a x))-7 \arcsin (a x) \text {Si}(7 \arcsin (a x))}{64 a^7 \arcsin (a x)} \]
-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*ArcSin[a*x]] + 25*ArcSin[a*x]*SinIntegra l[5*ArcSin[a*x]] - 7*ArcSin[a*x]*SinIntegral[7*ArcSin[a*x]])/(a^7*ArcSin[a *x])
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5142, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^6}{\arcsin (a x)^2} \, dx\) |
\(\Big \downarrow \) 5142 |
\(\displaystyle \frac {\int \left (-\frac {5 a x}{64 \arcsin (a x)}+\frac {27 \sin (3 \arcsin (a x))}{64 \arcsin (a x)}-\frac {25 \sin (5 \arcsin (a x))}{64 \arcsin (a x)}+\frac {7 \sin (7 \arcsin (a x))}{64 \arcsin (a x)}\right )d\arcsin (a x)}{a^7}-\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {5}{64} \text {Si}(\arcsin (a x))+\frac {27}{64} \text {Si}(3 \arcsin (a x))-\frac {25}{64} \text {Si}(5 \arcsin (a x))+\frac {7}{64} \text {Si}(7 \arcsin (a x))}{a^7}-\frac {x^6 \sqrt {1-a^2 x^2}}{a \arcsin (a x)}\) |
-((x^6*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) + ((-5*SinIntegral[ArcSin[a*x]] )/64 + (27*SinIntegral[3*ArcSin[a*x]])/64 - (25*SinIntegral[5*ArcSin[a*x]] )/64 + (7*SinIntegral[7*ArcSin[a*x]])/64)/a^7
3.1.51.3.1 Defintions of rubi rules used
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] - Simp [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]
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 | \(\frac {-\frac {5 \sqrt {-a^{2} x^{2}+1}}{64 \arcsin \left (a x \right )}-\frac {5 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right )}{64}+\frac {9 \cos \left (3 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {27 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right )}{64}-\frac {5 \cos \left (5 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}-\frac {25 \,\operatorname {Si}\left (5 \arcsin \left (a x \right )\right )}{64}+\frac {\cos \left (7 \arcsin \left (a x \right )\right )}{64 \arcsin \left (a x \right )}+\frac {7 \,\operatorname {Si}\left (7 \arcsin \left (a x \right )\right )}{64}}{a^{7}}\) | \(105\) |
1/a^7*(-5/64/arcsin(a*x)*(-a^2*x^2+1)^(1/2)-5/64*Si(arcsin(a*x))+9/64/arcs in(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)))
\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{2}} \,d x } \]
\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int \frac {x^{6}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx \]
\[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int { \frac {x^{6}}{\arcsin \left (a x\right )^{2}} \,d x } \]
-(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, s qrt(a*x + 1)*sqrt(-a*x + 1)))
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 \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=-\frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} - \frac {3 \, {\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} + \frac {7 \, \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {25 \, \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {27 \, \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a^{7}} - \frac {5 \, \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a^{7}} + \frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{7} \arcsin \left (a x\right )} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{7} \arcsin \left (a x\right )} \]
-(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))
Timed out. \[ \int \frac {x^6}{\arcsin (a x)^2} \, dx=\int \frac {x^6}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]