Integrand size = 10, antiderivative size = 165 \[ \int x^3 \arccos (a x)^n \, dx=\frac {2^{-4-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-2 i \arccos (a x))}{a^4}+\frac {2^{-4-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^4}+\frac {2^{-2 (3+n)} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-4 i \arccos (a x))}{a^4}+\frac {2^{-2 (3+n)} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,4 i \arccos (a x))}{a^4} \]
2^(-4-n)*arccos(a*x)^n*GAMMA(1+n,-2*I*arccos(a*x))/a^4/((-I*arccos(a*x))^n )+2^(-4-n)*arccos(a*x)^n*GAMMA(1+n,2*I*arccos(a*x))/a^4/((I*arccos(a*x))^n )+arccos(a*x)^n*GAMMA(1+n,-4*I*arccos(a*x))/(2^(6+2*n))/a^4/((-I*arccos(a* x))^n)+arccos(a*x)^n*GAMMA(1+n,4*I*arccos(a*x))/(2^(6+2*n))/a^4/((I*arccos (a*x))^n)
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.79 \[ \int x^3 \arccos (a x)^n \, dx=\frac {2^{-2 (3+n)} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left (2^{2+n} (i \arccos (a x))^n \Gamma (1+n,-2 i \arccos (a x))+2^{2+n} (-i \arccos (a x))^n \Gamma (1+n,2 i \arccos (a x))+(i \arccos (a x))^n \Gamma (1+n,-4 i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,4 i \arccos (a x))\right )}{a^4} \]
(ArcCos[a*x]^n*(2^(2 + n)*(I*ArcCos[a*x])^n*Gamma[1 + n, (-2*I)*ArcCos[a*x ]] + 2^(2 + n)*((-I)*ArcCos[a*x])^n*Gamma[1 + n, (2*I)*ArcCos[a*x]] + (I*A rcCos[a*x])^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]] + ((-I)*ArcCos[a*x])^n*Gamm a[1 + n, (4*I)*ArcCos[a*x]]))/(2^(2*(3 + n))*a^4*(ArcCos[a*x]^2)^n)
Time = 0.37 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5147, 4906, 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.
\(\displaystyle \int x^3 \arccos (a x)^n \, dx\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle -\frac {\int a^3 x^3 \sqrt {1-a^2 x^2} \arccos (a x)^nd\arccos (a x)}{a^4}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {\int \left (\frac {1}{4} \sin (2 \arccos (a x)) \arccos (a x)^n+\frac {1}{8} \sin (4 \arccos (a x)) \arccos (a x)^n\right )d\arccos (a x)}{a^4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-2^{-n-4} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-2 i \arccos (a x))-2^{-2 (n+3)} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-4 i \arccos (a x))-2^{-n-4} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,2 i \arccos (a x))-2^{-2 (n+3)} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,4 i \arccos (a x))}{a^4}\) |
-((-((2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]])/((-I)*Arc Cos[a*x])^n) - (2^(-4 - n)*ArcCos[a*x]^n*Gamma[1 + n, (2*I)*ArcCos[a*x]])/ (I*ArcCos[a*x])^n - (ArcCos[a*x]^n*Gamma[1 + n, (-4*I)*ArcCos[a*x]])/(2^(2 *(3 + n))*((-I)*ArcCos[a*x])^n) - (ArcCos[a*x]^n*Gamma[1 + n, (4*I)*ArcCos [a*x]])/(2^(2*(3 + n))*(I*ArcCos[a*x])^n))/a^4)
3.2.30.3.1 Defintions of rubi rules used
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (b*c^(m + 1))^(-1) Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b], x], x , a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.90 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{8 a^{4}}-\frac {2^{-5-n} \sqrt {\pi }\, \left (\frac {2^{2+n} \arccos \left (a x \right )^{1+n} \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{1-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 4 \arccos \left (a x \right )\right ) \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-2-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-\sin \left (4 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a^{4}}\) | \(287\) |
-1/8*Pi^(1/2)/a^4*(2/Pi^(1/2)/(2+n)*arccos(a*x)^(1+n)*sin(2*arccos(a*x))-2 ^(1/2-n)/Pi^(1/2)/(2+n)*arccos(a*x)^(1/2)*LommelS1(n+3/2,3/2,2*arccos(a*x) )*sin(2*arccos(a*x))-3*2^(-3/2-n)/Pi^(1/2)/(2+n)/arccos(a*x)^(1/2)*(4/3+2/ 3*n)*(2*arccos(a*x)*cos(2*arccos(a*x))-sin(2*arccos(a*x)))*LommelS1(n+1/2, 1/2,2*arccos(a*x)))-2^(-5-n)*Pi^(1/2)/a^4*(2^(2+n)/Pi^(1/2)/(2+n)*arccos(a *x)^(1+n)*sin(4*arccos(a*x))-2^(1-n)/Pi^(1/2)/(2+n)*arccos(a*x)^(1/2)*Lomm elS1(n+3/2,3/2,4*arccos(a*x))*sin(4*arccos(a*x))-3*2^(-2-n)/Pi^(1/2)/(2+n) /arccos(a*x)^(1/2)*(4/3+2/3*n)*(4*arccos(a*x)*cos(4*arccos(a*x))-sin(4*arc cos(a*x)))*LommelS1(n+1/2,1/2,4*arccos(a*x)))
\[ \int x^3 \arccos (a x)^n \, dx=\int { x^{3} \arccos \left (a x\right )^{n} \,d x } \]
\[ \int x^3 \arccos (a x)^n \, dx=\int x^{3} \operatorname {acos}^{n}{\left (a x \right )}\, dx \]
Exception generated. \[ \int x^3 \arccos (a x)^n \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^3 \arccos (a x)^n \, dx=\int { x^{3} \arccos \left (a x\right )^{n} \,d x } \]
Timed out. \[ \int x^3 \arccos (a x)^n \, dx=\int x^3\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \]