Integrand size = 8, antiderivative size = 83 \[ \int x \arccos (a x)^n \, dx=\frac {2^{-3-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-2 i \arccos (a x))}{a^2}+\frac {2^{-3-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^2} \] Output:
2^(-3-n)*arccos(a*x)^n*GAMMA(1+n,-2*I*arccos(a*x))/a^2/((-I*arccos(a*x))^n )+2^(-3-n)*arccos(a*x)^n*GAMMA(1+n,2*I*arccos(a*x))/a^2/((I*arccos(a*x))^n )
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int x \arccos (a x)^n \, dx=\frac {2^{-3-n} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left ((i \arccos (a x))^n \Gamma (1+n,-2 i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,2 i \arccos (a x))\right )}{a^2} \] Input:
Integrate[x*ArcCos[a*x]^n,x]
Output:
(2^(-3 - n)*ArcCos[a*x]^n*((I*ArcCos[a*x])^n*Gamma[1 + n, (-2*I)*ArcCos[a* x]] + ((-I)*ArcCos[a*x])^n*Gamma[1 + n, (2*I)*ArcCos[a*x]]))/(a^2*(ArcCos[ a*x]^2)^n)
Time = 0.40 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5147, 4906, 27, 3042, 3789, 2612}
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.
\(\displaystyle \int x \arccos (a x)^n \, dx\) |
\(\Big \downarrow \) 5147 |
\(\displaystyle -\frac {\int a x \sqrt {1-a^2 x^2} \arccos (a x)^nd\arccos (a x)}{a^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {\int \frac {1}{2} \arccos (a x)^n \sin (2 \arccos (a x))d\arccos (a x)}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \arccos (a x)^n \sin (2 \arccos (a x))d\arccos (a x)}{2 a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \arccos (a x)^n \sin (2 \arccos (a x))d\arccos (a x)}{2 a^2}\) |
\(\Big \downarrow \) 3789 |
\(\displaystyle -\frac {\frac {1}{2} i \int e^{-2 i \arccos (a x)} \arccos (a x)^nd\arccos (a x)-\frac {1}{2} i \int e^{2 i \arccos (a x)} \arccos (a x)^nd\arccos (a x)}{2 a^2}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle -\frac {-2^{-n-2} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-2 i \arccos (a x))-2^{-n-2} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,2 i \arccos (a x))}{2 a^2}\) |
Input:
Int[x*ArcCos[a*x]^n,x]
Output:
-1/2*(-((2^(-2 - n)*ArcCos[a*x]^n*Gamma[1 + n, (-2*I)*ArcCos[a*x]])/((-I)* ArcCos[a*x])^n) - (2^(-2 - n)*ArcCos[a*x]^n*Gamma[1 + n, (2*I)*ArcCos[a*x] ])/(I*ArcCos[a*x])^n)/a^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
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 = 0.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \cos \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{4 a^{2}}\) | \(138\) |
Input:
int(x*arccos(a*x)^n,x,method=_RETURNVERBOSE)
Output:
-1/4*Pi^(1/2)/a^2*(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*cos(2*arccos(a*x))*arccos(a*x)-sin(2*arccos(a*x)))*LommelS1(n+1/2, 1/2,2*arccos(a*x)))
\[ \int x \arccos (a x)^n \, dx=\int { x \arccos \left (a x\right )^{n} \,d x } \] Input:
integrate(x*arccos(a*x)^n,x, algorithm="fricas")
Output:
integral(x*arccos(a*x)^n, x)
\[ \int x \arccos (a x)^n \, dx=\int x \operatorname {acos}^{n}{\left (a x \right )}\, dx \] Input:
integrate(x*acos(a*x)**n,x)
Output:
Integral(x*acos(a*x)**n, x)
Exception generated. \[ \int x \arccos (a x)^n \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*arccos(a*x)^n,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
\[ \int x \arccos (a x)^n \, dx=\int { x \arccos \left (a x\right )^{n} \,d x } \] Input:
integrate(x*arccos(a*x)^n,x, algorithm="giac")
Output:
integrate(x*arccos(a*x)^n, x)
Timed out. \[ \int x \arccos (a x)^n \, dx=\int x\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \] Input:
int(x*acos(a*x)^n,x)
Output:
int(x*acos(a*x)^n, x)
\[ \int x \arccos (a x)^n \, dx=\int \mathit {acos} \left (a x \right )^{n} x d x \] Input:
int(x*acos(a*x)^n,x)
Output:
int(acos(a*x)**n*x,x)