Integrand size = 8, antiderivative size = 83 \[ \int x \arccos (a x)^n \, dx=\frac {2^{-3-n} (-i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,-2 i \arccos (a x))}{a^2}+\frac {2^{-3-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^2} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4732, 4491, 12, 3389, 2212} \[ \int x \arccos (a x)^n \, dx=\frac {2^{-n-3} \arccos (a x)^n (-i \arccos (a x))^{-n} \Gamma (n+1,-2 i \arccos (a x))}{a^2}+\frac {2^{-n-3} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (n+1,2 i \arccos (a x))}{a^2} \]
[In]
[Out]
Rule 12
Rule 2212
Rule 3389
Rule 4491
Rule 4732
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^n \cos (x) \sin (x) \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{2} x^n \sin (2 x) \, dx,x,\arccos (a x)\right )}{a^2} \\ & = -\frac {\text {Subst}\left (\int x^n \sin (2 x) \, dx,x,\arccos (a x)\right )}{2 a^2} \\ & = -\frac {i \text {Subst}\left (\int e^{-2 i x} x^n \, dx,x,\arccos (a x)\right )}{4 a^2}+\frac {i \text {Subst}\left (\int e^{2 i x} x^n \, dx,x,\arccos (a x)\right )}{4 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}+\frac {2^{-3-n} (i \arccos (a x))^{-n} \arccos (a x)^n \Gamma (1+n,2 i \arccos (a x))}{a^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int x \arccos (a x)^n \, dx=\frac {2^{-3-n} \arccos (a x)^n \left (\arccos (a x)^2\right )^{-n} \left ((i \arccos (a x))^n \Gamma (1+n,-2 i \arccos (a x))+(-i \arccos (a x))^n \Gamma (1+n,2 i \arccos (a x))\right )}{a^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.10 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \operatorname {LommelS1}\left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \operatorname {LommelS1}\left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{4 a^{2}}\) | \(138\) |
[In]
[Out]
\[ \int x \arccos (a x)^n \, dx=\int { x \arccos \left (a x\right )^{n} \,d x } \]
[In]
[Out]
\[ \int x \arccos (a x)^n \, dx=\int x \operatorname {acos}^{n}{\left (a x \right )}\, dx \]
[In]
[Out]
Exception generated. \[ \int x \arccos (a x)^n \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int x \arccos (a x)^n \, dx=\int { x \arccos \left (a x\right )^{n} \,d x } \]
[In]
[Out]
Timed out. \[ \int x \arccos (a x)^n \, dx=\int x\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \]
[In]
[Out]