Optimal. Leaf size=165 \[ \frac{2^{-n-4} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )}{a^4} \]
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Rubi [A] time = 0.175624, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4636, 4406, 3308, 2181} \[ \frac{2^{-n-4} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-n-4} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-2 (n+3)} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )}{a^4} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^3 \cos ^{-1}(a x)^n \, dx &=-\frac{\operatorname{Subst}\left (\int x^n \cos ^3(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} x^n \sin (2 x)+\frac{1}{8} x^n \sin (4 x)\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int x^n \sin (4 x) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac{\operatorname{Subst}\left (\int x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}+\frac{i \operatorname{Subst}\left (\int e^{4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}-\frac{i \operatorname{Subst}\left (\int e^{-2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}+\frac{i \operatorname{Subst}\left (\int e^{2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}\\ &=\frac{2^{-4-n} \left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{2^{-4-n} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,2 i \cos ^{-1}(a x)\right )}{a^4}+\frac{4^{-3-n} \left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-4 i \cos ^{-1}(a x)\right )}{a^4}+\frac{4^{-3-n} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,4 i \cos ^{-1}(a x)\right )}{a^4}\\ \end{align*}
Mathematica [A] time = 0.101475, size = 130, normalized size = 0.79 \[ \frac{2^{-2 (n+3)} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (2^{n+2} \left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )+\left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,4 i \cos ^{-1}(a x)\right )+2^{n+2} \left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-4 i \cos ^{-1}(a x)\right )\right )}{a^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.215, size = 287, normalized size = 1.7 \begin{align*} -{\frac{\sqrt{\pi }}{8\,{a}^{4}} \left ( 2\,{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{1+n}\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }}-{\frac{\sin \left ( 2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }{2}^{{\frac{1}{2}}-n}\sqrt{\arccos \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},2\,\arccos \left ( ax \right ) \right ) }-3\,{\frac{{2}^{-3/2-n} \left ( 4/3+2/3\,n \right ) \left ( 2\,\arccos \left ( ax \right ) \cos \left ( 2\,\arccos \left ( ax \right ) \right ) -\sin \left ( 2\,\arccos \left ( ax \right ) \right ) \right ){\it LommelS1} \left ( n+1/2,1/2,2\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) \sqrt{\arccos \left ( ax \right ) }}} \right ) }-{\frac{{2}^{-5-n}\sqrt{\pi }}{{a}^{4}} \left ({\frac{{2}^{2+n} \left ( \arccos \left ( ax \right ) \right ) ^{1+n}\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }}-{\frac{{2}^{1-n}\sin \left ( 4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) }\sqrt{\arccos \left ( ax \right ) }{\it LommelS1} \left ( n+{\frac{3}{2}},{\frac{3}{2}},4\,\arccos \left ( ax \right ) \right ) }-3\,{\frac{{2}^{-2-n} \left ( 4/3+2/3\,n \right ) \left ( 4\,\arccos \left ( ax \right ) \cos \left ( 4\,\arccos \left ( ax \right ) \right ) -\sin \left ( 4\,\arccos \left ( ax \right ) \right ) \right ){\it LommelS1} \left ( n+1/2,1/2,4\,\arccos \left ( ax \right ) \right ) }{\sqrt{\pi } \left ( 2+n \right ) \sqrt{\arccos \left ( ax \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \arccos \left (a x\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acos}^{n}{\left (a x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \arccos \left (a x\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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