Optimal. Leaf size=83 \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (n+1,2 i \cos ^{-1}(a x)\right )}{a^2} \]
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Rubi [A] time = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4636, 4406, 12, 3308, 2181} \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-2 i \cos ^{-1}(a x)\right )}{a^2}+\frac {2^{-n-3} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text {Gamma}\left (n+1,2 i \cos ^{-1}(a x)\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2181
Rule 3308
Rule 4406
Rule 4636
Rubi steps
\begin {align*} \int x \cos ^{-1}(a x)^n \, dx &=-\frac {\operatorname {Subst}\left (\int x^n \cos (x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{2} x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{a^2}\\ &=-\frac {\operatorname {Subst}\left (\int x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{2 a^2}\\ &=-\frac {i \operatorname {Subst}\left (\int e^{-2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{4 a^2}+\frac {i \operatorname {Subst}\left (\int e^{2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{4 a^2}\\ &=\frac {2^{-3-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^2}+\frac {2^{-3-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^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 74, normalized size = 0.89 \[ \frac {2^{-n-3} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,2 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,-2 i \cos ^{-1}(a x)\right )\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \arccos \left (a x\right )^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \arccos \left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 138, normalized size = 1.66 \[ -\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {acos}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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