Optimal. Leaf size=75 \[ \frac {\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \Gamma \left (n+1,-i \cos ^{-1}(a x)\right )}{2 a}+\frac {\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (n+1,i \cos ^{-1}(a x)\right )}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4624, 3308, 2181} \[ \frac {\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text {Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{2 a}+\frac {\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text {Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3308
Rule 4624
Rubi steps
\begin {align*} \int \cos ^{-1}(a x)^n \, dx &=-\frac {\operatorname {Subst}\left (\int x^n \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {i \operatorname {Subst}\left (\int e^{-i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{2 a}+\frac {i \operatorname {Subst}\left (\int e^{i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{2 a}\\ &=\frac {\left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-i \cos ^{-1}(a x)\right )}{2 a}+\frac {\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,i \cos ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 70, normalized size = 0.93 \[ \frac {\cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \Gamma \left (n+1,-i \cos ^{-1}(a x)\right )\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\arccos \left (a x\right )^{n}, x\right ) \]
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 \[ \int \arccos \left (a x\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.12, size = 148, normalized size = 1.97 \[ -\frac {2^{n} \sqrt {\pi }\, \left (\frac {\arccos \left (a x \right )^{1+n} 2^{-n} \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-1-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (a x \arccos \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a} \]
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 \[ \int {\mathrm {acos}\left (a\,x\right )}^n \,d x \]
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 \[ \int \operatorname {acos}^{n}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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