Optimal. Leaf size=163 \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )}{8 a^3} \]
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Rubi [A] time = 0.14896, antiderivative size = 163, 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} \[ \frac{\cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-n-1} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )}{8 a^3} \]
Antiderivative was successfully verified.
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Rule 4636
Rule 4406
Rule 3308
Rule 2181
Rubi steps
\begin{align*} \int x^2 \cos ^{-1}(a x)^n \, dx &=-\frac{\operatorname{Subst}\left (\int x^n \cos ^2(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{4} x^n \sin (x)+\frac{1}{4} x^n \sin (3 x)\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac{\operatorname{Subst}\left (\int x^n \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac{\operatorname{Subst}\left (\int x^n \sin (3 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{i \operatorname{Subst}\left (\int e^{-i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac{i \operatorname{Subst}\left (\int e^{i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}-\frac{i \operatorname{Subst}\left (\int e^{-3 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac{i \operatorname{Subst}\left (\int e^{3 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}\\ &=\frac{\left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{\left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-1-n} \left (-i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,-3 i \cos ^{-1}(a x)\right )}{8 a^3}+\frac{3^{-1-n} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \Gamma \left (1+n,3 i \cos ^{-1}(a x)\right )}{8 a^3}\\ \end{align*}
Mathematica [A] time = 0.194486, size = 152, normalized size = 0.93 \[ \frac{\frac{1}{8} 3^{-n-1} \cos ^{-1}(a x)^n \left (\cos ^{-1}(a x)^2\right )^{-n} \left (\left (-i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,3 i \cos ^{-1}(a x)\right )+\left (i \cos ^{-1}(a x)\right )^n \text{Gamma}\left (n+1,-3 i \cos ^{-1}(a x)\right )\right )+\frac{1}{4} \left (\frac{1}{2} \cos ^{-1}(a x)^n \left (-i \cos ^{-1}(a x)\right )^{-n} \text{Gamma}\left (n+1,-i \cos ^{-1}(a x)\right )+\frac{1}{2} \left (i \cos ^{-1}(a x)\right )^{-n} \cos ^{-1}(a x)^n \text{Gamma}\left (n+1,i \cos ^{-1}(a x)\right )\right )}{a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.152, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \arccos \left ( ax \right ) \right ) ^{n}\, dx \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^{2} \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^{2} \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^{2} \arccos \left (a x\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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