Optimal. Leaf size=163 \[ \frac {(-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-i \text {ArcCos}(a x))}{8 a^3}+\frac {(i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,i \text {ArcCos}(a x))}{8 a^3}+\frac {3^{-1-n} (-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-3 i \text {ArcCos}(a x))}{8 a^3}+\frac {3^{-1-n} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,3 i \text {ArcCos}(a x))}{8 a^3} \]
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Rubi [A]
time = 0.11, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4732, 4491,
3389, 2212} \begin {gather*} \frac {\text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-i \text {ArcCos}(a x))}{8 a^3}+\frac {3^{-n-1} \text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-3 i \text {ArcCos}(a x))}{8 a^3}+\frac {(i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,i \text {ArcCos}(a x))}{8 a^3}+\frac {3^{-n-1} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,3 i \text {ArcCos}(a x))}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 4491
Rule 4732
Rubi steps
\begin {align*} \int x^2 \cos ^{-1}(a x)^n \, dx &=-\frac {\text {Subst}\left (\int x^n \cos ^2(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^3}\\ &=-\frac {\text {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 {\text {Subst}\left (\int x^n \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}-\frac {\text {Subst}\left (\int x^n \sin (3 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac {i \text {Subst}\left (\int e^{-i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac {i \text {Subst}\left (\int e^{i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}-\frac {i \text {Subst}\left (\int e^{-3 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^3}+\frac {i \text {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*}
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Mathematica [A]
time = 0.14, size = 152, normalized size = 0.93 \begin {gather*} \frac {\frac {1}{4} \left (\frac {1}{2} (-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-i \text {ArcCos}(a x))+\frac {1}{2} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,i \text {ArcCos}(a x))\right )+\frac {1}{8} 3^{-1-n} \text {ArcCos}(a x)^n \left (\text {ArcCos}(a x)^2\right )^{-n} \left ((i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-3 i \text {ArcCos}(a x))+(-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,3 i \text {ArcCos}(a x))\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int x^{2} \arccos \left (a x \right )^{n}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \operatorname {acos}^{n}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\mathrm {acos}\left (a\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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