Optimal. Leaf size=165 \[ \frac {2^{-4-n} (-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-4-n} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (3+n)} (-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-4 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (3+n)} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,4 i \text {ArcCos}(a x))}{a^4} \]
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Rubi [A]
time = 0.13, antiderivative size = 165, 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 {2^{-n-4} \text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (n+3)} \text {ArcCos}(a x)^n (-i \text {ArcCos}(a x))^{-n} \text {Gamma}(n+1,-4 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-n-4} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,2 i \text {ArcCos}(a x))}{a^4}+\frac {2^{-2 (n+3)} (i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,4 i \text {ArcCos}(a x))}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 4491
Rule 4732
Rubi steps
\begin {align*} \int x^3 \cos ^{-1}(a x)^n \, dx &=-\frac {\text {Subst}\left (\int x^n \cos ^3(x) \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a^4}\\ &=-\frac {\text {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 {\text {Subst}\left (\int x^n \sin (4 x) \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int x^n \sin (2 x) \, dx,x,\cos ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {i \text {Subst}\left (\int e^{-4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}+\frac {i \text {Subst}\left (\int e^{4 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{16 a^4}-\frac {i \text {Subst}\left (\int e^{-2 i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{8 a^4}+\frac {i \text {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*}
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Mathematica [A]
time = 0.07, size = 130, normalized size = 0.79 \begin {gather*} \frac {2^{-2 (3+n)} \text {ArcCos}(a x)^n \left (\text {ArcCos}(a x)^2\right )^{-n} \left (2^{2+n} (i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-2 i \text {ArcCos}(a x))+2^{2+n} (-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,2 i \text {ArcCos}(a x))+(i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-4 i \text {ArcCos}(a x))+(-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,4 i \text {ArcCos}(a x))\right )}{a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.73, size = 287, normalized size = 1.74
method | result | size |
default | \(-\frac {\sqrt {\pi }\, \left (\frac {2 \arccos \left (a x \right )^{1+n} \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{\frac {1}{2}-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 2 \arccos \left (a x \right )\right ) \sin \left (2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-\frac {3}{2}-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (2 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-\sin \left (2 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 2 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{8 a^{4}}-\frac {2^{-5-n} \sqrt {\pi }\, \left (\frac {2^{2+n} \arccos \left (a x \right )^{1+n} \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{1-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, 4 \arccos \left (a x \right )\right ) \sin \left (4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-2-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-\sin \left (4 \arccos \left (a x \right )\right )\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, 4 \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a^{4}}\) | \(287\) |
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^{3} \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^3\,{\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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