Optimal. Leaf size=75 \[ \frac {(-i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,-i \text {ArcCos}(a x))}{2 a}+\frac {(i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(1+n,i \text {ArcCos}(a x))}{2 a} \]
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Rubi [A]
time = 0.03, 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 = {4720, 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))}{2 a}+\frac {(i \text {ArcCos}(a x))^{-n} \text {ArcCos}(a x)^n \text {Gamma}(n+1,i \text {ArcCos}(a x))}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 4720
Rubi steps
\begin {align*} \int \cos ^{-1}(a x)^n \, dx &=-\frac {\text {Subst}\left (\int x^n \sin (x) \, dx,x,\cos ^{-1}(a x)\right )}{a}\\ &=-\frac {i \text {Subst}\left (\int e^{-i x} x^n \, dx,x,\cos ^{-1}(a x)\right )}{2 a}+\frac {i \text {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 = 0.03, size = 70, normalized size = 0.93 \begin {gather*} \frac {\text {ArcCos}(a x)^n \left (\text {ArcCos}(a x)^2\right )^{-n} \left ((i \text {ArcCos}(a x))^n \text {Gamma}(1+n,-i \text {ArcCos}(a x))+(-i \text {ArcCos}(a x))^n \text {Gamma}(1+n,i \text {ArcCos}(a x))\right )}{2 a} \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 = 0.17, size = 148, normalized size = 1.97
method | result | size |
default | \(-\frac {2^{n} \sqrt {\pi }\, \left (\frac {\arccos \left (a x \right )^{1+n} 2^{-n} \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {2^{-n} \sqrt {\arccos \left (a x \right )}\, \LommelS 1 \left (n +\frac {3}{2}, \frac {3}{2}, \arccos \left (a x \right )\right ) \sqrt {-a^{2} x^{2}+1}}{\sqrt {\pi }\, \left (2+n \right )}-\frac {3 \,2^{-1-n} \left (\frac {4}{3}+\frac {2 n}{3}\right ) \left (a x \arccos \left (a x \right )-\sqrt {-a^{2} x^{2}+1}\right ) \LommelS 1 \left (n +\frac {1}{2}, \frac {1}{2}, \arccos \left (a x \right )\right )}{\sqrt {\pi }\, \left (2+n \right ) \sqrt {\arccos \left (a x \right )}}\right )}{a}\) | \(148\) |
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 \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 {\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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