Optimal. Leaf size=176 \[ -\frac {\text {ArcCos}(a x)^4}{x}-8 i a \text {ArcCos}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+12 i a \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-12 i a \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )-24 a \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+24 a \text {ArcCos}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right )-24 i a \text {PolyLog}\left (4,-i e^{i \text {ArcCos}(a x)}\right )+24 i a \text {PolyLog}\left (4,i e^{i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4804,
4266, 2611, 6744, 2320, 6724} \begin {gather*} -8 i a \text {ArcCos}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+12 i a \text {ArcCos}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-12 i a \text {ArcCos}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )-24 a \text {ArcCos}(a x) \text {Li}_3\left (-i e^{i \text {ArcCos}(a x)}\right )+24 a \text {ArcCos}(a x) \text {Li}_3\left (i e^{i \text {ArcCos}(a x)}\right )-24 i a \text {Li}_4\left (-i e^{i \text {ArcCos}(a x)}\right )+24 i a \text {Li}_4\left (i e^{i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^4}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^4}{x^2} \, dx &=-\frac {\cos ^{-1}(a x)^4}{x}-(4 a) \int \frac {\cos ^{-1}(a x)^3}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\cos ^{-1}(a x)^4}{x}+(4 a) \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(12 a) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(12 a) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(24 i a) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )+(24 a) \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )-(24 a) \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-(24 i a) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+(24 i a) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {\cos ^{-1}(a x)^4}{x}-8 i a \cos ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-12 i a \cos ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-24 a \cos ^{-1}(a x) \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+24 a \cos ^{-1}(a x) \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-24 i a \text {Li}_4\left (-i e^{i \cos ^{-1}(a x)}\right )+24 i a \text {Li}_4\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(549\) vs. \(2(176)=352\).
time = 0.75, size = 549, normalized size = 3.12 \begin {gather*} a \left (-\frac {7 i \pi ^4}{16}-\frac {1}{2} i \pi ^3 \text {ArcCos}(a x)+\frac {3}{2} i \pi ^2 \text {ArcCos}(a x)^2-2 i \pi \text {ArcCos}(a x)^3+i \text {ArcCos}(a x)^4-\frac {\text {ArcCos}(a x)^4}{a x}+3 \pi ^2 \text {ArcCos}(a x) \log \left (1-i e^{-i \text {ArcCos}(a x)}\right )-6 \pi \text {ArcCos}(a x)^2 \log \left (1-i e^{-i \text {ArcCos}(a x)}\right )-\frac {1}{2} \pi ^3 \log \left (1+i e^{-i \text {ArcCos}(a x)}\right )+4 \text {ArcCos}(a x)^3 \log \left (1+i e^{-i \text {ArcCos}(a x)}\right )+\frac {1}{2} \pi ^3 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-3 \pi ^2 \text {ArcCos}(a x) \log \left (1+i e^{i \text {ArcCos}(a x)}\right )+6 \pi \text {ArcCos}(a x)^2 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )-4 \text {ArcCos}(a x)^3 \log \left (1+i e^{i \text {ArcCos}(a x)}\right )+\frac {1}{2} \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \text {ArcCos}(a x))\right )\right )+12 i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{-i \text {ArcCos}(a x)}\right )+3 i \pi (\pi -4 \text {ArcCos}(a x)) \text {PolyLog}\left (2,i e^{-i \text {ArcCos}(a x)}\right )+3 i \pi ^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-12 i \pi \text {ArcCos}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )+12 i \text {ArcCos}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )+24 \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{-i \text {ArcCos}(a x)}\right )-12 \pi \text {PolyLog}\left (3,i e^{-i \text {ArcCos}(a x)}\right )+12 \pi \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )-24 \text {ArcCos}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )-24 i \text {PolyLog}\left (4,-i e^{-i \text {ArcCos}(a x)}\right )-24 i \text {PolyLog}\left (4,-i e^{i \text {ArcCos}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\arccos \left (a x \right )^{4}}{x^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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 \frac {\operatorname {acos}^{4}{\left (a x \right )}}{x^{2}}\, 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 \frac {{\mathrm {acos}\left (a\,x\right )}^4}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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