Optimal. Leaf size=122 \[ -\frac {\text {ArcCos}(a x)^3}{x}-6 i a \text {ArcCos}(a x)^2 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+6 i a \text {ArcCos}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-6 i a \text {ArcCos}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )-6 a \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+6 a \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4724, 4804,
4266, 2611, 2320, 6724} \begin {gather*} -6 i a \text {ArcCos}(a x)^2 \text {ArcTan}\left (e^{i \text {ArcCos}(a x)}\right )+6 i a \text {ArcCos}(a x) \text {Li}_2\left (-i e^{i \text {ArcCos}(a x)}\right )-6 i a \text {ArcCos}(a x) \text {Li}_2\left (i e^{i \text {ArcCos}(a x)}\right )-6 a \text {Li}_3\left (-i e^{i \text {ArcCos}(a x)}\right )+6 a \text {Li}_3\left (i e^{i \text {ArcCos}(a x)}\right )-\frac {\text {ArcCos}(a x)^3}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4266
Rule 4724
Rule 4804
Rule 6724
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^3}{x^2} \, dx &=-\frac {\cos ^{-1}(a x)^3}{x}-(3 a) \int \frac {\cos ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\cos ^{-1}(a x)^3}{x}+(3 a) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(6 a) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(6 a) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(6 i a) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(6 i a) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+(6 a) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {\cos ^{-1}(a x)^3}{x}-6 i a \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+6 i a \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-6 i a \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-6 a \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+6 a \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 139, normalized size = 1.14 \begin {gather*} -\frac {\text {ArcCos}(a x)^3}{x}+3 a \left (\text {ArcCos}(a x)^2 \left (\log \left (1-i e^{i \text {ArcCos}(a x)}\right )-\log \left (1+i e^{i \text {ArcCos}(a x)}\right )\right )+2 i \text {ArcCos}(a x) \left (\text {PolyLog}\left (2,-i e^{i \text {ArcCos}(a x)}\right )-\text {PolyLog}\left (2,i e^{i \text {ArcCos}(a x)}\right )\right )-2 \text {PolyLog}\left (3,-i e^{i \text {ArcCos}(a x)}\right )+2 \text {PolyLog}\left (3,i e^{i \text {ArcCos}(a x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.37, size = 0, normalized size = 0.00 \[\int \frac {\arccos \left (a x \right )^{3}}{x^{2}}\, dx\]
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}^{3}{\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 )}^3}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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