Optimal. Leaf size=74 \[ 2 i a \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-2 i a \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4628, 4710, 4181, 2279, 2391} \[ 2 i a \text {PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-2 i a \text {PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4181
Rule 4628
Rule 4710
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\cos ^{-1}(a x)^2}{x}-(2 a) \int \frac {\cos ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\cos ^{-1}(a x)^2}{x}+(2 a) \operatorname {Subst}\left (\int x \sec (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-(2 a) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+(2 a) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+(2 i a) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )-(2 i a) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {\cos ^{-1}(a x)^2}{x}-4 i a \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+2 i a \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-2 i a \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 98, normalized size = 1.32 \[ 2 i a \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-2 i a \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x) \left (\cos ^{-1}(a x)+2 a x \left (\log \left (1+i e^{i \cos ^{-1}(a x)}\right )-\log \left (1-i e^{i \cos ^{-1}(a x)}\right )\right )\right )}{x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\arccos \left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 135, normalized size = 1.82 \[ -\frac {\arccos \left (a x \right )^{2}}{x}-2 a \arccos \left (a x \right ) \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 a \arccos \left (a x \right ) \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+2 i a \dilog \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i a \dilog \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right ) \]
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 \[ \frac {2 \, a x \int \frac {\sqrt {-a x + 1} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{\sqrt {a x + 1} {\left (a x - 1\right )} x}\,{d x} - \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}}{x} \]
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 \[ \int \frac {{\mathrm {acos}\left (a\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acos}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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