Optimal. Leaf size=192 \[ i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {a^2 \cos ^{-1}(a x)}{x}+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\cos ^{-1}(a x)^3}{3 x^3} \]
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Rubi [A] time = 0.30, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4628, 4702, 4710, 4181, 2531, 2282, 6589, 266, 63, 208} \[ i a^3 \cos ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )-a^3 \text {PolyLog}\left (3,-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text {PolyLog}\left (3,i e^{i \cos ^{-1}(a x)}\right )+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {a^2 \cos ^{-1}(a x)}{x}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^3}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2282
Rule 2531
Rule 4181
Rule 4628
Rule 4702
Rule 4710
Rule 6589
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^3}{x^4} \, dx &=-\frac {\cos ^{-1}(a x)^3}{3 x^3}-a \int \frac {\cos ^{-1}(a x)^2}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac {\cos ^{-1}(a x)}{x^2} \, dx-\frac {1}{2} a^3 \int \frac {\cos ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}+\frac {1}{2} a^3 \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\cos ^{-1}(a x)\right )-a^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-a^3 \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+a^3 \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )+a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )-\left (i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )+\left (i a^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-a^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )+a^3 \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )\\ &=-\frac {a^2 \cos ^{-1}(a x)}{x}+\frac {a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x^2}-\frac {\cos ^{-1}(a x)^3}{3 x^3}-i a^3 \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )+a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i a^3 \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i a^3 \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-a^3 \text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+a^3 \text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.83, size = 165, normalized size = 0.86 \[ -\frac {\cos ^{-1}(a x) \left (12 a^2 x^2+4 \cos ^{-1}(a x)^2-3 \cos ^{-1}(a x) \sin \left (2 \cos ^{-1}(a x)\right )\right )}{12 x^3}+a^3 \left (\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+i \cos ^{-1}(a x) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )-\text {Li}_3\left (-i e^{i \cos ^{-1}(a x)}\right )+\text {Li}_3\left (i e^{i \cos ^{-1}(a x)}\right )-i \cos ^{-1}(a x)^2 \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x\right )^{3}}{x^{4}}, x\right ) \]
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 \[ \int \frac {\arccos \left (a x\right )^{3}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 272, normalized size = 1.42 \[ \frac {a \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \arccos \left (a x \right )}{x}-\frac {\arccos \left (a x \right )^{3}}{3 x^{3}}+\frac {a^{3} \arccos \left (a x \right )^{2} \ln \left (1-i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}-i a^{3} \arccos \left (a x \right ) \polylog \left (2, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )+a^{3} \polylog \left (3, i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-\frac {a^{3} \arccos \left (a x \right )^{2} \ln \left (1+i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )}{2}+i a^{3} \arccos \left (a x \right ) \polylog \left (2, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-a^{3} \polylog \left (3, -i \left (i \sqrt {-a^{2} x^{2}+1}+a x \right )\right )-2 i a^{3} \arctan \left (i \sqrt {-a^{2} x^{2}+1}+a x \right ) \]
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 \[ \frac {3 \, a x^{3} \int \frac {\sqrt {-a x + 1} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2}}{\sqrt {a x + 1} {\left (a x - 1\right )} x^{3}}\,{d x} - \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}}{3 \, x^{3}} \]
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 \[ \int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^4} \,d x \]
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 \[ \int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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