Optimal. Leaf size=102 \[ -\frac {3}{2} i a^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^3}{2 x^2} \]
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Rubi [A] time = 0.18, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4628, 4682, 4626, 3719, 2190, 2279, 2391} \[ -\frac {3}{2} i a^2 \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}(a x)}\right )+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^3}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4626
Rule 4628
Rule 4682
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cos ^{-1}(a x)^3}{2 x^2}-\frac {1}{2} (3 a) \int \frac {\cos ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+\left (3 a^2\right ) \int \frac {\cos ^{-1}(a x)}{x} \, dx\\ &=\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+\left (6 i a^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )+\frac {1}{2} \left (3 i a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a x)}\right )\\ &=-\frac {3}{2} i a^2 \cos ^{-1}(a x)^2+\frac {3 a \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^2}{2 x}-\frac {\cos ^{-1}(a x)^3}{2 x^2}+3 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {3}{2} i a^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 92, normalized size = 0.90 \[ \frac {1}{2} \left (-3 i a^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(a x)}\right )+\frac {3 a \left (\sqrt {1-a^2 x^2}-i a x\right ) \cos ^{-1}(a x)^2}{x}+6 a^2 \cos ^{-1}(a x) \log \left (1+e^{2 i \cos ^{-1}(a x)}\right )-\frac {\cos ^{-1}(a x)^3}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arccos \left (a x\right )^{3}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\arccos \left (a x\right )^{3}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 113, normalized size = 1.11 \[ -\frac {3 i a^{2} \arccos \left (a x \right )^{2}}{2}-\frac {\arccos \left (a x \right )^{3}}{2 x^{2}}+3 a^{2} \arccos \left (a x \right ) \ln \left (1+\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )-\frac {3 i a^{2} \polylog \left (2, -\left (i \sqrt {-a^{2} x^{2}+1}+a x \right )^{2}\right )}{2}+\frac {3 a \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{2 x} \]
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 \[ \frac {\frac {3}{4} \, {\left (\sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} + 4 \, x \int \frac {3 \, \sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} + 2 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )}{4 \, {\left (a^{2} x^{4} - x^{2}\right )}}\,{d x}\right )} a x - \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}}{2 \, x^{2}} \]
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 \[ \int \frac {{\mathrm {acos}\left (a\,x\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acos}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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