Optimal. Leaf size=58 \[ \frac{a^3 \sqrt{1-a^2 x^2}}{6 x}+\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\cos ^{-1}(a x)}{4 x^4} \]
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Rubi [A] time = 0.0221729, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4628, 271, 264} \[ \frac{a^3 \sqrt{1-a^2 x^2}}{6 x}+\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\cos ^{-1}(a x)}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^5} \, dx &=-\frac{\cos ^{-1}(a x)}{4 x^4}-\frac{1}{4} a \int \frac{1}{x^4 \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a \sqrt{1-a^2 x^2}}{12 x^3}-\frac{\cos ^{-1}(a x)}{4 x^4}-\frac{1}{6} a^3 \int \frac{1}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a \sqrt{1-a^2 x^2}}{12 x^3}+\frac{a^3 \sqrt{1-a^2 x^2}}{6 x}-\frac{\cos ^{-1}(a x)}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0221487, size = 41, normalized size = 0.71 \[ \frac{a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+1\right )-3 \cos ^{-1}(a x)}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 58, normalized size = 1. \begin{align*}{a}^{4} \left ( -{\frac{\arccos \left ( ax \right ) }{4\,{a}^{4}{x}^{4}}}+{\frac{1}{12\,{a}^{3}{x}^{3}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{6\,ax}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43934, size = 68, normalized size = 1.17 \begin{align*} \frac{1}{12} \,{\left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x} + \frac{\sqrt{-a^{2} x^{2} + 1}}{x^{3}}\right )} a - \frac{\arccos \left (a x\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02855, size = 88, normalized size = 1.52 \begin{align*} \frac{{\left (2 \, a^{3} x^{3} + a x\right )} \sqrt{-a^{2} x^{2} + 1} - 3 \, \arccos \left (a x\right )}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.67278, size = 102, normalized size = 1.76 \begin{align*} - \frac{a \left (\begin{cases} - \frac{2 i a^{2} \sqrt{a^{2} x^{2} - 1}}{3 x} - \frac{i \sqrt{a^{2} x^{2} - 1}}{3 x^{3}} & \text{for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac{2 a^{2} \sqrt{- a^{2} x^{2} + 1}}{3 x} - \frac{\sqrt{- a^{2} x^{2} + 1}}{3 x^{3}} & \text{otherwise} \end{cases}\right )}{4} - \frac{\operatorname{acos}{\left (a x \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13634, size = 176, normalized size = 3.03 \begin{align*} -\frac{1}{96} \,{\left (\frac{{\left (a^{4} + \frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{2}}{x^{2}}\right )} a^{6} x^{3}}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}{\left | a \right |}} - \frac{\frac{9 \,{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )} a^{4}}{x} + \frac{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}^{3}}{x^{3}}}{a^{2}{\left | a \right |}}\right )} a - \frac{\arccos \left (a x\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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