Optimal. Leaf size=80 \[ \frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}+\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{5 x^5} \]
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Rubi [A] time = 0.0466105, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4628, 266, 51, 63, 208} \[ \frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}+\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 4628
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^6} \, dx &=-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{5} a \int \frac{1}{x^5 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{10} a \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}-\frac{1}{80} \left (3 a^5\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}+\frac{1}{40} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )\\ &=\frac{a \sqrt{1-a^2 x^2}}{20 x^4}+\frac{3 a^3 \sqrt{1-a^2 x^2}}{40 x^2}-\frac{\cos ^{-1}(a x)}{5 x^5}+\frac{3}{40} a^5 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.061937, size = 72, normalized size = 0.9 \[ \frac{1}{40} \left (\frac{a \sqrt{1-a^2 x^2} \left (3 a^2 x^2+2\right )}{x^4}+3 a^5 \log \left (\sqrt{1-a^2 x^2}+1\right )-3 a^5 \log (x)-\frac{8 \cos ^{-1}(a x)}{x^5}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 73, normalized size = 0.9 \begin{align*}{a}^{5} \left ( -{\frac{\arccos \left ( ax \right ) }{5\,{a}^{5}{x}^{5}}}+{\frac{1}{20\,{a}^{4}{x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{40\,{a}^{2}{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3}{40}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45141, size = 111, normalized size = 1.39 \begin{align*} \frac{1}{40} \,{\left (3 \, a^{4} \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} a^{2}}{x^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{x^{4}}\right )} a - \frac{\arccos \left (a x\right )}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19651, size = 289, normalized size = 3.61 \begin{align*} \frac{3 \, a^{5} x^{5} \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{5} x^{5} \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right ) - 16 \, x^{5} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right ) + 16 \,{\left (x^{5} - 1\right )} \arccos \left (a x\right ) + 2 \,{\left (3 \, a^{3} x^{3} + 2 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{80 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.43766, size = 184, normalized size = 2.3 \begin{align*} - \frac{a \left (\begin{cases} - \frac{3 a^{4} \operatorname{acosh}{\left (\frac{1}{a x} \right )}}{8} + \frac{3 a^{3}}{8 x \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{a}{8 x^{3} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{5} \sqrt{-1 + \frac{1}{a^{2} x^{2}}}} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac{3 i a^{4} \operatorname{asin}{\left (\frac{1}{a x} \right )}}{8} - \frac{3 i a^{3}}{8 x \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i a}{8 x^{3} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} + \frac{i}{4 a x^{5} \sqrt{1 - \frac{1}{a^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{5} - \frac{\operatorname{acos}{\left (a x \right )}}{5 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15694, size = 120, normalized size = 1.5 \begin{align*} -\frac{1}{80} \, a^{5}{\left (\frac{2 \,{\left (3 \,{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{-a^{2} x^{2} + 1}\right )}}{a^{4} x^{4}} - 3 \, \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) + 3 \, \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right )\right )} - \frac{\arccos \left (a x\right )}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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