Optimal. Leaf size=60 \[ \frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]
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Rubi [A] time = 0.04213, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5264, 4628, 266, 51, 63, 208} \[ \frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5264
Rule 4628
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sec ^{-1}\left (\frac{a}{x}\right )}{x^4} \, dx &=\int \frac{\cos ^{-1}\left (\frac{x}{a}\right )}{x^4} \, dx\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx}{3 a}\\ &=-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,x^2\right )}{6 a}\\ &=\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,x^2\right )}{12 a^3}\\ &=\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a}\\ &=\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\cos ^{-1}\left (\frac{x}{a}\right )}{3 x^3}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}\\ \end{align*}
Mathematica [A] time = 0.0430254, size = 69, normalized size = 1.15 \[ \frac{a^2 x \sqrt{1-\frac{x^2}{a^2}}+x^3 \log \left (\sqrt{1-\frac{x^2}{a^2}}+1\right )-2 a^3 \sec ^{-1}\left (\frac{a}{x}\right )-x^3 \log (x)}{6 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 98, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arcsec} \left ({\frac{a}{x}}\right )}+{\frac{1}{6\,{a}^{3}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}}+{\frac{x}{6\,{a}^{4}}\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}}\ln \left ({\frac{a}{x}}+\sqrt{-1+{\frac{{a}^{2}}{{x}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{x}^{2}}{{a}^{2}} \left ( -1+{\frac{{a}^{2}}{{x}^{2}}} \right ) }}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40078, size = 294, normalized size = 4.9 \begin{align*} -\frac{4 \, a^{3} x^{3} \arctan \left (-\frac{x^{2} \sqrt{\frac{a^{2} - x^{2}}{x^{2}}}}{a^{2} - x^{2}}\right ) - x^{3} \log \left (x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} + a\right ) + x^{3} \log \left (x \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} - a\right ) - 2 \, a x^{2} \sqrt{\frac{a^{2} - x^{2}}{x^{2}}} - 4 \,{\left (a^{3} x^{3} - a^{3}\right )} \operatorname{arcsec}\left (\frac{a}{x}\right )}{12 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asec}{\left (\frac{a}{x} \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1037, size = 108, normalized size = 1.8 \begin{align*} \frac{a{\left (\frac{\log \left ({\left | a + \sqrt{a^{2} - x^{2}} \right |}\right )}{a^{3}} - \frac{\log \left ({\left | -a + \sqrt{a^{2} - x^{2}} \right |}\right )}{a^{3}} + \frac{2 \, \sqrt{a^{2} - x^{2}}}{a^{2} x^{2}}\right )}}{12 \,{\left | a \right |}} - \frac{\arccos \left (\frac{x}{a}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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