Optimal. Leaf size=60 \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]
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Rubi [A] time = 0.0418601, 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 = {5265, 4627, 266, 51, 63, 208} \[ -\frac{\sqrt{1-\frac{x^2}{a^2}}}{6 a x^2}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{x^2}{a^2}}\right )}{6 a^3}-\frac{\sin ^{-1}\left (\frac{x}{a}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5265
Rule 4627
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\csc ^{-1}\left (\frac{a}{x}\right )}{x^4} \, dx &=\int \frac{\sin ^{-1}\left (\frac{x}{a}\right )}{x^4} \, dx\\ &=-\frac{\sin ^{-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{\sin ^{-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{\sin ^{-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{\sin ^{-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{\sin ^{-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 = 0.0519034, size = 69, normalized size = 1.15 \[ -\frac{a^2 x \sqrt{1-\frac{x^2}{a^2}}+x^3 \log \left (\sqrt{1-\frac{x^2}{a^2}}+1\right )+2 a^3 \csc ^{-1}\left (\frac{a}{x}\right )-x^3 \log (x)}{6 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 98, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,{x}^{3}}{\rm arccsc} \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 [A] time = 3.68391, size = 198, normalized size = 3.3 \begin{align*} -\frac{4 \, a^{3} \operatorname{arccsc}\left (\frac{a}{x}\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}}}}{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{acsc}{\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 = 1.1126, 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{\arcsin \left (\frac{x}{a}\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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