Optimal. Leaf size=118 \[ \frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{4 a x^4}+\frac{1}{4} a \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )+\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2} \]
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Rubi [A] time = 0.0611727, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6335, 30, 259, 266, 51, 63, 208} \[ \frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{4 a x^4}+\frac{1}{4} a \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )+\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6335
Rule 30
Rule 259
Rule 266
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{x^3} \, dx &=-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac{\int \frac{1}{x^5} \, dx}{a}-\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^5 \sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{a}\\ &=\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{1}{x^5 \sqrt{1-a^2 x^4}} \, dx}{a}\\ &=\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}-\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-a^2 x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{4 a x^4}-\frac{1}{8} \left (a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^4\right )\\ &=\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{4 a x^4}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^4}\right )}{4 a}\\ &=\frac{1}{4 a x^4}-\frac{e^{\text{sech}^{-1}\left (a x^2\right )}}{2 x^2}+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{4 a x^4}+\frac{1}{4} a \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^4}\right )\\ \end{align*}
Mathematica [A] time = 0.255935, size = 105, normalized size = 0.89 \[ -\frac{-\frac{a^2 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right ) \tan ^{-1}\left (\sqrt{a^2 x^4-1}\right )}{\sqrt{a^2 x^4-1}}+\frac{\sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )}{x^4}+\frac{1}{x^4}}{4 a} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.299, size = 129, normalized size = 1.1 \begin{align*}{\frac{{\it csgn} \left ({a}^{-1} \right ) }{4\,{x}^{2}}\sqrt{-{\frac{a{x}^{2}-1}{a{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+1}{a{x}^{2}}}} \left ( \ln \left ( 2\,{\frac{1}{{a}^{2}{x}^{2}} \left ({\it csgn} \left ({a}^{-1} \right ) a\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}+1 \right ) } \right ){x}^{4}a-\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}{\it csgn} \left ({a}^{-1} \right ) \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}}-{\frac{1}{4\,{x}^{4}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\frac{1}{4} \, a^{2} \log \left (\frac{2 \, \sqrt{-a^{2} x^{4} + 1}}{x^{2}} + \frac{2}{x^{2}}\right ) - \frac{1}{4} \, \sqrt{-a^{2} x^{4} + 1} a^{2} - \frac{{\left (-a^{2} x^{4} + 1\right )}^{\frac{3}{2}}}{4 \, x^{4}}}{a} - \frac{1}{4 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03432, size = 319, normalized size = 2.7 \begin{align*} \frac{a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} + 1\right ) - a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 1\right ) - 2 \, a x^{2} \sqrt{\frac{a x^{2} + 1}{a x^{2}}} \sqrt{-\frac{a x^{2} - 1}{a x^{2}}} - 2}{8 \, a x^{4}} \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*} \frac{\int \frac{1}{x^{5}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x^{2}}} \sqrt{1 + \frac{1}{a x^{2}}}}{x^{3}}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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