Optimal. Leaf size=46 \[ -\frac{1}{2} \sqrt{\frac{1}{a^2 x^4}+1}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )-\frac{1}{2 a x^2} \]
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Rubi [A] time = 0.0327176, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6336, 30, 266, 50, 63, 208} \[ -\frac{1}{2} \sqrt{\frac{1}{a^2 x^4}+1}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{\frac{1}{a^2 x^4}+1}\right )-\frac{1}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x} \, dx &=\frac{\int \frac{1}{x^3} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x} \, dx\\ &=-\frac{1}{2 a x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a^2}}}{x} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a^2}}} \, dx,x,\frac{1}{x^4}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}-\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{-a^2+a^2 x^2} \, dx,x,\sqrt{1+\frac{1}{a^2 x^4}}\right )\\ &=-\frac{1}{2} \sqrt{1+\frac{1}{a^2 x^4}}-\frac{1}{2 a x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a^2 x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0453229, size = 22, normalized size = 0.48 \[ \tanh ^{-1}\left (e^{\text{csch}^{-1}\left (a x^2\right )}\right )-\frac{1}{2} e^{\text{csch}^{-1}\left (a x^2\right )} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.272, size = 86, normalized size = 1.9 \begin{align*} -{\frac{1}{2}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( -\ln \left ({x}^{2}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}} \right ){x}^{2}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00797, size = 73, normalized size = 1.59 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{1}{a^{2} x^{4}} + 1} - \frac{1}{2 \, a x^{2}} + \frac{1}{4} \, \log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right ) - \frac{1}{4} \, \log \left (\sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55955, size = 163, normalized size = 3.54 \begin{align*} -\frac{a x^{2} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - a x^{2}\right ) + a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + a x^{2} + 1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.13401, size = 54, normalized size = 1.17 \begin{align*} - \frac{a x^{2}}{2 \sqrt{a^{2} x^{4} + 1}} + \frac{\operatorname{asinh}{\left (a x^{2} \right )}}{2} - \frac{1}{2 a x^{2}} - \frac{1}{2 a x^{2} \sqrt{a^{2} x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22897, size = 80, normalized size = 1.74 \begin{align*} \frac{{\left (a \log \left (a + \sqrt{a^{2} + \frac{1}{x^{4}}}\right ) - a \log \left (-a + \sqrt{a^{2} + \frac{1}{x^{4}}}\right ) - 2 \, \sqrt{a^{2} + \frac{1}{x^{4}}}\right )}{\left | a \right |}}{4 \, a^{2}} - \frac{1}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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