Optimal. Leaf size=42 \[ -\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{4 x^2}-\frac{1}{4 a x^4}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right ) \]
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Rubi [A] time = 0.039321, antiderivative size = 42, 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, 335, 275, 195, 215} \[ -\frac{\sqrt{\frac{1}{a^2 x^4}+1}}{4 x^2}-\frac{1}{4 a x^4}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right ) \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 275
Rule 195
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^3} \, dx &=\frac{\int \frac{1}{x^5} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^3} \, dx\\ &=-\frac{1}{4 a x^4}-\operatorname{Subst}\left (\int x \sqrt{1+\frac{x^4}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4 a x^4}-\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{4 x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^4}}}{4 x^2}-\frac{1}{4} a \text{csch}^{-1}\left (a x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0443945, size = 24, normalized size = 0.57 \[ -\frac{1}{8} a \left (2 \text{csch}^{-1}\left (a x^2\right )+e^{2 \text{csch}^{-1}\left (a x^2\right )}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.287, size = 114, normalized size = 2.7 \begin{align*} -{\frac{1}{4\,{x}^{2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}} \left ( \ln \left ( 2\,{\frac{1}{{a}^{2}{x}^{2}} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ){x}^{4}+\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}\sqrt{{a}^{-2}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}}}}}}{\frac{1}{\sqrt{{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 [B] time = 1.07214, size = 124, normalized size = 2.95 \begin{align*} -\frac{a^{2} x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1}}{4 \,{\left (a^{2} x^{4}{\left (\frac{1}{a^{2} x^{4}} + 1\right )} - 1\right )}} - \frac{1}{8} \, a \log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} + 1\right ) + \frac{1}{8} \, a \log \left (a x^{2} \sqrt{\frac{1}{a^{2} x^{4}} + 1} - 1\right ) - \frac{1}{4 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.51988, size = 227, normalized size = 5.4 \begin{align*} -\frac{a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1\right ) - a^{2} x^{4} \log \left (a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} - 1\right ) + 2 \, a x^{2} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 2}{8 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.86306, size = 39, normalized size = 0.93 \begin{align*} - \frac{a \operatorname{asinh}{\left (\frac{1}{a x^{2}} \right )}}{4} - \frac{\sqrt{1 + \frac{1}{a^{2} x^{4}}}}{4 x^{2}} - \frac{1}{4 a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12455, size = 103, normalized size = 2.45 \begin{align*} -\frac{a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{4} + 1} + 1\right ) - a^{4}{\left | a \right |} \log \left (\sqrt{a^{2} x^{4} + 1} - 1\right ) + \frac{2 \,{\left (\sqrt{a^{2} x^{4} + 1} a^{4}{\left | a \right |} + a^{5}\right )}}{a^{2} x^{4}}}{8 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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