Optimal. Leaf size=31 \[ -\frac{1}{6} a^2 \left (\frac{1}{a^2 x^4}+1\right )^{3/2}-\frac{1}{6 a x^6} \]
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Rubi [A] time = 0.0216713, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6336, 30, 261} \[ -\frac{1}{6} a^2 \left (\frac{1}{a^2 x^4}+1\right )^{3/2}-\frac{1}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 261
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}\left (a x^2\right )}}{x^5} \, dx &=\frac{\int \frac{1}{x^7} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^4}}}{x^5} \, dx\\ &=-\frac{1}{6} a^2 \left (1+\frac{1}{a^2 x^4}\right )^{3/2}-\frac{1}{6 a x^6}\\ \end{align*}
Mathematica [A] time = 0.0534463, size = 39, normalized size = 1.26 \[ -\frac{a x^2 \sqrt{\frac{1}{a^2 x^4}+1} \left (a^2 x^4+1\right )+1}{6 a x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.273, size = 42, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{x}^{4}+1}{6\,{x}^{4}}\sqrt{{\frac{{a}^{2}{x}^{4}+1}{{a}^{2}{x}^{4}}}}}-{\frac{1}{6\,{x}^{6}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03725, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{6} \, a^{2}{\left (\frac{1}{a^{2} x^{4}} + 1\right )}^{\frac{3}{2}} - \frac{1}{6 \, a x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.51877, size = 105, normalized size = 3.39 \begin{align*} -\frac{a^{3} x^{6} +{\left (a^{3} x^{6} + a x^{2}\right )} \sqrt{\frac{a^{2} x^{4} + 1}{a^{2} x^{4}}} + 1}{6 \, a x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.2093, size = 34, normalized size = 1.1 \begin{align*} -\frac{{\left (a^{2} + \frac{1}{x^{4}}\right )}^{\frac{3}{2}}{\left | a \right |}}{6 \, a^{2}} - \frac{1}{6 \, a x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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