Optimal. Leaf size=51 \[ -\frac{1}{5} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{5/2}+\frac{1}{3} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{5 a x^5} \]
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Rubi [A] time = 0.0384009, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6336, 30, 266, 43} \[ -\frac{1}{5} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{5/2}+\frac{1}{3} a^4 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x^5} \, dx &=\frac{\int \frac{1}{x^6} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^5} \, dx\\ &=-\frac{1}{5 a x^5}-\frac{1}{2} \operatorname{Subst}\left (\int x \sqrt{1+\frac{x}{a^2}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{1}{5 a x^5}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-a^2 \sqrt{1+\frac{x}{a^2}}+a^2 \left (1+\frac{x}{a^2}\right )^{3/2}\right ) \, dx,x,\frac{1}{x^2}\right )\\ &=\frac{1}{3} a^4 \left (1+\frac{1}{a^2 x^2}\right )^{3/2}-\frac{1}{5} a^4 \left (1+\frac{1}{a^2 x^2}\right )^{5/2}-\frac{1}{5 a x^5}\\ \end{align*}
Mathematica [A] time = 0.0419801, size = 46, normalized size = 0.9 \[ \frac{a x \sqrt{\frac{1}{a^2 x^2}+1} \left (2 a^4 x^4-a^2 x^2-3\right )-3}{15 a x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.197, size = 52, normalized size = 1. \begin{align*}{\frac{ \left ({a}^{2}{x}^{2}+1 \right ) \left ( 2\,{a}^{2}{x}^{2}-3 \right ) }{15\,{x}^{4}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{5\,a{x}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988547, size = 55, normalized size = 1.08 \begin{align*} -\frac{1}{5} \, a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \, a^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.43858, size = 124, normalized size = 2.43 \begin{align*} \frac{2 \, a^{5} x^{5} +{\left (2 \, a^{5} x^{5} - a^{3} x^{3} - 3 \, a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 3}{15 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.04264, size = 65, normalized size = 1.27 \begin{align*} \frac{2 a^{3} \sqrt{a^{2} x^{2} + 1}}{15 x} - \frac{a \sqrt{a^{2} x^{2} + 1}}{15 x^{3}} - \frac{\sqrt{a^{2} x^{2} + 1}}{5 a x^{5}} - \frac{1}{5 a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25469, size = 167, normalized size = 3.27 \begin{align*} \frac{4 \,{\left (15 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{6} a^{4} \mathrm{sgn}\left (x\right ) + 5 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{4} a^{4} \mathrm{sgn}\left (x\right ) + 5 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} a^{4} \mathrm{sgn}\left (x\right ) - a^{4} \mathrm{sgn}\left (x\right )\right )}}{15 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{5}} - \frac{1}{5 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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