Optimal. Leaf size=31 \[ -\frac{1}{3} a^2 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{3 a x^3} \]
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Rubi [A] time = 0.0219248, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6336, 30, 261} \[ -\frac{1}{3} a^2 \left (\frac{1}{a^2 x^2}+1\right )^{3/2}-\frac{1}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 261
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x^3} \, dx &=\frac{\int \frac{1}{x^4} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^3} \, dx\\ &=-\frac{1}{3} a^2 \left (1+\frac{1}{a^2 x^2}\right )^{3/2}-\frac{1}{3 a x^3}\\ \end{align*}
Mathematica [A] time = 0.0336995, size = 37, normalized size = 1.19 \[ -\frac{a x \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^2+1\right )+1}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.179, size = 42, normalized size = 1.4 \begin{align*} -{\frac{{a}^{2}{x}^{2}+1}{3\,{x}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}}}-{\frac{1}{3\,{x}^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03324, size = 34, normalized size = 1.1 \begin{align*} -\frac{1}{3} \, a^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} - \frac{1}{3 \, a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.41047, size = 103, normalized size = 3.32 \begin{align*} -\frac{a^{3} x^{3} +{\left (a^{3} x^{3} + a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{3 \, a x^{3}} \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 [B] time = 1.19317, size = 93, normalized size = 3. \begin{align*} \frac{2 \,{\left (3 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{4} a^{2} \mathrm{sgn}\left (x\right ) + a^{2} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} + 1}\right )}^{2} - 1\right )}^{3}} - \frac{1}{3 \, a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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