Optimal. Leaf size=65 \[ -\frac{a^2 \sqrt{\frac{1}{a^2 x^2}+1}}{8 x}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{4 x^3}+\frac{1}{8} a^3 \text{csch}^{-1}(a x)-\frac{1}{4 a x^4} \]
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Rubi [A] time = 0.0434752, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6336, 30, 335, 279, 321, 215} \[ -\frac{a^2 \sqrt{\frac{1}{a^2 x^2}+1}}{8 x}-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{4 x^3}+\frac{1}{8} a^3 \text{csch}^{-1}(a x)-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 6336
Rule 30
Rule 335
Rule 279
Rule 321
Rule 215
Rubi steps
\begin{align*} \int \frac{e^{\text{csch}^{-1}(a x)}}{x^4} \, dx &=\frac{\int \frac{1}{x^5} \, dx}{a}+\int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^4} \, dx\\ &=-\frac{1}{4 a x^4}-\operatorname{Subst}\left (\int x^2 \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{4 x^3}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1+\frac{1}{a^2 x^2}}}{8 x}+\frac{1}{8} a^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4 a x^4}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1+\frac{1}{a^2 x^2}}}{8 x}+\frac{1}{8} a^3 \text{csch}^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0490127, size = 53, normalized size = 0.82 \[ \frac{-a x \sqrt{\frac{1}{a^2 x^2}+1} \left (a^2 x^2+2\right )+a^4 x^4 \sinh ^{-1}\left (\frac{1}{a x}\right )-2}{8 a x^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.183, size = 173, normalized size = 2.7 \begin{align*}{\frac{{a}^{2}}{8\,{x}^{3}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{-2}}{x}^{2}{a}^{2}-\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}\sqrt{{a}^{-2}}{x}^{4}{a}^{2}+\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ){x}^{4}-2\, \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{3/2}\sqrt{{a}^{-2}} \right ){\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+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 = 1.02206, size = 174, normalized size = 2.68 \begin{align*} \frac{1}{16} \, a^{3} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - \frac{1}{16} \, a^{3} \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right ) - \frac{a^{6} x^{3}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{\frac{3}{2}} + a^{4} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{8 \,{\left (a^{4} x^{4}{\left (\frac{1}{a^{2} x^{2}} + 1\right )}^{2} - 2 \, a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} + 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 = 2.58623, size = 250, normalized size = 3.85 \begin{align*} \frac{a^{4} x^{4} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{4} x^{4} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) -{\left (a^{3} x^{3} + 2 \, a x\right )} \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - 2}{8 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.32647, size = 83, normalized size = 1.28 \begin{align*} \frac{a^{3} \operatorname{asinh}{\left (\frac{1}{a x} \right )}}{8} - \frac{a^{2}}{8 x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{3}{8 x^{3} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} - \frac{1}{4 a x^{4}} - \frac{1}{4 a^{2} x^{5} \sqrt{1 + \frac{1}{a^{2} x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15207, size = 139, normalized size = 2.14 \begin{align*} \frac{a^{6}{\left | a \right |} \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) \mathrm{sgn}\left (x\right ) - a^{6}{\left | a \right |} \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right ) \mathrm{sgn}\left (x\right ) - \frac{2 \,{\left ({\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{6}{\left | a \right |} \mathrm{sgn}\left (x\right ) + \sqrt{a^{2} x^{2} + 1} a^{6}{\left | a \right |} \mathrm{sgn}\left (x\right ) + 2 \, a^{7}\right )}}{a^{4} x^{4}}}{16 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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