Optimal. Leaf size=88 \[ \frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a-\frac{9}{x}\right )+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{3}{8} a^4 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.0895651, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6169, 833, 780, 216} \[ \frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a-\frac{9}{x}\right )+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{3}{8} a^4 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 833
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{x^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3 \left (1-\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{x^2 \left (\frac{3}{a}-\frac{4 x}{a^2}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}-\frac{1}{12} a^4 \operatorname{Subst}\left (\int \frac{x \left (\frac{8}{a^2}-\frac{9 x}{a^3}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a-\frac{9}{x}\right )-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}+\frac{1}{8} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{24} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a-\frac{9}{x}\right )-\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}+\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{3 x^2}+\frac{3}{8} a^4 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0939583, size = 59, normalized size = 0.67 \[ \frac{1}{24} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^3 x^3-9 a^2 x^2+8 a x-6\right )}{x^3}+9 a^3 \sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.138, size = 308, normalized size = 3.5 \begin{align*} -{\frac{ax+1}{24\,{x}^{4}}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -24\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{5}{a}^{5}+24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-9\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{4}{a}^{4}-9\,{a}^{4}{x}^{4}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}+24\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}-24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-15\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+8\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa-6\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51904, size = 234, normalized size = 2.66 \begin{align*} -\frac{1}{12} \,{\left (9 \, a^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - \frac{39 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 31 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 49 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 9 \, a^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{4 \,{\left (a x - 1\right )}}{a x + 1} + \frac{6 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + \frac{{\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93733, size = 180, normalized size = 2.05 \begin{align*} -\frac{18 \, a^{4} x^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (16 \, a^{4} x^{4} + 7 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x - 6\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{24 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a x - 1}{a x + 1}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18818, size = 348, normalized size = 3.95 \begin{align*} -\frac{3}{4} \, a^{4} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right ) + \frac{9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{7} a^{4} \mathrm{sgn}\left (a x + 1\right ) + 33 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{5} a^{4} \mathrm{sgn}\left (a x + 1\right ) + 48 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{4} a^{3}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) - 33 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{3} a^{4} \mathrm{sgn}\left (a x + 1\right ) + 64 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} a^{3}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right ) - 9 \,{\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )} a^{4} \mathrm{sgn}\left (a x + 1\right ) + 16 \, a^{3}{\left | a \right |} \mathrm{sgn}\left (a x + 1\right )}{12 \,{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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