Optimal. Leaf size=133 \[ -\frac{27}{4} a^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{3}{x}\right )-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{51}{8} a^4 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.8388, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6169, 1633, 1593, 12, 852, 1635, 1815, 27, 743, 641, 216} \[ -\frac{27}{4} a^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{3}{x}\right )-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{51}{8} a^4 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 1633
Rule 1593
Rule 12
Rule 852
Rule 1635
Rule 1815
Rule 27
Rule 743
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3 \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}} \left (a x^3-x^4\right )}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(a-x) x^3 \sqrt{1-\frac{x^2}{a^2}}}{\left (1+\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{a^2 x^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )}{a^2}\\ &=-\operatorname{Subst}\left (\int \frac{x^3 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1+\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^3 \left (1-\frac{x}{a}\right )^3}{\left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2 \left (-3 a^3+a^2 x-a x^2\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{1}{4} a^2 \operatorname{Subst}\left (\int \frac{12 a-28 x+\frac{27 x^2}{a}-\frac{12 x^3}{a^2}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}+\frac{1}{12} a^4 \operatorname{Subst}\left (\int \frac{-\frac{36}{a}+\frac{108 x}{a^2}-\frac{81 x^2}{a^3}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}+\frac{1}{12} a^4 \operatorname{Subst}\left (\int -\frac{9 (2 a-3 x)^2}{a^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{1}{4} (3 a) \operatorname{Subst}\left (\int \frac{(2 a-3 x)^2}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{3}{x}\right )-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}+\frac{1}{8} \left (3 a^3\right ) \operatorname{Subst}\left (\int \frac{-17+\frac{18 x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27}{4} a^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{3}{x}\right )-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{1}{8} \left (51 a^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{27}{4} a^4 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{9}{8} a^3 \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a-\frac{3}{x}\right )-\frac{a \left (a-\frac{1}{x}\right )^3}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a \sqrt{1-\frac{1}{a^2 x^2}}}{4 x^3}-\frac{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}{x^2}-\frac{51}{8} a^4 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0503231, size = 75, normalized size = 0.56 \[ -\frac{a \sqrt{1-\frac{1}{a^2 x^2}} \left (80 a^4 x^4+29 a^3 x^3-11 a^2 x^2+6 a x-2\right )}{8 x^3 (a x+1)}-\frac{51}{8} a^4 \sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.138, size = 690, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53044, size = 261, normalized size = 1.96 \begin{align*} \frac{1}{4} \,{\left (51 \, a^{3} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 16 \, a^{3} \sqrt{\frac{a x - 1}{a x + 1}} - \frac{77 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 149 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 123 \, a^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 35 \, 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 = 1.51496, size = 184, normalized size = 1.38 \begin{align*} \frac{102 \, a^{4} x^{4} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (80 \, a^{4} x^{4} + 29 \, a^{3} x^{3} - 11 \, a^{2} x^{2} + 6 \, a x - 2\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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