Optimal. Leaf size=91 \[ -\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )}-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{9}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.453295, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6169, 1633, 1593, 12, 793, 665, 216} \[ -\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )}-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{9}{2} a^2 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 1633
Rule 1593
Rule 12
Rule 793
Rule 665
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \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{\left (-a x-x^2\right ) \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-x) x \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 \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 \left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}+(3 a) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{\left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )}+\frac{1}{2} (9 a) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )}+\frac{1}{2} (9 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{9}{2} a^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}{\left (a-\frac{1}{x}\right )^3}-\frac{3 a^3 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}{2 \left (a-\frac{1}{x}\right )}+\frac{9}{2} a^2 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0930147, size = 56, normalized size = 0.62 \[ \frac{1}{2} a \left (\frac{\sqrt{1-\frac{1}{a^2 x^2}} \left (-14 a^2 x^2+5 a x+1\right )}{x (a x-1)}+9 a \sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.171, size = 642, normalized size = 7.1 \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 = 1.52817, size = 149, normalized size = 1.64 \begin{align*} -{\left (9 \, a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{15 \,{\left (a x - 1\right )} a}{a x + 1} + \frac{9 \,{\left (a x - 1\right )}^{2} a}{{\left (a x + 1\right )}^{2}} + 4 \, a}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 2 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + \sqrt{\frac{a x - 1}{a x + 1}}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94519, size = 194, normalized size = 2.13 \begin{align*} -\frac{18 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (14 \, a^{3} x^{3} + 9 \, a^{2} x^{2} - 6 \, a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \,{\left (a x^{3} - x^{2}\right )}} \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{1}{x^{3} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18949, size = 146, normalized size = 1.6 \begin{align*} -{\left (9 \, a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{4 \, a}{\sqrt{\frac{a x - 1}{a x + 1}}} + \frac{\frac{5 \,{\left (a x - 1\right )} a \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 7 \, a \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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