Optimal. Leaf size=105 \[ \frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{5 c^2 \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.325673, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {6177, 1805, 1807, 1809, 844, 216, 266, 63, 208} \[ \frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 x \sqrt{1-\frac{1}{a^2 x^2}}+\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{5 c^2 \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1805
Rule 1807
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^5}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-c^5+\frac{5 c^5 x}{a}+\frac{5 c^5 x^2}{a^2}-\frac{c^5 x^3}{a^3}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\operatorname{Subst}\left (\int \frac{-\frac{5 c^5}{a}-\frac{5 c^5 x}{a^2}+\frac{c^5 x^2}{a^3}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{a^2 \operatorname{Subst}\left (\int \frac{\frac{5 c^5}{a^3}+\frac{5 c^5 x}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{5 c^2 \csc ^{-1}(a x)}{a}+\frac{\left (5 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a}\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{5 c^2 \csc ^{-1}(a x)}{a}-\left (5 a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{16 c^2 \left (a-\frac{1}{x}\right )}{a^2 \sqrt{1-\frac{1}{a^2 x^2}}}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{5 c^2 \csc ^{-1}(a x)}{a}-\frac{5 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [C] time = 0.388522, size = 424, normalized size = 4.04 \[ \frac{c^2 \left (7 \sqrt{2} a x (a x-1)^3 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{5}{2},\frac{7}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+5 \sqrt{2} (a x-1)^4 (a x+1) \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{7}{2},\frac{9}{2},\frac{1}{2} \left (1-\frac{1}{a x}\right )\right )+35 a^6 x^6 \sqrt{\frac{1}{a x}+1}-595 a^5 x^5 \sqrt{\frac{1}{a x}+1}+280 a^4 x^4 \sqrt{\frac{1}{a x}+1}+315 a^3 x^3 \sqrt{\frac{1}{a x}+1}-35 a^2 x^2 \sqrt{\frac{1}{a x}+1}+910 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-105 a^5 x^5 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )+910 a^4 x^4 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )-105 a^4 x^4 \sqrt{1-\frac{1}{a x}} \sin ^{-1}\left (\frac{1}{a x}\right )-175 a^5 x^5 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{\frac{1}{a x}+1} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\right )}{35 a^5 x^4 \sqrt{1-\frac{1}{a x}} (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.137, size = 600, normalized size = 5.7 \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 = 1.54951, size = 201, normalized size = 1.91 \begin{align*} -{\left (\frac{4 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac{10 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{5 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{5 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{16 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86976, size = 286, normalized size = 2.72 \begin{align*} -\frac{10 \, a c^{2} x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + 5 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 5 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c^{2} x^{2} + 18 \, a c^{2} x + c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x} \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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