Optimal. Leaf size=62 \[ \frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right )}{a}-\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^2 \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.0960085, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6177, 813, 844, 216, 266, 63, 208} \[ \frac{c^2 x \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right )}{a}-\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}+\frac{c^2 \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 813
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac{c}{a x}\right )^2 \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right ) x}{a}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{\frac{2 c}{a}+\frac{2 c x}{a^2}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}} \left (a+\frac{1}{x}\right ) x}{a}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{c^2 \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}} \left (a+\frac{1}{x}\right ) x}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}+\frac{c^2 \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}} \left (a+\frac{1}{x}\right ) x}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}-\left (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}} \left (a+\frac{1}{x}\right ) x}{a}+\frac{c^2 \csc ^{-1}(a x)}{a}-\frac{c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [B] time = 0.162609, size = 158, normalized size = 2.55 \[ \frac{c^2 \left (2 a^3 x^3+2 a^2 x^2-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{\sqrt{1-\frac{1}{a x}}}{\sqrt{2}}\right )+a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sin ^{-1}\left (\frac{1}{a x}\right )-2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-2 a x-2\right )}{2 a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.126, size = 166, normalized size = 2.7 \begin{align*}{\frac{{c}^{2} \left ( ax-1 \right ) }{{a}^{2}x} \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) - \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}+\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2} \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52721, size = 169, normalized size = 2.73 \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{2 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63475, size = 278, normalized size = 4.48 \begin{align*} -\frac{2 \, a c^{2} x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c^{2} x^{2} + 2 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c^{2} \left (\int \frac{a^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int \frac{1}{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{2 a}{x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20774, size = 165, normalized size = 2.66 \begin{align*} -a{\left (\frac{2 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{4 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2}{\left (\frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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