Optimal. Leaf size=77 \[ c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{3 c^2 \csc ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.238034, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6177, 1807, 1809, 844, 216, 266, 63, 208} \[ c^2 x \sqrt{1-\frac{1}{a^2 x^2}}-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}-\frac{3 c^2 \csc ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 1807
Rule 1809
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 &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c-\frac{c x}{a}\right )^3}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=c^2 \sqrt{1-\frac{1}{a^2 x^2}} x+\frac{\operatorname{Subst}\left (\int \frac{\frac{3 c^3}{a}-\frac{3 c^3 x}{a^2}+\frac{c^3 x^2}{a^3}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{3 c^3}{a^3}+\frac{3 c^3 x}{a^4}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{c^2 \sqrt{1-\frac{1}{a^2 x^2}}}{a}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{\left (3 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 (3 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}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3 c^2 \csc ^{-1}(a x)}{a}+\frac{\left (3 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}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3 c^2 \csc ^{-1}(a x)}{a}-\left (3 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}+c^2 \sqrt{1-\frac{1}{a^2 x^2}} x-\frac{3 c^2 \csc ^{-1}(a x)}{a}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.197717, size = 55, normalized size = 0.71 \[ \frac{c^2 \left (\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)-3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-3 \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.131, size = 227, normalized size = 3. \begin{align*}{\frac{ \left ( ax+1 \right ){c}^{2}}{{a}^{2}x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-3\,\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}-3\,ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +4\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-4\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{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 [A] time = 1.56537, size = 170, normalized size = 2.21 \begin{align*} -{\left (\frac{4 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} - \frac{6 \, c^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{3 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{3 \, 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 [A] time = 1.91077, size = 266, normalized size = 3.45 \begin{align*} \frac{6 \, a c^{2} x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) - 3 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) + 3 \, a c^{2} x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (a^{2} c^{2} x^{2} - 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 a^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{2}}\, dx + \int - \frac{2 a \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20112, size = 176, normalized size = 2.29 \begin{align*} \frac{6 \, c^{2} \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{3 \, c^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c^{2} \mathrm{sgn}\left (a x + 1\right )}{a} - \frac{2 \, c^{2} \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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