Optimal. Leaf size=105 \[ -\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
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Rubi [A] time = 0.292632, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^2}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^2} \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^2 \left (c-\frac{c x}{a}\right )^3} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^3}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{\operatorname{Subst}\left (\int \frac{-3 c^3-\frac{9 c^3 x}{a}-\frac{8 c^3 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{3 c^3+\frac{9 c^3 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c^2}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c^2}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{3 a^2 c^2 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{9 a+\frac{11}{x}}{3 a^2 c^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^2}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.0625943, size = 94, normalized size = 0.9 \[ \frac{3 a^3 x^3-16 a^2 x^2+9 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-5 a x+14}{3 a^2 c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.139, size = 339, normalized size = 3.2 \begin{align*}{\frac{1}{3\,a \left ( ax-1 \right ) ^{2}{c}^{2}} \left ( 9\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{3}{a}^{4}+9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-27\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}-6\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+27\,\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}+5\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-9\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02613, size = 185, normalized size = 1.76 \begin{align*} \frac{1}{3} \, a{\left (\frac{\frac{11 \,{\left (a x - 1\right )}}{a x + 1} - \frac{18 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60619, size = 309, normalized size = 2.94 \begin{align*} \frac{9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{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*} \frac{a^{2} \int \frac{x^{2}}{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 2 a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} + \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16618, size = 200, normalized size = 1.9 \begin{align*} \frac{1}{3} \, a{\left (\frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{9 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac{{\left (a x + 1\right )}{\left (\frac{12 \,{\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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