Optimal. Leaf size=181 \[ \frac{x \sqrt{1-\frac{1}{a x}}}{c^2 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]
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Rubi [A] time = 0.123586, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {6194, 103, 21, 99, 152, 12, 92, 208} \[ \frac{x \sqrt{1-\frac{1}{a x}}}{c^2 \left (\frac{1}{a x}+1\right )^{5/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{\frac{1}{a x}+1}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (\frac{1}{a x}+1\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 103
Rule 21
Rule 99
Rule 152
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{a}-\frac{3 x}{a^2}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}}}{x \left (1+\frac{x}{a}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{6 \operatorname{Subst}\left (\int \frac{-\frac{5}{2}+\frac{2 x}{a}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{15}{2 a}+\frac{9 x}{2 a^2}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{5 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{(2 a) \operatorname{Subst}\left (\int -\frac{15}{2 a^2 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{5 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a}-\frac{x^2}{a}} \, dx,x,\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a^2 c^2}\\ &=\frac{6 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{5/2}}+\frac{9 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{24 \sqrt{1-\frac{1}{a x}}}{5 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c^2 \left (1+\frac{1}{a x}\right )^{5/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.167981, size = 78, normalized size = 0.43 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (5 a^3 x^3+39 a^2 x^2+57 a x+24\right )}{5 (a x+1)^3}-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.151, size = 438, normalized size = 2.4 \begin{align*} -{\frac{1}{40\,a \left ( ax+1 \right ) ^{2}{c}^{2} \left ( ax-1 \right ) } \left ( 120\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{4}{a}^{5}-125\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}+480\,\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}+85\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-500\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+720\,\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}+148\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-750\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+480\,\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}+67\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-500\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+120\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -125\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\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 = 1.0334, size = 217, normalized size = 1.2 \begin{align*} -\frac{1}{20} \, a{\left (\frac{40 \, \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} - \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 85 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} + \frac{60 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{60 \, \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 = 1.24718, size = 315, normalized size = 1.74 \begin{align*} -\frac{15 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \,{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (5 \, a^{3} x^{3} + 39 \, a^{2} x^{2} + 57 \, a x + 24\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\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(-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 [A] time = 1.18331, size = 80, normalized size = 0.44 \begin{align*} \frac{3 \, \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{c^{2}{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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