Optimal. Leaf size=138 \[ -\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]
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Rubi [A] time = 0.390016, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]
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 )^4} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (c-\frac{c x}{a}\right )^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^3}{x^2 \left (1-\frac{x^2}{a^2}\right )^{7/2}} \, dx,x,\frac{1}{x}\right )}{c^7}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}+\frac{\operatorname{Subst}\left (\int \frac{-5 c^3-\frac{15 c^3 x}{a}-\frac{16 c^3 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{5 c^7}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{15 c^3+\frac{45 c^3 x}{a}+\frac{42 c^3 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-15 c^3-\frac{45 c^3 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{15 c^7}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}-\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^4}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\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^4}\\ &=-\frac{4 \left (a+\frac{1}{x}\right )}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}}-\frac{5 a+\frac{7}{x}}{5 a^2 c^4 \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{15 a+\frac{19}{x}}{5 a^2 c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c^4}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.0778986, size = 104, normalized size = 0.75 \[ \frac{5 a^4 x^4-34 a^3 x^3+18 a^2 x^2+15 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+33 a x-24}{5 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.148, size = 436, normalized size = 3.2 \begin{align*}{\frac{ax+1}{40\,a{c}^{4} \left ( ax-1 \right ) ^{4}}\sqrt{{\frac{ax-1}{ax+1}}} \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 ){\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 = 1.09118, size = 207, normalized size = 1.5 \begin{align*} \frac{1}{20} \, a{\left (\frac{\frac{9 \,{\left (a x - 1\right )}}{a x + 1} + \frac{75 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - \frac{125 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 1}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}} + \frac{60 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{60 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93405, size = 385, normalized size = 2.79 \begin{align*} \frac{15 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \,{\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (5 \, a^{4} x^{4} - 34 \, a^{3} x^{3} + 18 \, a^{2} x^{2} + 33 \, a x - 24\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{5 \,{\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\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^{4} \int \frac{x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{4} x^{4} - 4 a^{3} x^{3} + 6 a^{2} x^{2} - 4 a x + 1}\, dx}{c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19832, size = 80, normalized size = 0.58 \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^{4}{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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