Optimal. Leaf size=111 \[ -\frac{x \left (4 a+\frac{3}{x}\right )}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{8 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]
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Rubi [A] time = 0.150347, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 857, 823, 807, 266, 63, 208} \[ -\frac{x \left (4 a+\frac{3}{x}\right )}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{8 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 857
Rule 823
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \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 ) \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^3}\\ &=-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{4 c}{a^2}-\frac{3 c x}{a^3}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}-\frac{\left (4 a+\frac{3}{x}\right ) x}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{a^4 \operatorname{Subst}\left (\int \frac{-\frac{8 c}{a^4}-\frac{3 c x}{a^5}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=\frac{8 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}-\frac{\left (4 a+\frac{3}{x}\right ) x}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c^4}\\ &=\frac{8 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}-\frac{\left (4 a+\frac{3}{x}\right ) x}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\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{8 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}-\frac{\left (4 a+\frac{3}{x}\right ) x}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{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{8 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 c^4}-\frac{a x}{3 c^4 \sqrt{1-\frac{1}{a^2 x^2}} \left (a-\frac{1}{x}\right )}-\frac{\left (4 a+\frac{3}{x}\right ) x}{3 a c^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.0665271, size = 94, normalized size = 0.85 \[ \frac{3 a^3 x^3-7 a^2 x^2+3 a x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1) \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-5 a x+8}{3 a^2 c^4 x \sqrt{1-\frac{1}{a^2 x^2}} (a x-1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.141, size = 523, normalized size = 4.7 \begin{align*}{\frac{1}{24\,a{c}^{4} \left ( ax-1 \right ) ^{4}} \left ( 24\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ){x}^{5}{a}^{6}+45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{5}{a}^{5}-24\,\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}-21\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{3}{a}^{3}-45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{4}{a}^{4}-48\,\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}-11\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}{x}^{2}{a}^{2}-90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+48\,\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}+5\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa+90\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+24\,\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}+19\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -45\,\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 = 1.02483, size = 216, normalized size = 1.95 \begin{align*} \frac{1}{12} \, a{\left (\frac{\frac{17 \,{\left (a x - 1\right )}}{a x + 1} - \frac{42 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82364, size = 306, normalized size = 2.76 \begin{align*} \frac{3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \,{\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} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c^{4} x^{2} - 2 \, 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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (c - \frac{c}{a x}\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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