Optimal. Leaf size=180 \[ \frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]
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Rubi [A] time = 0.115933, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{\frac{1}{a x}+1}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{\frac{1}{a x}+1}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 103
Rule 152
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{1}{a}-\frac{3 x}{a^2}}{x \left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{a \operatorname{Subst}\left (\int \frac{\frac{3}{a^2}+\frac{8 x}{a^3}}{x \left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^2 \operatorname{Subst}\left (\int \frac{-\frac{3}{a^3}-\frac{11 x}{a^4}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{a^3 \operatorname{Subst}\left (\int -\frac{3}{a^4 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c^2}\\ &=-\frac{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{\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{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\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{4}{3 a c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}-\frac{11}{3 a c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}+\frac{8 \sqrt{1-\frac{1}{a x}}}{3 a c^2 \sqrt{1+\frac{1}{a x}}}+\frac{x}{c^2 \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}}}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c^2}\\ \end{align*}
Mathematica [A] time = 0.153513, size = 83, normalized size = 0.46 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^3 x^3-7 a^2 x^2-5 a x+8\right )}{3 (a x-1)^2 (a x+1)}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.149, size = 530, normalized size = 2.9 \begin{align*} -{\frac{1}{24\,a \left ( ax+1 \right ) ^{2} \left ( ax-1 \right ) ^{2}{c}^{2}} \left ( -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}^{5}{a}^{6}+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}+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}+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}^{3}{a}^{4}-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}-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}-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\,\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}+45\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \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 = 1.0066, size = 216, normalized size = 1.2 \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^{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{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} + \frac{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37424, size = 306, normalized size = 1.7 \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^{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^{4} \int \frac{x^{4}}{a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}} - 2 a^{2} x^{2} \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 = 1.14055, size = 231, normalized size = 1.28 \begin{align*} \frac{1}{12} \, a{\left (\frac{12 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac{12 \, \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{18 \,{\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{3 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac{24 \, \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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