Optimal. Leaf size=233 \[ \frac{1}{5} a^4 c^2 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{20} a^3 c^2 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{7}{60} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{7}{24} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{7}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{7 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a} \]
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Rubi [A] time = 0.212644, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac{1}{5} a^4 c^2 x^5 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{9/2}-\frac{1}{20} a^3 c^2 x^4 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{7/2}-\frac{7}{60} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{7}{24} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{7}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{7 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6191
Rule 6195
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\left (a^4 c^2\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^2 x^4 \, dx\\ &=-\left (\left (a^4 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{7/2}}{x^6} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{1}{5} \left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{7/2}}{x^5 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{1}{20} \left (7 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{5/2}}{x^4 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{7}{60} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{1}{12} \left (7 a c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^3 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{7}{24} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{7}{60} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{1}{8} \left (7 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{7}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{7}{24} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{7}{60} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5+\frac{\left (7 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 a}\\ &=-\frac{7}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{7}{24} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{7}{60} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{\left (7 c^2\right ) \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 )}{8 a^2}\\ &=-\frac{7}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{7}{24} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{7}{60} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{20} a^3 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{7/2} x^4+\frac{1}{5} a^4 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{9/2} x^5-\frac{7 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{8 a}\\ \end{align*}
Mathematica [A] time = 0.102832, size = 79, normalized size = 0.34 \[ \frac{c^2 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (24 a^4 x^4+90 a^3 x^3+112 a^2 x^2+15 a x-136\right )-105 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{120 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.184, size = 192, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax-1 \right ) ^{2}{c}^{2}}{120\,a \left ( ax+1 \right ) } \left ( 24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+90\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+16\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}+105\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+120\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-105\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08232, size = 350, normalized size = 1.5 \begin{align*} -\frac{1}{120} \, a{\left (\frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (105 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 490 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 896 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 790 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 105 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{5 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{10 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{10 \,{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac{5 \,{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac{{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67084, size = 301, normalized size = 1.29 \begin{align*} -\frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (24 \, a^{5} c^{2} x^{5} + 114 \, a^{4} c^{2} x^{4} + 202 \, a^{3} c^{2} x^{3} + 127 \, a^{2} c^{2} x^{2} - 121 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{120 \, a} \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*} c^{2} \left (\int - \frac{2 a^{2} x^{2}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{a^{4} x^{4}}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx + \int \frac{1}{\frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18678, size = 316, normalized size = 1.36 \begin{align*} -\frac{1}{120} \, a{\left (\frac{105 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{105 \, c^{2} \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{2 \,{\left (\frac{790 \,{\left (a x - 1\right )} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{896 \,{\left (a x - 1\right )}^{2} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{490 \,{\left (a x - 1\right )}^{3} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} - \frac{105 \,{\left (a x - 1\right )}^{4} c^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + 105 \, c^{2} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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