Optimal. Leaf size=233 \[ \frac{1}{5} a^4 c^2 x^5 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{4} a^3 c^2 x^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{1}{4} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a} \]
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Rubi [A] time = 0.199331, 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} \[ \frac{1}{5} a^4 c^2 x^5 \left (1-\frac{1}{a x}\right )^{5/2} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{4} a^3 c^2 x^4 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{5/2}+\frac{1}{4} a^2 c^2 x^3 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{5/2}-\frac{1}{8} a c^2 x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{3}{8} c^2 x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{3 c^2 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 6191
Rule 6195
Rule 94
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\left (a^4 c^2\right ) \int e^{-\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{\left (1-\frac{x}{a}\right )^{5/2} \left (1+\frac{x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\left (a^3 c^2\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \left (1+\frac{x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{1}{4} \left (3 a^2 c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{1}{4} \left (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{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{1}{8} \left (3 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{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5+\frac{\left (3 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{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{\left (3 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{3}{8} c^2 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x-\frac{1}{8} a c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2+\frac{1}{4} a^2 c^2 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{5/2} x^3-\frac{1}{4} a^3 c^2 \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{5/2} x^4+\frac{1}{5} a^4 c^2 \left (1-\frac{1}{a x}\right )^{5/2} \left (1+\frac{1}{a x}\right )^{5/2} x^5-\frac{3 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 = 0.10375, size = 79, normalized size = 0.34 \[ \frac{c^2 \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-15 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{40 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.138, size = 183, normalized size = 0.8 \begin{align*} -{\frac{ \left ( ax+1 \right ){c}^{2}}{120\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -24\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+30\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+40\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-16\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}\sqrt{{a}^{2}}-45\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+45\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a \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 = 1.13966, size = 350, normalized size = 1.5 \begin{align*} -\frac{1}{40} \, a{\left (\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac{2 \,{\left (15 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 70 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 128 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 70 \, c^{2} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 15 \, 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 = 2.00816, size = 285, normalized size = 1.22 \begin{align*} -\frac{15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (8 \, a^{5} c^{2} x^{5} - 2 \, a^{4} c^{2} x^{4} - 26 \, a^{3} c^{2} x^{3} + 9 \, a^{2} c^{2} x^{2} + 33 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{40 \, 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 - 2 a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int a^{4} x^{4} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int \sqrt{\frac{a x}{a x + 1} - \frac{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 = 1.16409, size = 170, normalized size = 0.73 \begin{align*} \frac{3 \, c^{2} \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{8 \,{\left | a \right |}} + \frac{1}{40} \, \sqrt{a^{2} x^{2} - 1}{\left ({\left (25 \, c^{2} \mathrm{sgn}\left (a x + 1\right ) - 2 \,{\left (8 \, a c^{2} \mathrm{sgn}\left (a x + 1\right ) -{\left (4 \, a^{3} c^{2} x \mathrm{sgn}\left (a x + 1\right ) - 5 \, a^{2} c^{2} \mathrm{sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x + \frac{8 \, c^{2} \mathrm{sgn}\left (a x + 1\right )}{a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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