Optimal. Leaf size=145 \[ -\frac{1}{3} a^2 c x^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{1}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{2 a} \]
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Rubi [A] time = 0.114582, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6191, 6195, 94, 92, 208} \[ -\frac{1}{3} a^2 c x^3 \left (1-\frac{1}{a x}\right )^{3/2} \left (\frac{1}{a x}+1\right )^{3/2}+\frac{1}{2} a c x^2 \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{1}{2} c x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{2 a} \]
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 ) \, dx &=-\left (\left (a^2 c\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right ) x^2 \, dx\right )\\ &=\left (a^2 c\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \sqrt{1+\frac{x}{a}}}{x^4} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{3} a^2 c \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3-(a c) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x^2 \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{1}{2} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3+\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=-\frac{1}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{1}{2} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3-\frac{c \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 )}{2 a^2}\\ &=-\frac{1}{2} c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x+\frac{1}{2} a c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} x^2-\frac{1}{3} a^2 c \left (1-\frac{1}{a x}\right )^{3/2} \left (1+\frac{1}{a x}\right )^{3/2} x^3-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{2 a}\\ \end{align*}
Mathematica [A] time = 0.0875979, size = 61, normalized size = 0.42 \[ \frac{c \left (a x \sqrt{1-\frac{1}{a^2 x^2}} \left (-2 a^2 x^2+3 a x+2\right )-3 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )\right )}{6 a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.134, size = 119, normalized size = 0.8 \begin{align*}{\frac{c \left ( ax+1 \right ) }{6\,a}\sqrt{{\frac{ax-1}{ax+1}}} \left ( 3\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-3\,\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 = 1.07162, size = 231, normalized size = 1.59 \begin{align*} \frac{1}{6} \, a{\left (\frac{2 \,{\left (3 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 8 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 3 \, c \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{3 \,{\left (a x - 1\right )} a^{2}}{a x + 1} - \frac{3 \,{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac{{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}} - \frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82624, size = 216, normalized size = 1.49 \begin{align*} -\frac{3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \, c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (2 \, a^{3} c x^{3} - a^{2} c x^{2} - 5 \, a c x - 2 \, c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{6 \, 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 \left (\int a^{2} x^{2} \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 = 1.15411, size = 111, normalized size = 0.77 \begin{align*} \frac{c \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{2 \,{\left | a \right |}} - \frac{1}{6} \, \sqrt{a^{2} x^{2} - 1}{\left ({\left (2 \, a c x \mathrm{sgn}\left (a x + 1\right ) - 3 \, c \mathrm{sgn}\left (a x + 1\right )\right )} x - \frac{2 \, c \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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