Optimal. Leaf size=144 \[ \frac{x \sqrt{1-\frac{1}{a x}}}{c \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c} \]
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Rubi [A] time = 0.0981175, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6194, 99, 152, 12, 92, 208} \[ \frac{x \sqrt{1-\frac{1}{a x}}}{c \left (\frac{1}{a x}+1\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{\frac{1}{a x}+1}}+\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (\frac{1}{a x}+1\right )^{3/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 99
Rule 152
Rule 12
Rule 92
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{c-\frac{c}{a^2 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x}{a}}}{x^2 \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{a}+\frac{2 x}{a^2}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{-\frac{9}{a^2}+\frac{5 x}{a^3}}{x \sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{a^2 \operatorname{Subst}\left (\int -\frac{9}{a^3 x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{3 c}\\ &=\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{3 \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}\\ &=\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{3 \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}\\ &=\frac{5 \sqrt{1-\frac{1}{a x}}}{3 a c \left (1+\frac{1}{a x}\right )^{3/2}}+\frac{14 \sqrt{1-\frac{1}{a x}}}{3 a c \sqrt{1+\frac{1}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} x}{c \left (1+\frac{1}{a x}\right )^{3/2}}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.130275, size = 69, normalized size = 0.48 \[ \frac{\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^2 x^2+19 a x+14\right )}{(a x+1)^2}-\frac{9 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}}{3 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.131, size = 346, normalized size = 2.4 \begin{align*} -{\frac{1}{3\,a \left ( ax+1 \right ) c \left ( ax-1 \right ) } \left ( 9\,\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}-9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}+27\,\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}+6\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+27\,\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}+5\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-27\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+9\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -9\,\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 = 1.07444, size = 189, normalized size = 1.31 \begin{align*} -\frac{1}{3} \, a{\left (\frac{6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )} a^{2} c}{a x + 1} - a^{2} c} - \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 12 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c} + \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31951, size = 235, normalized size = 1.63 \begin{align*} -\frac{9 \,{\left (a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 9 \,{\left (a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (3 \, a^{2} x^{2} + 19 \, a x + 14\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{2} c x + a c\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 [A] time = 1.15778, size = 80, normalized size = 0.56 \begin{align*} \frac{3 \, \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{c{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} \mathrm{sgn}\left (a x + 1\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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