Optimal. Leaf size=108 \[ c x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}+\frac{2 c \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{a}+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.0763682, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {6194, 97, 154, 21, 105, 41, 216, 92, 208} \[ c x \sqrt{\frac{1}{a x}+1} \left (1-\frac{1}{a x}\right )^{3/2}+\frac{2 c \sqrt{\frac{1}{a x}+1} \sqrt{1-\frac{1}{a x}}}{a}+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6194
Rule 97
Rule 154
Rule 21
Rule 105
Rule 41
Rule 216
Rule 92
Rule 208
Rubi steps
\begin{align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right ) \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^{3/2} \sqrt{1+\frac{x}{a}}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x-c \operatorname{Subst}\left (\int \frac{\left (-\frac{1}{a}-\frac{2 x}{a^2}\right ) \sqrt{1-\frac{x}{a}}}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{a}+c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x-(a c) \operatorname{Subst}\left (\int \frac{-\frac{1}{a^2}-\frac{x}{a^3}}{x \sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{a}+c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x+\frac{c \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{x \sqrt{1-\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{a}+c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x+\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x}{a}} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\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 )}{a}\\ &=\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{a}+c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x+\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}-\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 )}{a^2}\\ &=\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{a}+c \left (1-\frac{1}{a x}\right )^{3/2} \sqrt{1+\frac{1}{a x}} x+\frac{c \csc ^{-1}(a x)}{a}-\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.083748, size = 55, normalized size = 0.51 \[ \frac{c \left (\sqrt{1-\frac{1}{a^2 x^2}} (a x+1)-\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.13, size = 166, normalized size = 1.5 \begin{align*} -{\frac{c \left ( ax+1 \right ) }{{a}^{2}x}\sqrt{{\frac{ax-1}{ax+1}}} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}-\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) x{a}^{2}-ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \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 = 1.51355, size = 158, normalized size = 1.46 \begin{align*} -a{\left (\frac{4 \, c \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{{\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - a^{2}} + \frac{2 \, c \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a^{2}} + \frac{c \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac{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 = 1.42899, size = 262, normalized size = 2.43 \begin{align*} -\frac{2 \, a c x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + a c x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - a c x \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) -{\left (a^{2} c x^{2} + 2 \, a c x + c\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} x} \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{c \left (\int a^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}\, dx + \int - \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{x^{2}}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14917, size = 163, normalized size = 1.51 \begin{align*} -\frac{2 \, c \arctan \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1}\right ) \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{c \log \left ({\left | -x{\left | a \right |} + \sqrt{a^{2} x^{2} - 1} \right |}\right ) \mathrm{sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac{\sqrt{a^{2} x^{2} - 1} c \mathrm{sgn}\left (a x + 1\right )}{a} + \frac{2 \, c \mathrm{sgn}\left (a x + 1\right )}{{\left ({\left (x{\left | a \right |} - \sqrt{a^{2} x^{2} - 1}\right )}^{2} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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