Optimal. Leaf size=107 \[ c x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{a}+\frac{c \csc ^{-1}(a x)}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.0711992, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6194, 97, 154, 21, 105, 41, 216, 92, 208} \[ c x \sqrt{1-\frac{1}{a x}} \left (\frac{1}{a x}+1\right )^{3/2}-\frac{2 c \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{a}+\frac{c \csc ^{-1}(a x)}{a}+\frac{c \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a} \]
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{\sqrt{1-\frac{x}{a}} \left (1+\frac{x}{a}\right )^{3/2}}{x^2} \, dx,x,\frac{1}{x}\right )\right )\\ &=c \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 \sqrt{1-\frac{1}{a x}} \left (1+\frac{1}{a x}\right )^{3/2} 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 = 0.0782458, size = 53, normalized size = 0.5 \[ \frac{c \left (\sqrt{1-\frac{1}{a^2 x^2}} (a x-1)+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.153, size = 163, normalized size = 1.5 \begin{align*}{\frac{c \left ( ax-1 \right ) }{{a}^{2}x} \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{{\frac{ax-1}{ax+1}}}}}{\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 = 1.51892, size = 158, normalized size = 1.48 \begin{align*} -{\left (\frac{4 \, c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{\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 )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4371, size = 248, normalized size = 2.32 \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} - 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 \frac{a^{2}}{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx + \int - \frac{1}{x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17049, size = 170, normalized size = 1.59 \begin{align*} -a{\left (\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 ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac{4 \,{\left (a x - 1\right )} c \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )} a^{2}{\left (\frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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