Optimal. Leaf size=146 \[ \frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
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Rubi [A] time = 0.272623, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {6182, 6180, 84, 156, 63, 208, 206} \[ \frac{2 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6180
Rule 84
Rule 156
Rule 63
Rule 208
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{\left (a \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{-\frac{1}{a}-\frac{3 x}{a^2}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{a \sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (8 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{\left (2 a \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.369214, size = 218, normalized size = 1.49 \[ \frac{2 a x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}}{a x-1}+\sqrt{c} \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )-2 \sqrt{2} \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )-\sqrt{c} \log (1-a x)+2 \sqrt{2} \sqrt{c} \log \left ((a x-1)^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.18, size = 161, normalized size = 1.1 \begin{align*} -{\frac{ax-1}{ax+1}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,\sqrt{a}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) x-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) \sqrt{{a}^{-1}}xa-2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}\sqrt{{a}^{-1}} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}}}{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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