Optimal. Leaf size=215 \[ \frac{a x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{7 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}} \]
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Rubi [A] time = 0.150193, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6182, 6179, 98, 151, 156, 63, 208, 206} \[ \frac{a x \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{7 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 6182
Rule 6179
Rule 98
Rule 151
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}}} \, dx &=\frac{\sqrt{1-\frac{1}{a x}} \int \frac{e^{3 \coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a x}}} \, dx}{\sqrt{c-\frac{c}{a x}}}\\ &=-\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{x^2 \left (1-\frac{x}{a}\right )^2} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a x}}}\\ &=\frac{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{\sqrt{1-\frac{1}{a x}} \operatorname{Subst}\left (\int \frac{-\frac{7}{2 a}-\frac{5 x}{2 a^2}}{x \left (1-\frac{x}{a}\right )^2 \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{c-\frac{c}{a x}}}\\ &=-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}-\frac{\left (a \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\frac{7}{a^2}+\frac{3 x}{a^3}}{x \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{c-\frac{c}{a x}}}\\ &=-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}-\frac{\left (5 \sqrt{1-\frac{1}{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^2 \sqrt{c-\frac{c}{a x}}}-\frac{\left (7 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a \sqrt{c-\frac{c}{a x}}}\\ &=-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}-\frac{\left (7 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{-a+a x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{c-\frac{c}{a x}}}-\frac{\left (10 \sqrt{1-\frac{1}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{c-\frac{c}{a x}}}\\ &=-\frac{3 \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{a \sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}} x}{\left (a-\frac{1}{x}\right ) \sqrt{c-\frac{c}{a x}}}+\frac{7 \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\sqrt{1+\frac{1}{a x}}\right )}{a \sqrt{c-\frac{c}{a x}}}-\frac{5 \sqrt{2} \sqrt{1-\frac{1}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{a \sqrt{c-\frac{c}{a x}}}\\ \end{align*}
Mathematica [A] time = 0.099784, size = 115, normalized size = 0.53 \[ \frac{\sqrt{1-\frac{1}{a x}} \left (a x \sqrt{\frac{1}{a x}+1} (a x-3)+7 (a x-1) \tanh ^{-1}\left (\sqrt{\frac{1}{a x}+1}\right )-5 \sqrt{2} (a x-1) \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )\right )}{a (a x-1) \sqrt{c-\frac{c}{a x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.181, size = 259, normalized size = 1.2 \begin{align*}{\frac{x}{ \left ( 2\,ax+2 \right ) c}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 2\,{a}^{5/2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}x-5\,{a}^{3/2}\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+7\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ){a}^{2}\sqrt{{a}^{-1}}x-6\,\sqrt{ \left ( ax+1 \right ) x}{a}^{3/2}\sqrt{{a}^{-1}}-7\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1}{\sqrt{a}}} \right ) a\sqrt{{a}^{-1}}+5\,\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ) \sqrt{a} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\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{1}{\sqrt{c - \frac{c}{a 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{1}{\sqrt{c - \frac{c}{a 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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