Optimal. Leaf size=144 \[ \frac{x \sqrt{\frac{1}{a x}+1}}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{\frac{1}{a x}+1}}{3 a c \sqrt{1-\frac{1}{a x}}}-\frac{5 \sqrt{\frac{1}{a x}+1}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c} \]
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Rubi [A] time = 0.101411, 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} \[ \frac{x \sqrt{\frac{1}{a x}+1}}{c \left (1-\frac{1}{a x}\right )^{3/2}}-\frac{14 \sqrt{\frac{1}{a x}+1}}{3 a c \sqrt{1-\frac{1}{a x}}}-\frac{5 \sqrt{\frac{1}{a x}+1}}{3 a c \left (1-\frac{1}{a x}\right )^{3/2}}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}\right )}{a c} \]
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 \left (1-\frac{x}{a}\right )^{5/2} \sqrt{1+\frac{x}{a}}} \, 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 \left (1-\frac{x}{a}\right )^{3/2} \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{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 = 0.124484, 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 = 0.184, size = 346, normalized size = 2.4 \begin{align*}{\frac{1}{3\, \left ( ax-1 \right ) ac \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 ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06804, size = 180, normalized size = 1.25 \begin{align*} \frac{1}{3} \, a{\left (\frac{\frac{11 \,{\left (a x - 1\right )}}{a x + 1} - \frac{18 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - a^{2} c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}} + \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 = 1.58576, size = 301, normalized size = 2.09 \begin{align*} \frac{9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 9 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) +{\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{3 \,{\left (a^{3} c x^{2} - 2 \, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{a^{2} \int \frac{x^{2}}{\frac{a^{3} x^{3} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{a x \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} + \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1}}\, dx}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.195, size = 200, normalized size = 1.39 \begin{align*} \frac{1}{3} \, a{\left (\frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{9 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac{{\left (a x + 1\right )}{\left (\frac{12 \,{\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{6 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c{\left (\frac{a x - 1}{a x + 1} - 1\right )}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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