Optimal. Leaf size=80 \[ \frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
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Rubi [A] time = 0.284056, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6175, 6178, 852, 1805, 12, 266, 63, 208} \[ \frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 6175
Rule 6178
Rule 852
Rule 1805
Rule 12
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{c-a c x} \, dx &=-\frac{\int \frac{e^{3 \coth ^{-1}(a x)}}{\left (1-\frac{1}{a x}\right ) x} \, dx}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x \left (1-\frac{x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^4}{x \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{\operatorname{Subst}\left (\int \frac{-3-\frac{4 x}{a}+\frac{3 x^2}{a^2}}{x \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{\operatorname{Subst}\left (\int \frac{3}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 a c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{2 a c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}} x}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{c}\\ &=\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}+\frac{4}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}} x}-\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.0819563, size = 63, normalized size = 0.79 \[ \frac{\frac{4 x \sqrt{1-\frac{1}{a^2 x^2}} (2 a x-1)}{(a x-1)^2}-\frac{3 \log \left (a x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{a}}{3 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.206, size = 345, normalized size = 4.3 \begin{align*} -{\frac{1}{3\,ac \left ( ax-1 \right ) \left ( ax+1 \right ) } \left ( 3\,\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}+3\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-9\,\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}-3\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+9\,\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}+ \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}+9\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-3\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -3\,\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 = 1.0276, size = 128, normalized size = 1.6 \begin{align*} -\frac{1}{3} \, a{\left (\frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{2} c} - \frac{2 \,{\left (\frac{3 \,{\left (a x - 1\right )}}{a x + 1} + 1\right )}}{a^{2} c \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6133, size = 284, normalized size = 3.55 \begin{align*} -\frac{3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 3 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) - 4 \,{\left (2 \, a^{2} x^{2} + a x - 1\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{\int \frac{1}{\frac{a^{2} x^{2} \sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a x + 1} - \frac{2 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 = 1.27822, size = 146, normalized size = 1.82 \begin{align*} -\frac{1}{3} \, a{\left (\frac{3 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{3 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac{2 \,{\left (a x + 1\right )}{\left (\frac{3 \,{\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}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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