Optimal. Leaf size=105 \[ -\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{4 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
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Rubi [A] time = 0.301814, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac{8 \left (a+\frac{1}{x}\right )}{3 a^2 c \left (1-\frac{1}{a^2 x^2}\right )^{3/2}}-\frac{4 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{c}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 6177
Rule 852
Rule 1805
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{c-\frac{c}{a x}} \, dx &=-\left (c^3 \operatorname{Subst}\left (\int \frac{\left (1-\frac{x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac{c x}{a}\right )^4} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\left (c+\frac{c x}{a}\right )^4}{x^2 \left (1-\frac{x^2}{a^2}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{c^5}\\ &=-\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 c^4-\frac{12 c^4 x}{a}-\frac{13 c^4 x^2}{a^2}}{x^2 \left (1-\frac{x^2}{a^2}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\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 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\operatorname{Subst}\left (\int \frac{3 c^4+\frac{12 c^4 x}{a}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{3 c^5}\\ &=-\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 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c}-\frac{4 \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 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\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 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c}+\frac{(4 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 \left (3 a+\frac{4}{x}\right )}{3 a^2 c \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{c}+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a c}\\ \end{align*}
Mathematica [A] time = 0.122189, size = 70, normalized size = 0.67 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (3 a^2 x^2-26 a x+19\right )}{(a x-1)^2}+12 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{3 a c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.173, size = 346, normalized size = 3.3 \begin{align*}{\frac{1}{3\, \left ( ax-1 \right ) ac \left ( ax+1 \right ) } \left ( 12\,\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}+12\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{3}{a}^{3}-36\,\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}-9\,\sqrt{{a}^{2}} \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}xa-36\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+36\,\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}+7\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+36\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa-12\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) -12\,\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 = 1.05289, size = 180, normalized size = 1.71 \begin{align*} \frac{2}{3} \, a{\left (\frac{\frac{8 \,{\left (a x - 1\right )}}{a x + 1} - \frac{12 \,{\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{6 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{6 \, \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 = 1.53813, size = 304, normalized size = 2.9 \begin{align*} \frac{12 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 12 \,{\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} - 23 \, a^{2} x^{2} - 7 \, a x + 19\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 \int \frac{x}{\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 = 1.22507, size = 200, normalized size = 1.9 \begin{align*} \frac{2}{3} \, a{\left (\frac{6 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac{6 \, \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{9 \,{\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{3 \, \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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