Optimal. Leaf size=90 \[ \frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}+\frac{9 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2} \]
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Rubi [A] time = 0.842227, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {6169, 6742, 266, 51, 63, 208, 264, 651} \[ \frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}-\frac{3 x \sqrt{1-\frac{1}{a^2 x^2}}}{a}-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}+\frac{9 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 6742
Rule 266
Rule 51
Rule 63
Rule 208
Rule 264
Rule 651
Rubi steps
\begin{align*} \int e^{-3 \coth ^{-1}(a x)} x \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^3 \left (1+\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (\frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}}-\frac{3}{a x^2 \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{a^2 x \sqrt{1-\frac{x^2}{a^2}}}-\frac{4}{a^2 (a+x) \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{(a+x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{a}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )-\frac{2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+4 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{4 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{a^2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=-\frac{4 \sqrt{1-\frac{1}{a^2 x^2}}}{a \left (a+\frac{1}{x}\right )}-\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{9 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0616541, size = 66, normalized size = 0.73 \[ \frac{\frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (a^2 x^2-5 a x-14\right )}{a x+1}+9 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 421, normalized size = 4.7 \begin{align*}{\frac{1}{2\,{a}^{2} \left ( ax-1 \right ) } \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}-10\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}+2\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ){x}^{2}{a}^{3}+10\,\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}+4\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-20\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}+20\,\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}-10\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) a+10\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995983, size = 204, normalized size = 2.27 \begin{align*} -\frac{1}{2} \, a{\left (\frac{2 \,{\left (7 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 5 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{3}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} - \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{3}} + \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{3}} + \frac{8 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72874, size = 186, normalized size = 2.07 \begin{align*} \frac{{\left (a^{2} x^{2} - 5 \, a x - 14\right )} \sqrt{\frac{a x - 1}{a x + 1}} + 9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - 9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \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*} \int x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}\, dx \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*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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