Optimal. Leaf size=114 \[ \frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4} \]
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Rubi [A] time = 0.123382, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6169, 835, 807, 266, 63, 208} \[ \frac{1}{4} x^4 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x^3 \sqrt{1-\frac{1}{a^2 x^2}}}{3 a}+\frac{3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{8 a^2}+\frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{3 a^3}+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} x^3 \, dx &=-\operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{x^5 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-\frac{4}{a}-\frac{3 x}{a^2}}{x^4 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{1}{12} \operatorname{Subst}\left (\int \frac{\frac{9}{a^2}+\frac{8 x}{a^3}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{1}{24} \operatorname{Subst}\left (\int \frac{-\frac{16}{a^3}-\frac{9 x}{a^4}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^3}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a^4}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^3}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4-\frac{3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{16 a^4}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^3}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^2}\\ &=\frac{2 \sqrt{1-\frac{1}{a^2 x^2}} x}{3 a^3}+\frac{3 \sqrt{1-\frac{1}{a^2 x^2}} x^2}{8 a^2}+\frac{\sqrt{1-\frac{1}{a^2 x^2}} x^3}{3 a}+\frac{1}{4} \sqrt{1-\frac{1}{a^2 x^2}} x^4+\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.0688987, size = 68, normalized size = 0.6 \[ \frac{a x \sqrt{1-\frac{1}{a^2 x^2}} \left (6 a^3 x^3+8 a^2 x^2+9 a x+16\right )+9 \log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{24 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.187, size = 193, normalized size = 1.7 \begin{align*}{\frac{ax-1}{24\,{a}^{4}} \left ( 6\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+15\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa+8\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}+24\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }-15\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ) a+24\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02973, size = 274, normalized size = 2.4 \begin{align*} \frac{1}{24} \, a{\left (\frac{2 \,{\left (9 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 49 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 31 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 39 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{4 \,{\left (a x - 1\right )} a^{5}}{a x + 1} - \frac{6 \,{\left (a x - 1\right )}^{2} a^{5}}{{\left (a x + 1\right )}^{2}} + \frac{4 \,{\left (a x - 1\right )}^{3} a^{5}}{{\left (a x + 1\right )}^{3}} - \frac{{\left (a x - 1\right )}^{4} a^{5}}{{\left (a x + 1\right )}^{4}} - a^{5}} + \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{5}} - \frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{5}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68643, size = 227, normalized size = 1.99 \begin{align*} \frac{{\left (6 \, a^{4} x^{4} + 14 \, a^{3} x^{3} + 17 \, a^{2} x^{2} + 25 \, a x + 16\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 )}{24 \, a^{4}} \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 \frac{x^{3}}{\sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18111, size = 246, normalized size = 2.16 \begin{align*} \frac{1}{24} \, a{\left (\frac{9 \, \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{5}} - \frac{9 \, \log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{5}} - \frac{2 \,{\left (\frac{31 \,{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{49 \,{\left (a x - 1\right )}^{2} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac{9 \,{\left (a x - 1\right )}^{3} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} - 39 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{5}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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