Optimal. Leaf size=63 \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6169, 835, 807, 266, 63, 208} \[ \frac {1}{2} x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{a}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 6169
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} x \, dx &=-\operatorname {Subst}\left (\int \frac {1+\frac {x}{a}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{2} \operatorname {Subst}\left (\int \frac {-\frac {2}{a}-\frac {x}{a^2}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2-\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}\\ &=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2-\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 {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^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 {\sqrt {1-\frac {1}{a^2 x^2}} x}{a}+\frac {1}{2} \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 49, normalized size = 0.78 \[ \frac {a x \sqrt {1-\frac {1}{a^2 x^2}} (a x+2)+\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 73, normalized size = 1.16 \[ \frac {{\left (a^{2} x^{2} + 3 \, a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}} + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 118, normalized size = 1.87 \[ \frac {1}{2} \, a {\left (\frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{3}} - \frac {2 \, {\left (\frac {{\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - 3 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{3} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 152, normalized size = 2.41 \[ \frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a +2 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{2 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 128, normalized size = 2.03 \[ \frac {1}{2} \, a {\left (\frac {2 \, {\left (\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {2 \, {\left (a x - 1\right )} a^{3}}{a x + 1} - \frac {{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 98, normalized size = 1.56 \[ \frac {3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{a^2+\frac {a^2\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {2\,a^2\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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