Optimal. Leaf size=22 \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6169, 844, 216, 266, 63, 208} \[ \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 6169
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {1+\frac {x}{a}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\csc ^{-1}(a x)-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )\\ &=-\csc ^{-1}(a x)+a^2 \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.64 \[ \log \left (x \left (\sqrt {\frac {a^2 x^2-1}{a^2 x^2}}+1\right )\right )-\sin ^{-1}\left (\frac {1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.67, size = 57, normalized size = 2.59 \[ 2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 70, normalized size = 3.18 \[ a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} - \frac {\log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 131, normalized size = 5.95 \[ \frac {\left (a x -1\right ) \left (\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}+a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 69, normalized size = 3.14 \[ a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 37, normalized size = 1.68 \[ 2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \]
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 \[ \int \frac {1}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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