Optimal. Leaf size=46 \[ -\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
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Rubi [A] time = 0.76, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {6169, 6742, 216, 266, 63, 208, 651} \[ -\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}+\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 651
Rule 6169
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x \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}{a \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{x \sqrt {1-\frac {x^2}{a^2}}}-\frac {4}{(a+x) \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{(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 )\\ &=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\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 )\\ &=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\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 )\\ &=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\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 = 0.06, size = 55, normalized size = 1.20 \[ -\frac {4 a x \sqrt {1-\frac {1}{a^2 x^2}}}{a x+1}+\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )-\sin ^{-1}\left (\frac {1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.81, size = 74, normalized size = 1.61 \[ -4 \, \sqrt {\frac {a x - 1}{a x + 1}} + 2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 369, normalized size = 8.02 \[ \frac {\left (-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{2} a^{2}-a^{2} x^{2} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-2 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -2 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -2 a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+2 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}+a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{\sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 89, normalized size = 1.93 \[ a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} - \frac {4 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 54, normalized size = 1.17 \[ 2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-4\,\sqrt {\frac {a\,x-1}{a\,x+1}} \]
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 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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