Optimal. Leaf size=22 \[ -\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right ) \]
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Rubi [A]
time = 0.03, 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 = {6304, 858, 222,
272, 65, 214} \begin {gather*} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 6304
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx &=-\text {Subst}\left (\int \frac {1+\frac {x}{a}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {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} \text {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 \text {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 \begin {gather*} -\text {ArcSin}\left (\frac {1}{a x}\right )+\log \left (x \left (1+\sqrt {\frac {-1+a^2 x^2}{a^2 x^2}}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(130\) vs.
\(2(20)=40\).
time = 0.07, size = 131, normalized size = 5.95
method | result | size |
default | \(\frac {\left (a x -1\right ) \left (\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}+a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs.
\(2 (20) = 40\).
time = 0.47, size = 69, normalized size = 3.14 \begin {gather*} 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (20) = 40\).
time = 0.35, size = 57, normalized size = 2.59 \begin {gather*} 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 ) \end {gather*}
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 \begin {gather*} \int \frac {1}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 63 vs.
\(2 (20) = 40\).
time = 0.41, size = 63, normalized size = 2.86 \begin {gather*} \frac {2 \, \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} - \frac {a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
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 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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