Optimal. Leaf size=46 \[ -\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right ) \]
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Rubi [A]
time = 0.56, 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 = {6304, 6874,
222, 665, 272, 65, 214} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 665
Rule 6304
Rule 6874
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{x} \, dx &=-\text {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 )\\ &=-\text {Subst}\left (\int \left (-\frac {1}{a \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{(a-x) \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{x \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\left (4 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\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 )\\ &=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\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 )\\ &=-\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\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 )\\ &=-\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 = 0.04, size = 53, normalized size = 1.15 \begin {gather*} -\frac {4 a \sqrt {1-\frac {1}{a^2 x^2}} x}{-1+a x}+\text {ArcSin}\left (\frac {1}{a x}\right )+\log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\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. \(371\) vs.
\(2(42)=84\).
time = 0.08, size = 372, normalized size = 8.09
method | result | size |
default | \(-\frac {-\sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}-\sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a^{2} x^{2}+2 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+2 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +2 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x +2 \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) a x -\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}-\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}}{\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(372\) |
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. 90 vs.
\(2 (42) = 84\).
time = 0.46, size = 90, normalized size = 1.96 \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} + \frac {4}{a \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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs.
\(2 (42) = 84\).
time = 0.35, size = 104, normalized size = 2.26 \begin {gather*} -\frac {2 \, {\left (a x - 1\right )} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + 4 \, {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x - 1} \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 \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 54, normalized size = 1.17 \begin {gather*} 2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {4}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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