Optimal. Leaf size=62 \[ -\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A]
time = 0.59, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6303, 6874, 665,
270, 272, 65, 214} \begin {gather*} x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 270
Rule 272
Rule 665
Rule 6303
Rule 6874
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \, dx &=-\text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^2}{x^2 \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 {4}{a (a-x) \sqrt {1-\frac {x^2}{a^2}}}+\frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}}+\frac {3}{a x \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\frac {4 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+(3 a) \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 \sqrt {1-\frac {1}{a^2 x^2}}}{a-\frac {1}{x}}+\sqrt {1-\frac {1}{a^2 x^2}} x+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 54, normalized size = 0.87 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x (-5+a x)}{-1+a x}+\frac {3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a} \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. \(247\) vs.
\(2(56)=112\).
time = 0.09, size = 248, normalized size = 4.00
method | result | size |
risch | \(\frac {a x -1}{a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{\sqrt {a^{2}}}-\frac {4 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{a^{2} \left (x -\frac {1}{a}\right )}\right ) \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(133\) |
default | \(-\frac {-3 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+2 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+6 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +6 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}-3 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )}{a \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(248\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 110, normalized size = 1.77 \begin {gather*} -a {\left (\frac {2 \, {\left (\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 2\right )}}{a^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 92, normalized size = 1.48 \begin {gather*} \frac {3 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (a^{2} x^{2} - 4 \, a x - 5\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} x - a} \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}{\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 = 1.29, size = 59, normalized size = 0.95 \begin {gather*} \frac {2\,a\,x+12\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}-10}{2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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