Optimal. Leaf size=60 \[ x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rubi [A] time = 0.78, antiderivative size = 60, 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 = {6168, 6742, 264, 266, 63, 208, 651} \[ x \sqrt {1-\frac {1}{a^2 x^2}}+\frac {4 \sqrt {1-\frac {1}{a^2 x^2}}}{a+\frac {1}{x}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 264
Rule 266
Rule 651
Rule 6168
Rule 6742
Rubi steps
\begin {align*} \int e^{-3 \coth ^{-1}(a x)} \, dx &=-\operatorname {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 )\\ &=-\operatorname {Subst}\left (\int \left (\frac {1}{x^2 \sqrt {1-\frac {x^2}{a^2}}}-\frac {3}{a x \sqrt {1-\frac {x^2}{a^2}}}+\frac {4}{a (a+x) \sqrt {1-\frac {x^2}{a^2}}}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{(a+x) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\operatorname {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 \operatorname {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) \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 \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 = 0.05, size = 54, normalized size = 0.90 \[ \frac {x \sqrt {1-\frac {1}{a^2 x^2}} (a x+5)}{a x+1}-\frac {3 \log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 66, normalized size = 1.10 \[ \frac {{\left (a x + 5\right )} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} \]
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 = 248, normalized size = 4.13 \[ -\frac {\left (-3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+3 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}+2 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +6 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}-3 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+3 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \sqrt {a^{2}}\, \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 111, normalized size = 1.85 \[ -a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {4 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 78, normalized size = 1.30 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}+\frac {4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a}-\frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \int \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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