Optimal. Leaf size=74 \[ \frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \]
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Rubi [A]
time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6312, 837, 821,
272, 65, 214} \begin {gather*} -\frac {x \left (a-\frac {1}{x}\right )}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {2 x \sqrt {1-\frac {1}{a^2 x^2}}}{c^2}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 837
Rule 6312
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {c-\frac {c x}{a}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {a^2 \text {Subst}\left (\int \frac {\frac {2 c}{a^2}-\frac {c x}{a^3}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^2}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^2}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^2}\\ &=\frac {2 \sqrt {1-\frac {1}{a^2 x^2}} x}{c^2}-\frac {\left (a-\frac {1}{x}\right ) x}{a c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 69, normalized size = 0.93 \begin {gather*} \frac {-2+a x+a^2 x^2-a \sqrt {1-\frac {1}{a^2 x^2}} x \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(249\) vs.
\(2(68)=136\).
time = 0.10, size = 250, normalized size = 3.38
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a \,c^{2}}+\frac {\left (-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{2} \sqrt {a^{2}}}+\frac {\sqrt {a^{2} \left (x +\frac {1}{a}\right )^{2}-2 a \left (x +\frac {1}{a}\right )}}{a^{4} \left (x +\frac {1}{a}\right )}\right ) a^{2} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{2} \left (a x -1\right )}\) | \(138\) |
default | \(-\frac {\left (-3 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+2 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}+\left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-6 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x +4 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x -3 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+2 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{2 a \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x -1\right )}\) | \(250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 125, normalized size = 1.69 \begin {gather*} -a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2} c^{2}}{a x + 1} - a^{2} c^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 67, normalized size = 0.91 \begin {gather*} \frac {{\left (a x + 2\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a c^{2}} \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*} \frac {a^{2} \left (\int \left (- \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\right )\, dx + \int \frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx\right )}{c^{2}} \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 = 1.19, size = 90, normalized size = 1.22 \begin {gather*} \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2-\frac {a\,c^2\,\left (a\,x-1\right )}{a\,x+1}}+\frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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