Optimal. Leaf size=28 \[ \frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6310, 6313,
651} \begin {gather*} \frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 651
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^2 x^2} \, dx}{a^2 c^2}\\ &=-\frac {\text {Subst}\left (\int \frac {1-\frac {x}{a}}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^2 c^2}\\ &=\frac {a-\frac {1}{x}}{a^2 c^2 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 0.93 \begin {gather*} \frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^2 (1+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 35, normalized size = 1.25
method | result | size |
trager | \(\frac {\sqrt {-\frac {-a x +1}{a x +1}}}{a \,c^{2}}\) | \(25\) |
gosper | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\) | \(35\) |
default | \(\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )}{\left (a x -1\right ) a \,c^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 22, normalized size = 0.79 \begin {gather*} \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 22, normalized size = 0.79 \begin {gather*} \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{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 {\int \left (- \frac {\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 \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - a^{2} x^{2} - a x + 1}\, dx}{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 = 0.03, size = 22, normalized size = 0.79 \begin {gather*} \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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