Optimal. Leaf size=33 \[ -\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3} \]
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Rubi [A]
time = 0.07, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6310, 6313,
665} \begin {gather*} -\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{(c-a c x)^2} \, dx &=\frac {\int \frac {e^{\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 {\sqrt {1-\frac {x^2}{a^2}}}{\left (1-\frac {x}{a}\right )^3} \, dx,x,\frac {1}{x}\right )}{a^2 c^2}\\ &=-\frac {a^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}{3 c^2 \left (a-\frac {1}{x}\right )^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 34, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {1-\frac {1}{a^2 x^2}} x (1+a x)}{3 c^2 (-1+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 36, normalized size = 1.09
method | result | size |
gosper | \(-\frac {a x +1}{3 \left (a x -1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(36\) |
default | \(-\frac {a x +1}{3 \left (a x -1\right ) c^{2} \sqrt {\frac {a x -1}{a x +1}}\, a}\) | \(36\) |
trager | \(-\frac {\left (a x +1\right )^{2} \sqrt {-\frac {-a x +1}{a x +1}}}{3 a \,c^{2} \left (a x -1\right )^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 23, normalized size = 0.70 \begin {gather*} -\frac {1}{3 \, a c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 57, normalized size = 1.73 \begin {gather*} -\frac {{\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \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 \frac {1}{a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 49, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )}^{2} x^{2} + 1\right )}}{3 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{3} a c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 23, normalized size = 0.70 \begin {gather*} -\frac {1}{3\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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