Optimal. Leaf size=62 \[ \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6175, 6178, 793, 651} \[ \frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )} \]
Antiderivative was successfully verified.
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Rule 651
Rule 793
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{-\coth ^{-1}(a x)}}{(c-a c x)^3} \, dx &=-\frac {\int \frac {e^{-\coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^3 x^3} \, dx}{a^3 c^3}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {x}{a}\right )^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a^3 c^3}\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 a^2 c^3}\\ &=\frac {a \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )^2}-\frac {2 \sqrt {1-\frac {1}{a^2 x^2}}}{3 c^3 \left (a-\frac {1}{x}\right )}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 34, normalized size = 0.55 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} (a x-2)}{3 c^3 (a x-1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 57, normalized size = 0.92 \[ -\frac {{\left (a^{2} x^{2} - a x - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 45, normalized size = 0.73 \[ \frac {2 \, {\left (3 \, {\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}}{3 \, {\left ({\left (a + \sqrt {a^{2} - \frac {1}{x^{2}}}\right )} x - 1\right )}^{3} a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 41, normalized size = 0.66 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x -2\right ) \left (a x +1\right )}{3 \left (a x -1\right )^{2} c^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 39, normalized size = 0.63 \[ -\frac {\frac {3 \, {\left (a x - 1\right )}}{a x + 1} - 1}{6 \, a c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 38, normalized size = 0.61 \[ -\frac {\frac {a\,x-1}{a\,x+1}-\frac {1}{3}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \]
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 \[ - \frac {\int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{3} x^{3} - 3 a^{2} x^{2} + 3 a x - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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