Optimal. Leaf size=45 \[ \frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2}{a^2 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6177, 271, 191} \[ \frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2}{a^2 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{a^2 c^3}\\ &=-\frac {2}{a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {x}{c^3 \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.73 \[ \frac {a^2 x^2-2}{a^2 c^3 x \sqrt {1-\frac {1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 42, normalized size = 0.93 \[ \frac {{\left (a^{2} x^{2} - 2\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3} x - a c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 42, normalized size = 0.93 \[ \frac {{\left (\frac {\sqrt {a^{2} x^{2} - 1}}{c^{3}} - \frac {1}{\sqrt {a^{2} x^{2} - 1} c^{3}}\right )} \mathrm {sgn}\left (a x + 1\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 0.98 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a^{2} x^{2}-2\right ) \left (a x +1\right )}{a \left (a x -1\right )^{2} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.30, size = 92, normalized size = 2.04 \[ -\frac {1}{2} \, a {\left (\frac {\frac {5 \, {\left (a x - 1\right )}}{a x + 1} - 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 41, normalized size = 0.91 \[ \frac {a^2\,x^2-2}{\left (x\,a^2\,c^3+a\,c^3\right )\,\sqrt {\frac {a\,x-1}{a\,x+1}}} \]
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 \[ \frac {a^{3} \left (\int \left (- \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\right )\, dx + \int \frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx\right )}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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