Optimal. Leaf size=29 \[ -\frac {2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6174} \[ -\frac {2 (a x+1) e^{-3 \coth ^{-1}(a x)}}{a (c-a c x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=-\frac {2 e^{-3 \coth ^{-1}(a x)} (1+a x)}{a (c-a c x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.41 \[ -\frac {2 x \left (1-\frac {1}{a x}\right )^{3/2}}{\sqrt {\frac {1}{a x}+1} (c-a c x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 43, normalized size = 1.48 \[ -\frac {2 \, \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} x - a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 41, normalized size = 1.41 \[ \frac {{\left (\frac {\sqrt {2}}{a \sqrt {-c}} - \frac {2}{\sqrt {-a c x - c} a}\right )} {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 35, normalized size = 1.21 \[ -\frac {2 \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{a \left (-a c x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 45, normalized size = 1.55 \[ -\frac {2 \, {\left (a \sqrt {-c} x + \sqrt {-c}\right )} {\left (a x - 1\right )}}{{\left (a^{2} c^{2} x - a c^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 32, normalized size = 1.10 \[ \frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a\,c\,\sqrt {c-a\,c\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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