Optimal. Leaf size=38 \[ -\frac {4}{3 a (c-a c x)^{3/2}}+\frac {2}{a c \sqrt {c-a c x}} \]
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Rubi [A]
time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6302, 6265, 21,
45} \begin {gather*} \frac {2}{a c \sqrt {c-a c x}}-\frac {4}{3 a (c-a c x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 45
Rule 6265
Rule 6302
Rubi steps
\begin {align*} \int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx &=-\int \frac {e^{2 \tanh ^{-1}(a x)}}{(c-a c x)^{3/2}} \, dx\\ &=-\int \frac {1+a x}{(1-a x) (c-a c x)^{3/2}} \, dx\\ &=-\left (c \int \frac {1+a x}{(c-a c x)^{5/2}} \, dx\right )\\ &=-\left (c \int \left (\frac {2}{(c-a c x)^{5/2}}-\frac {1}{c (c-a c x)^{3/2}}\right ) \, dx\right )\\ &=-\frac {4}{3 a (c-a c x)^{3/2}}+\frac {2}{a c \sqrt {c-a c x}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 0.89 \begin {gather*} -\frac {2 (-1+3 a x) \sqrt {c-a c x}}{3 a c^2 (-1+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 33, normalized size = 0.87
method | result | size |
gosper | \(-\frac {2 \left (3 a x -1\right )}{3 a \left (-a c x +c \right )^{\frac {3}{2}}}\) | \(21\) |
trager | \(-\frac {2 \left (3 a x -1\right ) \sqrt {-a c x +c}}{3 c^{2} \left (a x -1\right )^{2} a}\) | \(31\) |
derivativedivides | \(-\frac {2 \left (\frac {2 c}{3 \left (-a c x +c \right )^{\frac {3}{2}}}-\frac {1}{\sqrt {-a c x +c}}\right )}{c a}\) | \(33\) |
default | \(-\frac {2 \left (\frac {2 c}{3 \left (-a c x +c \right )^{\frac {3}{2}}}-\frac {1}{\sqrt {-a c x +c}}\right )}{c a}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 26, normalized size = 0.68 \begin {gather*} -\frac {2 \, {\left (3 \, a c x - c\right )}}{3 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 44, normalized size = 1.16 \begin {gather*} -\frac {2 \, \sqrt {-a c x + c} {\left (3 \, a x - 1\right )}}{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 [A]
time = 16.77, size = 29, normalized size = 0.76 \begin {gather*} - \frac {4}{3 a \left (- a c x + c\right )^{\frac {3}{2}}} + \frac {2}{a c \sqrt {- a c x + c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 36, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (3 \, a c x - c\right )}}{3 \, {\left (a c x - c\right )} \sqrt {-a c x + c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 0.53 \begin {gather*} -\frac {6\,a\,x-2}{3\,a\,{\left (c-a\,c\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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