Optimal. Leaf size=37 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6265, 21,
65, 212} \begin {gather*} \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 65
Rule 212
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\\ &=-\frac {\int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx}{c}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a c^2}\\ &=\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.00 \begin {gather*} \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 29, normalized size = 0.78
method | result | size |
derivativedivides | \(\frac {\arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a \,c^{\frac {3}{2}}}\) | \(29\) |
default | \(\frac {\arctanh \left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}}{a \,c^{\frac {3}{2}}}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 52, normalized size = 1.41 \begin {gather*} -\frac {\sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{2 \, a c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 88, normalized size = 2.38 \begin {gather*} \left [\frac {\sqrt {2} \log \left (\frac {a x - \frac {2 \, \sqrt {2} \sqrt {-a c x + c}}{\sqrt {c}} - 3}{a x + 1}\right )}{2 \, a c^{\frac {3}{2}}}, \frac {\sqrt {2} \sqrt {-\frac {1}{c}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-\frac {1}{c}}}{a x - 1}\right )}{a c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 8.76, size = 41, normalized size = 1.11 \begin {gather*} - \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{a c \sqrt {- c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 36, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 28, normalized size = 0.76 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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