Integrand size = 20, antiderivative size = 76 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6302, 6265, 21, 52, 65, 212} \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}-\frac {4 \sqrt {c-a c x}}{a} \]
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Rule 21
Rule 52
Rule 65
Rule 212
Rule 6265
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x} \, dx \\ & = -\int \frac {(1-a x) \sqrt {c-a c x}}{1+a x} \, dx \\ & = -\frac {\int \frac {(c-a c x)^{3/2}}{1+a x} \, dx}{c} \\ & = -\frac {2 (c-a c x)^{3/2}}{3 a c}-2 \int \frac {\sqrt {c-a c x}}{1+a x} \, dx \\ & = -\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}-(4 c) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, dx \\ & = -\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {8 \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right )}{a} \\ & = -\frac {4 \sqrt {c-a c x}}{a}-\frac {2 (c-a c x)^{3/2}}{3 a c}+\frac {4 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {2 (-7+a x) \sqrt {c-a c x}+12 \sqrt {2} \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{3 a} \]
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Time = 0.53 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {2 \left (a x -7\right ) \left (a x -1\right ) c}{3 a \sqrt {-c \left (a x -1\right )}}+\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {c}}{a}\) | \(57\) |
derivativedivides | \(-\frac {2 \left (\frac {\left (-a c x +c \right )^{\frac {3}{2}}}{3}+2 c \sqrt {-a c x +c}-2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{c a}\) | \(59\) |
default | \(\frac {-\frac {2 \left (-a c x +c \right )^{\frac {3}{2}}}{3}-4 c \sqrt {-a c x +c}+4 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{a c}\) | \(59\) |
pseudoelliptic | \(\frac {4 \sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )+\frac {2 a x \sqrt {-c \left (a x -1\right )}}{3}-\frac {14 \sqrt {-c \left (a x -1\right )}}{3}}{a}\) | \(59\) |
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Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.57 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (3 \, \sqrt {2} \sqrt {c} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}, -\frac {2 \, {\left (6 \, \sqrt {2} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - \sqrt {-a c x + c} {\left (a x - 7\right )}\right )}}{3 \, a}\right ] \]
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Time = 2.58 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.30 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\begin {cases} - \frac {2 \cdot \left (\frac {2 \sqrt {2} c^{2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {- a c x + c}}{2 \sqrt {- c}} \right )}}{\sqrt {- c}} + 2 c \sqrt {- a c x + c} + \frac {\left (- a c x + c\right )^{\frac {3}{2}}}{3}\right )}{a c} & \text {for}\: a c \neq 0 \\\sqrt {c} \left (\begin {cases} - x & \text {for}\: a = 0 \\\frac {a x - 2 \log {\left (a x + 1 \right )} + 1}{a} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.04 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {2 \, {\left (3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right ) + {\left (-a c x + c\right )}^{\frac {3}{2}} + 6 \, \sqrt {-a c x + c} c\right )}}{3 \, a c} \]
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Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.01 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=-\frac {4 \, \sqrt {2} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{a \sqrt {-c}} - \frac {2 \, {\left ({\left (-a c x + c\right )}^{\frac {3}{2}} a^{2} c^{2} + 6 \, \sqrt {-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \]
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Time = 4.52 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.80 \[ \int e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x} \, dx=\frac {4\,\sqrt {2}\,\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}}{2\,\sqrt {c}}\right )}{a}-\frac {2\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,a\,c}-\frac {4\,\sqrt {c-a\,c\,x}}{a} \]
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