Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2320, 65, 214} \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rule 65
Rule 214
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,e^{c+d x}\right )}{d} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b e^{c+d x}}\right )}{b d} \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b e^{c+d x}}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
default | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,{\mathrm e}^{d x +c}}}{\sqrt {a}}\right )}{d \sqrt {a}}\) | \(26\) |
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none
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.59 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=\left [\frac {\log \left ({\left (b e^{\left (d x + c\right )} - 2 \, \sqrt {b e^{\left (d x + c\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-d x - c\right )}\right )}{\sqrt {a} d}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} \sqrt {-a}}{a}\right )}{a d}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (29) = 58\).
Time = 0.58 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=\begin {cases} \frac {\begin {cases} \frac {2 \operatorname {atan}{\left (\frac {\sqrt {a + b e^{c} e^{d x}}}{\sqrt {- a}} \right )}}{\sqrt {- a}} & \text {for}\: b e^{c} \neq 0 \\\frac {\log {\left (e^{d x} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases}}{d} & \text {for}\: d \neq 0 \\\frac {x}{\sqrt {a + b e^{c}}} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=\frac {\log \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a} - \sqrt {a}}{\sqrt {b e^{\left (d x + c\right )} + a} + \sqrt {a}}\right )}{\sqrt {a} d} \]
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none
Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=\frac {2 \, \arctan \left (\frac {\sqrt {b e^{\left (d x + c\right )} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} d} \]
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Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a+b e^{c+d x}}} \, dx=-\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}}{\sqrt {a}}\right )}{\sqrt {a}\,d} \]
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