Optimal. Leaf size=34 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.0294162, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2282, 63, 205} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 63
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-a+b e^{c+d x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{-a+b x}} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{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 \tan ^{-1}\left (\frac{\sqrt{-a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [A] time = 0.0132134, size = 34, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.191, size = 28, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{d\sqrt{a}}\arctan \left ({\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.815585, size = 207, normalized size = 6.09 \begin{align*} \left [-\frac{\sqrt{-a} \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 )}{a d}, \frac{2 \, \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right )}{\sqrt{a} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- a + b e^{c + d x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31754, size = 36, normalized size = 1.06 \begin{align*} \frac{2 \, \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right )}{\sqrt{a} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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