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 = 0.0536442, 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 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-a + b*E^(c + d*x)],x]
[Out]
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Rubi in Sympy [A] time = 6.77617, size = 42, normalized size = 1.24 \[ \frac{2 e^{- c - d x} e^{c + d x} \operatorname{atan}{\left (\frac{\sqrt{- a + b e^{c + d x}}}{\sqrt{a}} \right )}}{\sqrt{a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-a+b*exp(d*x+c))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0343188, 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.
[In] Integrate[1/Sqrt[-a + b*E^(c + d*x)],x]
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Maple [A] time = 0.011, size = 28, normalized size = 0.8 \[ 2\,{\frac{1}{d\sqrt{a}}\arctan \left ({\frac{\sqrt{-a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-a+b*exp(d*x+c))^(1/2),x)
[Out]
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*e^(d*x + c) - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239753, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left ({\left ({\left (b e^{\left (d x + c\right )} - 2 \, a\right )} \sqrt{-a} + 2 \, \sqrt{b e^{\left (d x + c\right )} - a} a\right )} e^{\left (-d x - c\right )}\right )}{\sqrt{-a} d}, -\frac{2 \, \arctan \left (\frac{\sqrt{a}}{\sqrt{b e^{\left (d x + c\right )} - a}}\right )}{\sqrt{a} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*e^(d*x + c) - a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- a + b e^{c + d x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-a+b*exp(d*x+c))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.228089, size = 36, normalized size = 1.06 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} - a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(b*e^(d*x + c) - a),x, algorithm="giac")
[Out]