3.681 \(\int \frac{1}{\sqrt{a+b e^{c+d x}}} \, dx\)

Optimal. Leaf size=32 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

[Out]

(-2*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/(Sqrt[a]*d)

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Rubi [A]  time = 0.0539488, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a + b*E^(c + d*x)],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/(Sqrt[a]*d)

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Rubi in Sympy [A]  time = 6.36645, size = 44, normalized size = 1.38 \[ - \frac{2 e^{- c - d x} e^{c + d x} \operatorname{atanh}{\left (\frac{\sqrt{a + b e^{c + d x}}}{\sqrt{a}} \right )}}{\sqrt{a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*exp(d*x+c))**(1/2),x)

[Out]

-2*exp(-c - d*x)*exp(c + d*x)*atanh(sqrt(a + b*exp(c + d*x))/sqrt(a))/(sqrt(a)*d
)

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Mathematica [A]  time = 0.0287575, size = 32, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a + b*E^(c + d*x)],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*E^(c + d*x)]/Sqrt[a]])/(Sqrt[a]*d)

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Maple [A]  time = 0.012, size = 26, normalized size = 0.8 \[ -2\,{\frac{1}{d\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*exp(d*x+c))^(1/2),x)

[Out]

-2*arctanh((a+b*exp(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)

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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]

Exception raised: ValueError

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Fricas [A]  time = 0.240798, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left ({\left ({\left (b e^{\left (d x + c\right )} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b e^{\left (d x + c\right )} + a} a\right )} e^{\left (-d x - c\right )}\right )}{\sqrt{a} d}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b e^{\left (d x + c\right )} + a} \sqrt{-a}}\right )}{\sqrt{-a} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*e^(d*x + c) + a),x, algorithm="fricas")

[Out]

[log(((b*e^(d*x + c) + 2*a)*sqrt(a) - 2*sqrt(b*e^(d*x + c) + a)*a)*e^(-d*x - c))
/(sqrt(a)*d), 2*arctan(a/(sqrt(b*e^(d*x + c) + a)*sqrt(-a)))/(sqrt(-a)*d)]

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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]

Integral(1/sqrt(a + b*exp(c + d*x)), x)

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GIAC/XCAS [A]  time = 0.227167, size = 39, normalized size = 1.22 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b e^{\left (d x + c\right )} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(b*e^(d*x + c) + a),x, algorithm="giac")

[Out]

2*arctan(sqrt(b*e^(d*x + c) + a)/sqrt(-a))/(sqrt(-a)*d)