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

Optimal. Leaf size=34 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{b e^{c+d x}-a}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

[Out]

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

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

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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]

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

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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]

[Out]

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

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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]

2*arctan((-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 = 0.239753, 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]

[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(sqrt(a)/sqrt(b*e^(d*x + c) - 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 = 0.228089, 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]

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