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

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 = 0.0273548, 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, Rules used = {2282, 63, 208} \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

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)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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 \tanh ^{-1}\left (\frac{\sqrt{a+b e^{c+d x}}}{\sqrt{a}}\right )}{\sqrt{a} d}\\ \end{align*}

Mathematica [A]  time = 0.0127239, 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 = 0.189, size = 26, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{d\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{{\rm e}^{dx+c}}}}{\sqrt{a}}} \right ) } \end{align*}

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. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.908636, size = 208, normalized size = 6.5 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*exp(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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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.

[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 [A]  time = 1.23535, size = 39, normalized size = 1.22 \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.

[In]

integrate(1/(a+b*exp(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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