3.490 \(\int \frac{2^x}{\sqrt{a+2^{2 x} b}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 4^x}}\right )}{\sqrt{b} \log (2)} \]

[Out]

ArcTanh[(2^x*Sqrt[b])/Sqrt[a + 4^x*b]]/(Sqrt[b]*Log[2])

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Rubi [A]  time = 0.0375751, antiderivative size = 31, 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 = {2249, 217, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 4^x}}\right )}{\sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^x/Sqrt[a + 2^(2*x)*b],x]

[Out]

ArcTanh[(2^x*Sqrt[b])/Sqrt[a + 4^x*b]]/(Sqrt[b]*Log[2])

Rule 2249

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[(d*e*Log[F])/(g*h*Log[G])]}, Dist[Denominator[m]/(g*h*Log[G]), Subst[Int[x^(Denominator[m]
- 1)*(a + b*F^(c*e - (d*e*f)/g)*x^Numerator[m])^p, x], x, G^((h*(f + g*x))/Denominator[m])], x] /; LtQ[m, -1]
|| GtQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{2^x}{\sqrt{a+2^{2 x} b}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{2^x}{\sqrt{a+4^x b}}\right )}{\log (2)}\\ &=\frac{\tanh ^{-1}\left (\frac{2^x \sqrt{b}}{\sqrt{a+4^x b}}\right )}{\sqrt{b} \log (2)}\\ \end{align*}

Mathematica [A]  time = 0.0043184, size = 33, normalized size = 1.06 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a+b 2^{2 x}}}\right )}{\sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^x/Sqrt[a + 2^(2*x)*b],x]

[Out]

ArcTanh[(2^x*Sqrt[b])/Sqrt[a + 2^(2*x)*b]]/(Sqrt[b]*Log[2])

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{2}^{x}{\frac{1}{\sqrt{a+{2}^{2\,x}b}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^x/(a+2^(2*x)*b)^(1/2),x)

[Out]

int(2^x/(a+2^(2*x)*b)^(1/2),x)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62163, size = 198, normalized size = 6.39 \begin{align*} \left [\frac{\log \left (-2 \, \sqrt{2^{2 \, x} b + a} 2^{x} \sqrt{b} - 2 \cdot 2^{2 \, x} b - a\right )}{2 \, \sqrt{b} \log \left (2\right )}, -\frac{\sqrt{-b} \arctan \left (\frac{2^{x} \sqrt{-b}}{\sqrt{2^{2 \, x} b + a}}\right )}{b \log \left (2\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log(-2*sqrt(2^(2*x)*b + a)*2^x*sqrt(b) - 2*2^(2*x)*b - a)/(sqrt(b)*log(2)), -sqrt(-b)*arctan(2^x*sqrt(-b)
/sqrt(2^(2*x)*b + a))/(b*log(2))]

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Sympy [A]  time = 0.823697, size = 85, normalized size = 2.74 \begin{align*} \frac{\begin{cases} \frac{\sqrt{- \frac{a}{b}} \operatorname{asin}{\left (2^{x} \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b < 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{asinh}{\left (2^{x} \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge b > 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{acosh}{\left (2^{x} \sqrt{- \frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: b > 0 \wedge a < 0 \end{cases}}{\log{\left (2 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**x/(a+2**(2*x)*b)**(1/2),x)

[Out]

Piecewise((sqrt(-a/b)*asin(2**x*sqrt(-b/a))/sqrt(a), (a > 0) & (b < 0)), (sqrt(a/b)*asinh(2**x*sqrt(b/a))/sqrt
(a), (a > 0) & (b > 0)), (sqrt(-a/b)*acosh(2**x*sqrt(-b/a))/sqrt(-a), (b > 0) & (a < 0)))/log(2)

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Giac [A]  time = 1.24596, size = 42, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | -2^{x} \sqrt{b} + \sqrt{2^{2 \, x} b + a} \right |}\right )}{\sqrt{b} \log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(abs(-2^x*sqrt(b) + sqrt(2^(2*x)*b + a)))/(sqrt(b)*log(2))