3.491 \(\int \frac{2^x}{\sqrt{a-4^x b}} \, dx\)

Optimal. Leaf size=32 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a-b 4^x}}\right )}{\sqrt{b} \log (2)} \]

[Out]

ArcTan[(2^x*Sqrt[b])/Sqrt[a - 4^x*b]]/(Sqrt[b]*Log[2])

________________________________________________________________________________________

Rubi [A]  time = 0.0398257, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2249, 217, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a-b 4^x}}\right )}{\sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^x/Sqrt[a - 4^x*b],x]

[Out]

ArcTan[(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 203

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

Rubi steps

\begin{align*} \int \frac{2^x}{\sqrt{a-4^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{\tan ^{-1}\left (\frac{2^x \sqrt{b}}{\sqrt{a-4^x b}}\right )}{\sqrt{b} \log (2)}\\ \end{align*}

Mathematica [A]  time = 0.0134453, size = 34, normalized size = 1.06 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a-b 2^{2 x}}}\right )}{\sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^x/Sqrt[a - 4^x*b],x]

[Out]

ArcTan[(2^x*Sqrt[b])/Sqrt[a - 2^(2*x)*b]]/(Sqrt[b]*Log[2])

________________________________________________________________________________________

Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{{2}^{x}{\frac{1}{\sqrt{a-{4}^{x}b}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^x/(a-4^x*b)^(1/2),x)

[Out]

int(2^x/(a-4^x*b)^(1/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2^{x}}{\sqrt{-4^{x} b + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a-4^x*b)^(1/2),x, algorithm="maxima")

[Out]

integrate(2^x/sqrt(-4^x*b + a), x)

________________________________________________________________________________________

Fricas [A]  time = 1.58186, size = 224, normalized size = 7. \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-2 \, \sqrt{-2^{2 \, x} b + a} 2^{x} \sqrt{-b} + 2 \cdot 2^{2 \, x} b - a\right )}{2 \, b \log \left (2\right )}, -\frac{\arctan \left (\frac{\sqrt{-2^{2 \, x} b + a} 2^{x} \sqrt{b}}{2^{2 \, x} b - a}\right )}{\sqrt{b} \log \left (2\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a-4^x*b)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.771352, size = 82, normalized size = 2.56 \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}\: a < 0 \wedge b < 0 \end{cases}}{\log{\left (2 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**x/(a-4**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), (a < 0) & (b < 0)))/log(2)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2^{x}}{\sqrt{-4^{x} b + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a-4^x*b)^(1/2),x, algorithm="giac")

[Out]

integrate(2^x/sqrt(-4^x*b + a), x)