3.481 \(\int \frac{2^x}{a+4^x b} \, dx\)

Optimal. Leaf size=30 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.0295758, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2249, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^x/(a + 4^x*b),x]

[Out]

ArcTan[(2^x*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*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 205

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

Rubi steps

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

Mathematica [A]  time = 0.0074805, size = 30, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^x/(a + 4^x*b),x]

[Out]

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

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Maple [B]  time = 0.029, size = 53, normalized size = 1.8 \begin{align*} -{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}-{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}+{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(-a*b)^(1/2)/ln(2)*ln(2^x-1/(-a*b)^(1/2)*a)+1/2/(-a*b)^(1/2)/ln(2)*ln(2^x+1/(-a*b)^(1/2)*a)

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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+4^x*b),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56519, size = 190, normalized size = 6.33 \begin{align*} \left [-\frac{\sqrt{-a b} \log \left (\frac{2^{2 \, x} b - 2 \, \sqrt{-a b} 2^{x} - a}{2^{2 \, x} b + a}\right )}{2 \, a b \log \left (2\right )}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{2^{x} b}\right )}{a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.326304, size = 29, normalized size = 0.97 \begin{align*} \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (2 i a + e^{\frac{x \log{\left (4 \right )}}{2}} \right )} \right )\right )}}{\log{\left (2 \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2**x/(a+4**x*b),x)

[Out]

RootSum(4*_z**2*a*b + 1, Lambda(_i, _i*log(2*_i*a + exp(x*log(4)/2))))/log(2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2^{x}}{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),x, algorithm="giac")

[Out]

integrate(2^x/(4^x*b + a), x)