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

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*b^(1/2)/a^(1/2))/ln(2)/a^(1/2)/b^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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 + 2^(2*x)*b),x]

[Out]

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

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]

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]

Rubi steps

\begin {align*} \int \frac {2^x}{a+2^{2 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*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]

Antiderivative was successfully verified.

[In]

Integrate[2^x/(a + 2^(2*x)*b),x]

[Out]

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

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fricas [A]  time = 0.41, size = 86, normalized size = 2.87 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {2^{2 \, x} b - 2 \, \sqrt {-a b} 2^{x} - a}{2^{2 \, x} b + a}\right )}{2 \, a b \log \relax (2)}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{2^{x} b}\right )}{a b \log \relax (2)}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2^x/(a+2^(2*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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giac [A]  time = 0.33, size = 21, normalized size = 0.70 \[ \frac {\arctan \left (\frac {2^{x} b}{\sqrt {a b}}\right )}{\sqrt {a b} \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(2^x*b/sqrt(a*b))/(sqrt(a*b)*log(2))

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maple [B]  time = 0.05, size = 53, normalized size = 1.77 \[ -\frac {\ln \left (-\frac {a}{\sqrt {-a b}}+2^{x}\right )}{2 \sqrt {-a b}\, \ln \relax (2)}+\frac {\ln \left (\frac {a}{\sqrt {-a b}}+2^{x}\right )}{2 \sqrt {-a b}\, \ln \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.07, size = 21, normalized size = 0.70 \[ \frac {\arctan \left (\frac {2^{x} b}{\sqrt {a b}}\right )}{\sqrt {a b} \log \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(2^x*b/sqrt(a*b))/(sqrt(a*b)*log(2))

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mupad [B]  time = 3.58, size = 22, normalized size = 0.73 \[ \frac {\mathrm {atan}\left (\frac {2^x\,\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,\ln \relax (2)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2^x/(a + 2^(2*x)*b),x)

[Out]

atan((2^x*b^(1/2))/a^(1/2))/(a^(1/2)*b^(1/2)*log(2))

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sympy [A]  time = 0.18, size = 24, normalized size = 0.80 \[ \frac {\operatorname {RootSum} {\left (4 z^{2} a b + 1, \left (i \mapsto i \log {\left (2^{x} + 2 i a \right )} \right )\right )}}{\log {\relax (2 )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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