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*Sqrt[b])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*Log[2])

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Rubi [A]  time = 0.0480266, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

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])

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Rubi in Sympy [A]  time = 7.9819, size = 27, normalized size = 0.9 \[ \frac{\operatorname{atan}{\left (\frac{2^{x} \sqrt{b}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2**x/(a+2**(2*x)*b),x)

[Out]

atan(2**x*sqrt(b)/sqrt(a))/(sqrt(a)*sqrt(b)*log(2))

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Mathematica [A]  time = 0.00686844, 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 + 2^(2*x)*b),x]

[Out]

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

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Maple [B]  time = 0.039, size = 53, normalized size = 1.8 \[ -{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}-{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}+{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

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(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257363, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{2 \cdot 2^{x} a b +{\left (2^{2 \, x} b - a\right )} \sqrt{-a b}}{2^{2 \, x} b + a}\right )}{2 \, \sqrt{-a b} \log \left (2\right )}, -\frac{\arctan \left (\frac{a}{\sqrt{a b} 2^{x}}\right )}{\sqrt{a b} \log \left (2\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.203482, size = 24, normalized size = 0.8 \[ \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (2^{x} + 2 i a \right )} \right )\right )}}{\log{\left (2 \right )}} \]

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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GIAC/XCAS [A]  time = 0.228125, size = 28, normalized size = 0.93 \[ \frac{\arctan \left (\frac{2^{x} b}{\sqrt{a b}}\right )}{\sqrt{a b}{\rm ln}\left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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