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

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

[Out]

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

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Rubi [A]  time = 0.0495007, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^x/(a - 4^x*b),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.05955, size = 27, normalized size = 0.9 \[ \frac{\operatorname{atanh}{\left (\frac{2^{x} \sqrt{b}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2**x/(a-4**x*b),x)

[Out]

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

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Mathematica [A]  time = 0.0101467, size = 30, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]  Integrate[2^x/(a - 4^x*b),x]

[Out]

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

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Maple [B]  time = 0.039, size = 49, normalized size = 1.6 \[{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}+{a{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{2\,\ln \left ( 2 \right ) }\ln \left ({2}^{x}-{a{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2^x/(a-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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271095, 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/(4^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)*log
(2)), -arctan(a/(sqrt(-a*b)*2^x))/(sqrt(-a*b)*log(2))]

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Sympy [A]  time = 0.220376, 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-4**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 [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{2^{x}}{4^{x} b - a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-2^x/(4^x*b - a), x)