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

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])

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Rubi [A]  time = 0.0647834, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} 2^x}{\sqrt{a-b 4^x}}\right )}{\sqrt{b} \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^x/Sqrt[a - 2^(2*x)*b],x]

[Out]

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

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Rubi in Sympy [A]  time = 7.61735, size = 29, normalized size = 0.91 \[ \frac{\operatorname{atan}{\left (\frac{2^{x} \sqrt{b}}{\sqrt{- 2^{2 x} b + a}} \right )}}{\sqrt{b} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(2**x/(a-2**(2*x)*b)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0254588, 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 - 2^(2*x)*b],x]

[Out]

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

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Maple [F]  time = 0.022, size = 0, normalized size = 0. \[ \int{{2}^{x}{\frac{1}{\sqrt{a-{2}^{2\,x}b}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2^x/(a-2^(2*x)*b)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.300708, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{-2^{2 \, x} b + a} 2^{x} b +{\left (2 \cdot 2^{2 \, x} b - a\right )} \sqrt{-b}\right )}{2 \, \sqrt{-b} \log \left (2\right )}, \frac{\arctan \left (\frac{2^{x} \sqrt{b}}{\sqrt{-2^{2 \, x} b + a}}\right )}{\sqrt{b} \log \left (2\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.18157, size = 87, normalized size = 2.72 \[ \frac{\begin{cases} \frac{\sqrt{\frac{a}{b}} \operatorname{asin}{\left (2^{x} \sqrt{\frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge - b < 0 \\\frac{\sqrt{- \frac{a}{b}} \operatorname{asinh}{\left (2^{x} \sqrt{- \frac{b}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge - b > 0 \\\frac{\sqrt{\frac{a}{b}} \operatorname{acosh}{\left (2^{x} \sqrt{\frac{b}{a}} \right )}}{\sqrt{- a}} & \text{for}\: - b > 0 \wedge a < 0 \end{cases}}{\log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2**x/(a-2**(2*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*s
qrt(b/a))/sqrt(-a), (a < 0) & (-b > 0)))/log(2)

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GIAC/XCAS [A]  time = 0.240397, size = 49, normalized size = 1.53 \[ -\frac{{\rm ln}\left ({\left | -2^{x} \sqrt{-b} + \sqrt{-2^{2 \, x} b + a} \right |}\right )}{\sqrt{-b}{\rm ln}\left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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