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