Optimal. Leaf size=30 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]
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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.
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Rule 205
Rule 2249
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.
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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.
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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.
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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.
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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.
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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.
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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.
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