Optimal. Leaf size=43 \[ \frac {2^x}{a \log (2)}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} 2^x}{\sqrt {b}}\right )}{a^{3/2} \log (2)} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2249, 193, 321, 205} \[ \frac {2^x}{a \log (2)}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} 2^x}{\sqrt {b}}\right )}{a^{3/2} \log (2)} \]
Antiderivative was successfully verified.
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Rule 193
Rule 205
Rule 321
Rule 2249
Rubi steps
\begin {align*} \int \frac {2^x}{a+2^{-2 x} b} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+\frac {b}{x^2}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{b+a x^2} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {2^x}{a \log (2)}-\frac {b \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,2^x\right )}{a \log (2)}\\ &=\frac {2^x}{a \log (2)}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {2^x \sqrt {a}}{\sqrt {b}}\right )}{a^{3/2} \log (2)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 0.93 \[ \frac {\frac {2^x}{a}-\frac {\sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} 2^x}{\sqrt {b}}\right )}{a^{3/2}}}{\log (2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 102, normalized size = 2.37 \[ \left [\frac {\sqrt {-\frac {b}{a}} \log \left (-\frac {2 \cdot 2^{x} a \sqrt {-\frac {b}{a}} - 2^{2 \, x} a + b}{2^{2 \, x} a + b}\right ) + 2 \cdot 2^{x}}{2 \, a \log \relax (2)}, -\frac {\sqrt {\frac {b}{a}} \arctan \left (\frac {2^{x} a \sqrt {\frac {b}{a}}}{b}\right ) - 2^{x}}{a \log \relax (2)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 38, normalized size = 0.88 \[ -\frac {b \arctan \left (\frac {2^{x} a}{\sqrt {a b}}\right )}{\sqrt {a b} a \log \relax (2)} + \frac {2^{x}}{a \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 74, normalized size = 1.72 \[ \frac {2^{x}}{\ln \relax (2) a}+\frac {\sqrt {-a b}\, \ln \left (2^{x}-\frac {\sqrt {-a b}}{a}\right )}{2 \ln \relax (2) a^{2}}-\frac {\sqrt {-a b}\, \ln \left (2^{x}+\frac {\sqrt {-a b}}{a}\right )}{2 \ln \relax (2) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.87, size = 39, normalized size = 0.91 \[ \frac {b \arctan \left (\frac {b}{\sqrt {a b} 2^{x}}\right )}{\sqrt {a b} a \log \relax (2)} + \frac {2^{x}}{a \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.54, size = 35, normalized size = 0.81 \[ \frac {2^x}{a\,\ln \relax (2)}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {2^x\,\sqrt {a}}{\sqrt {b}}\right )}{a^{3/2}\,\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 44, normalized size = 1.02 \[ \begin {cases} \frac {2^{x}}{a \log {\relax (2 )}} & \text {for}\: a \log {\relax (2 )} \neq 0 \\\frac {x}{a} & \text {otherwise} \end {cases} + \frac {\operatorname {RootSum} {\left (4 z^{2} a^{3} + b, \left (i \mapsto i \log {\left (\frac {2 i a^{2}}{b} + 2^{- x} \right )} \right )\right )}}{\log {\relax (2 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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