Optimal. Leaf size=30 \[ \frac {\tanh ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2281, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2281
Rubi steps
\begin {align*} \int \frac {2^x}{a-2^{2 x} b} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\tanh ^{-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 = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs.
\(2(22)=44\).
time = 0.02, size = 49, normalized size = 1.63
method | result | size |
risch | \(\frac {\ln \left (2^{x}+\frac {a}{\sqrt {b a}}\right )}{2 \sqrt {b a}\, \ln \left (2\right )}-\frac {\ln \left (2^{x}-\frac {a}{\sqrt {b a}}\right )}{2 \sqrt {b a}\, \ln \left (2\right )}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs.
\(2 (22) = 44\).
time = 0.50, size = 45, normalized size = 1.50 \begin {gather*} -\frac {\log \left (\frac {2^{x + 1} b - 2 \, \sqrt {a b}}{2^{x + 1} b + 2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 86, normalized size = 2.87 \begin {gather*} \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 \left (2\right )}, -\frac {\sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{2^{x} b}\right )}{a b \log \left (2\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 24, normalized size = 0.80 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.11, size = 24, normalized size = 0.80 \begin {gather*} -\frac {\arctan \left (\frac {2^{x} b}{\sqrt {-a b}}\right )}{\sqrt {-a b} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 22, normalized size = 0.73 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {2^x\,\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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