Optimal. Leaf size=32 \[ \frac {\tan ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a-4^x b}}\right )}{\sqrt {b} \log (2)} \]
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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2281, 223, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {b} 2^x}{\sqrt {a-b 4^x}}\right )}{\sqrt {b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 2281
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a-2^{2 x} b}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {2^x}{\sqrt {a-4^x b}}\right )}{\log (2)}\\ &=\frac {\tan ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a-4^x b}}\right )}{\sqrt {b} \log (2)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.06 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a-2^{2 x} b}}\right )}{\sqrt {b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {2^{x}}{\sqrt {a -2^{2 x} b}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 22, normalized size = 0.69 \begin {gather*} \frac {\arcsin \left (\frac {2^{x + 1} b}{2 \, \sqrt {a b}}\right )}{\sqrt {b} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 92, normalized size = 2.88 \begin {gather*} \left [-\frac {\sqrt {-b} \log \left (-2 \, \sqrt {-2^{2 \, x} b + a} 2^{x} \sqrt {-b} + 2 \cdot 2^{2 \, x} b - a\right )}{2 \, b \log \left (2\right )}, -\frac {\arctan \left (\frac {\sqrt {-2^{2 \, x} b + a} 2^{x} \sqrt {b}}{2^{2 \, x} b - a}\right )}{\sqrt {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 = 0.41, size = 82, normalized size = 2.56 \begin {gather*} \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}\: a < 0 \wedge b < 0 \end {cases}}{\log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.91, size = 36, normalized size = 1.12 \begin {gather*} -\frac {\log \left ({\left | -2^{x} \sqrt {-b} + \sqrt {-2^{2 \, x} b + a} \right |}\right )}{\sqrt {-b} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.63, size = 33, normalized size = 1.03 \begin {gather*} \frac {\ln \left (\sqrt {a-2^{2\,x}\,b}+2^x\,\sqrt {-b}\right )}{\sqrt {-b}\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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