Optimal. Leaf size=32 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a-b 4^x}}\right )}{\sqrt {b} \log (2)} \]
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Rubi [A] time = 0.04, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2249, 217, 203} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a-b 4^x}}\right )}{\sqrt {b} \log (2)} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 2249
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a-4^x b}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {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 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} 2^x}{\sqrt {a-b 2^{2 x}}}\right )}{\sqrt {b} \log (2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 92, normalized size = 2.88 \[ \left [-\frac {\sqrt {-b} \log \left (-2 \, \sqrt {-2^{2 \, x} b + a} 2^{x} \sqrt {-b} + 2 \cdot 2^{2 \, x} b - a\right )}{2 \, b \log \relax (2)}, -\frac {\arctan \left (\frac {\sqrt {-2^{2 \, x} b + a} 2^{x} \sqrt {b}}{2^{2 \, x} b - a}\right )}{\sqrt {b} \log \relax (2)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {-4^{x} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {-b 4^{x}+a}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {-4^{x} b + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.77, size = 33, normalized size = 1.03 \[ \frac {\ln \left (\sqrt {a-2^{2\,x}\,b}+2^x\,\sqrt {-b}\right )}{\sqrt {-b}\,\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.86, size = 82, normalized size = 2.56 \[ \frac {\begin {cases} \frac {\sqrt {\frac {a}{b}} \operatorname {asin}{\left (2^{x} \sqrt {\frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b > 0 \\\frac {\sqrt {- \frac {a}{b}} \operatorname {asinh}{\left (2^{x} \sqrt {- \frac {b}{a}} \right )}}{\sqrt {a}} & \text {for}\: a > 0 \wedge b < 0 \\\frac {\sqrt {\frac {a}{b}} \operatorname {acosh}{\left (2^{x} \sqrt {\frac {b}{a}} \right )}}{\sqrt {- a}} & \text {for}\: a < 0 \wedge b < 0 \end {cases}}{\log {\relax (2 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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