Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a+4^x b}}\right )}{\sqrt {b} \log (2)} \]
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Rubi [A]
time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2281, 223, 212}
\begin {gather*} \frac {\tanh ^{-1}\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 212
Rule 223
Rule 2281
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a+4^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 {\tanh ^{-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 = 0.06, size = 33, normalized size = 1.06 \begin {gather*} \frac {\tanh ^{-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 \[\int \frac {2^{x}}{\sqrt {a +4^{x} b}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 77, normalized size = 2.48 \begin {gather*} \left [\frac {\log \left (-2 \, \sqrt {2^{2 \, x} b + a} 2^{x} \sqrt {b} - 2 \cdot 2^{2 \, x} b - a\right )}{2 \, \sqrt {b} \log \left (2\right )}, -\frac {\sqrt {-b} \arctan \left (\frac {2^{x} \sqrt {-b}}{\sqrt {2^{2 \, x} b + a}}\right )}{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 = 0.38, size = 85, normalized size = 2.74 \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}\: b > 0 \wedge a < 0 \end {cases}}{\log {\left (2 \right )}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 28, normalized size = 0.90 \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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