Optimal. Leaf size=30 \[ \frac {2^x}{b \log (2)}-\frac {a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2248, 43} \[ \frac {2^x}{b \log (2)}-\frac {a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {2^{2 x}}{a+2^x b} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {2^x}{b \log (2)}-\frac {a \log \left (a+2^x b\right )}{b^2 \log (2)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.90 \[ \frac {\frac {2^x}{b}-\frac {a \log \left (a+b 2^x\right )}{b^2}}{\log (2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 25, normalized size = 0.83 \[ \frac {2^{x} b - a \log \left (2^{x} b + a\right )}{b^{2} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 31, normalized size = 1.03 \[ \frac {2^{x}}{b \log \relax (2)} - \frac {a \log \left ({\left | 2^{x} b + a \right |}\right )}{b^{2} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 35, normalized size = 1.17 \[ -\frac {a \ln \left (b \,{\mathrm e}^{\ln \relax (2) x}+a \right )}{\ln \relax (2) b^{2}}+\frac {{\mathrm e}^{\ln \relax (2) x}}{\ln \relax (2) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 30, normalized size = 1.00 \[ \frac {2^{x}}{b \log \relax (2)} - \frac {a \log \left (2^{x} b + a\right )}{b^{2} \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 26, normalized size = 0.87 \[ -\frac {a\,\ln \left (a+2^x\,b\right )-2^x\,b}{b^2\,\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 31, normalized size = 1.03 \[ - \frac {a \log {\left (2^{x} + \frac {a}{b} \right )}}{b^{2} \log {\relax (2 )}} + \begin {cases} \frac {2^{x}}{b \log {\relax (2 )}} & \text {for}\: b \log {\relax (2 )} \neq 0 \\\frac {x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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