Optimal. Leaf size=58 \[ \frac {b^2 x}{a^3}+\frac {2^{-1+2 x}}{a \log (2)}+\frac {2^x b}{a^2 \log (2)}+\frac {b^2 \log \left (a-2^{-x} b\right )}{a^3 \log (2)} \]
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Rubi [A]
time = 0.04, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2280, 46}
\begin {gather*} \frac {b^2 x}{a^3}+\frac {b^2 \log \left (a-b 2^{-x}\right )}{a^3 \log (2)}+\frac {b 2^x}{a^2 \log (2)}+\frac {2^{2 x-1}}{a \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2280
Rubi steps
\begin {align*} \int \frac {4^x}{a-2^{-x} b} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{x^3 (a-b x)} \, dx,x,2^{-x}\right )}{\log (2)}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {1}{a x^3}+\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}+\frac {b^3}{a^3 (a-b x)}\right ) \, dx,x,2^{-x}\right )}{\log (2)}\\ &=\frac {b^2 x}{a^3}+\frac {2^{-1+2 x}}{a \log (2)}+\frac {2^x b}{a^2 \log (2)}+\frac {b^2 \log \left (a-2^{-x} b\right )}{a^3 \log (2)}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 38, normalized size = 0.66 \begin {gather*} \frac {2^x a \left (2^x a+2 b\right )+2 b^2 \log \left (2^x a-b\right )}{a^3 \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 50, normalized size = 0.86
method | result | size |
risch | \(\frac {2^{2 x}}{2 a \ln \left (2\right )}+\frac {2^{x} b}{a^{2} \ln \left (2\right )}+\frac {b^{2} \ln \left (2^{x}-\frac {b}{a}\right )}{a^{3} \ln \left (2\right )}\) | \(50\) |
norman | \(\frac {b \,{\mathrm e}^{x \ln \left (2\right )}}{a^{2} \ln \left (2\right )}+\frac {{\mathrm e}^{2 x \ln \left (2\right )}}{2 a \ln \left (2\right )}+\frac {b^{2} \ln \left (a \,{\mathrm e}^{x \ln \left (2\right )}-b \right )}{a^{3} \ln \left (2\right )}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 58, normalized size = 1.00 \begin {gather*} \frac {b^{2} x}{a^{3}} + \frac {{\left (2^{-x + 1} b + a\right )} 2^{2 \, x - 1}}{a^{2} \log \left (2\right )} + \frac {b^{2} \log \left (-a + \frac {b}{2^{x}}\right )}{a^{3} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 41, normalized size = 0.71 \begin {gather*} \frac {2^{2 \, x} a^{2} + 2 \cdot 2^{x} a b + 2 \, b^{2} \log \left (2^{x} a - b\right )}{2 \, a^{3} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 90, normalized size = 1.55 \begin {gather*} \begin {cases} \frac {2^{2 x} a^{2} \log {\left (2 \right )} + 2 \cdot 2^{x} a b \log {\left (2 \right )}}{2 a^{3} \log {\left (2 \right )}^{2}} & \text {for}\: a^{3} \log {\left (2 \right )}^{2} \neq 0 \\x \left (- \frac {b^{2}}{a^{3}} + \frac {a^{2} + a b + b^{2}}{a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {b^{2} x}{a^{3}} + \frac {b^{2} \log {\left (- \frac {a}{b} + 2^{- x} \right )}}{a^{3} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {4^x}{a-\frac {b}{2^x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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