Optimal. Leaf size=31 \[ \frac {e^{2 x}}{2 b}-\frac {a \log \left (a+b e^{2 x}\right )}{2 b^2} \]
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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 = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac {e^{2 x}}{2 b}-\frac {a \log \left (a+b e^{2 x}\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{4 x}}{a+b e^{2 x}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac {e^{2 x}}{2 b}-\frac {a \log \left (a+b e^{2 x}\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.97 \[ \frac {1}{2} \left (\frac {e^{2 x}}{b}-\frac {a \log \left (a+b e^{2 x}\right )}{b^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 24, normalized size = 0.77 \[ \frac {b e^{\left (2 \, x\right )} - a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 26, normalized size = 0.84 \[ \frac {e^{\left (2 \, x\right )}}{2 \, b} - \frac {a \log \left ({\left | b e^{\left (2 \, x\right )} + a \right |}\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 0.84 \[ -\frac {a \ln \left (b \,{\mathrm e}^{2 x}+a \right )}{2 b^{2}}+\frac {{\mathrm e}^{2 x}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 25, normalized size = 0.81 \[ \frac {e^{\left (2 \, x\right )}}{2 \, b} - \frac {a \log \left (b e^{\left (2 \, x\right )} + a\right )}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 24, normalized size = 0.77 \[ -\frac {a\,\ln \left (a+b\,{\mathrm {e}}^{2\,x}\right )-b\,{\mathrm {e}}^{2\,x}}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 29, normalized size = 0.94 \[ - \frac {a \log {\left (\frac {a}{b} + e^{2 x} \right )}}{2 b^{2}} + \begin {cases} \frac {e^{2 x}}{2 b} & \text {for}\: 2 b \neq 0 \\\frac {x}{b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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