Optimal. Leaf size=36 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c e^x}{\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}} \]
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Rubi [A] time = 0.06, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2282, 1386, 618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c e^x}{\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 1386
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{a+b e^{-x}+c e^x} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \left (a+\frac {b}{x}+c x\right )} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{b+a x+c x^2} \, dx,x,e^x\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{a^2-4 b c-x^2} \, dx,x,a+2 c e^x\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {a+2 c e^x}{\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 36, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+2 c e^x}{\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 126, normalized size = 3.50 \[ \left [\frac {\log \left (\frac {2 \, c^{2} e^{\left (2 \, x\right )} + 2 \, a c e^{x} + a^{2} - 2 \, b c - \sqrt {a^{2} - 4 \, b c} {\left (2 \, c e^{x} + a\right )}}{c e^{\left (2 \, x\right )} + a e^{x} + b}\right )}{\sqrt {a^{2} - 4 \, b c}}, -\frac {2 \, \sqrt {-a^{2} + 4 \, b c} \arctan \left (-\frac {\sqrt {-a^{2} + 4 \, b c} {\left (2 \, c e^{x} + a\right )}}{a^{2} - 4 \, b c}\right )}{a^{2} - 4 \, b c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 35, normalized size = 0.97 \[ \frac {2 \, \arctan \left (\frac {2 \, c e^{x} + a}{\sqrt {-a^{2} + 4 \, b c}}\right )}{\sqrt {-a^{2} + 4 \, b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 1.00 \[ \frac {2 \arctan \left (\frac {2 c \,{\mathrm e}^{x}+a}{\sqrt {-a^{2}+4 b c}}\right )}{\sqrt {-a^{2}+4 b c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 35, normalized size = 0.97 \[ \frac {2\,\mathrm {atan}\left (\frac {a+2\,c\,{\mathrm {e}}^x}{\sqrt {4\,b\,c-a^2}}\right )}{\sqrt {4\,b\,c-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 36, normalized size = 1.00 \[ \operatorname {RootSum} {\left (z^{2} \left (a^{2} - 4 b c\right ) - 1, \left (i \mapsto i \log {\left (e^{x} + \frac {- i a^{2} + 4 i b c + a}{2 c} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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