Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} e^{m x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} m} \]
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Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2282, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} e^{m x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} m} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{b e^{-m x}+a e^{m x}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,e^{m x}\right )}{m}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} e^{m x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} m}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} e^{m x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 85, normalized size = 2.74 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {a e^{\left (2 \, m x\right )} - 2 \, \sqrt {-a b} e^{\left (m x\right )} - b}{a e^{\left (2 \, m x\right )} + b}\right )}{2 \, a b m}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} e^{\left (-m x\right )}}{a}\right )}{a b m}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 21, normalized size = 0.68 \[ \frac {\arctan \left (\frac {a e^{\left (m x\right )}}{\sqrt {a b}}\right )}{\sqrt {a b} m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.71 \[ \frac {\arctan \left (\frac {a \,{\mathrm e}^{m x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 23, normalized size = 0.74 \[ -\frac {\arctan \left (\frac {b e^{\left (-m x\right )}}{\sqrt {a b}}\right )}{\sqrt {a b} m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 21, normalized size = 0.68 \[ \frac {\mathrm {atan}\left (\frac {a\,{\mathrm {e}}^{m\,x}}{\sqrt {a\,b}}\right )}{m\,\sqrt {a\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 26, normalized size = 0.84 \[ \frac {\operatorname {RootSum} {\left (4 z^{2} a b + 1, \left (i \mapsto i \log {\left (- 2 i a + e^{- m x} \right )} \right )\right )}}{m} \]
Verification of antiderivative is not currently implemented for this CAS.
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