Optimal. Leaf size=30 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2282, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2282
Rubi steps
\begin {align*} \int \frac {1}{b f^{-x}+a f^x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,f^x\right )}{\log (f)}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} f^x}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 86, normalized size = 2.87 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {a f^{2 \, x} - 2 \, \sqrt {-a b} f^{x} - b}{a f^{2 \, x} + b}\right )}{2 \, a b \log \relax (f)}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a f^{x}}\right )}{a b \log \relax (f)}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 21, normalized size = 0.70 \[ \frac {\arctan \left (\frac {a f^{x}}{\sqrt {a b}}\right )}{\sqrt {a b} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.73 \[ \frac {\arctan \left (\frac {a \,f^{x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, \ln \relax (f )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 24, normalized size = 0.80 \[ -\frac {\arctan \left (\frac {b}{\sqrt {a b} f^{x}}\right )}{\sqrt {a b} \log \relax (f)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 21, normalized size = 0.70 \[ \frac {\mathrm {atan}\left (\frac {a\,f^x}{\sqrt {a\,b}}\right )}{\ln \relax (f)\,\sqrt {a\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 24, normalized size = 0.80 \[ \frac {\operatorname {RootSum} {\left (4 z^{2} a b + 1, \left (i \mapsto i \log {\left (2 i b + f^{x} \right )} \right )\right )}}{\log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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