Optimal. Leaf size=30 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0211419, antiderivative size = 30, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 2282
Rule 205
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*}
Mathematica [A] time = 0.0077143, size = 30, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} f^x}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 22, normalized size = 0.7 \begin{align*}{\frac{1}{\ln \left ( f \right ) }\arctan \left ({a{f}^{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62191, size = 190, normalized size = 6.33 \begin{align*} \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 \left (f\right )}, -\frac{\sqrt{a b} \arctan \left (\frac{\sqrt{a b}}{a f^{x}}\right )}{a b \log \left (f\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.263538, size = 26, normalized size = 0.87 \begin{align*} \frac{\operatorname{RootSum}{\left (4 z^{2} a b + 1, \left ( i \mapsto i \log{\left (- 2 i a + f^{- x} \right )} \right )\right )}}{\log{\left (f \right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21153, size = 28, normalized size = 0.93 \begin{align*} \frac{\arctan \left (\frac{a f^{x}}{\sqrt{a b}}\right )}{\sqrt{a b} \log \left (f\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]