Optimal. Leaf size=14 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {208} \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{a^2-b^2 x^2} \, dx &=\frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.01, size = 14, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {b x}{a}\right )}{a b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.17, size = 25, normalized size = 1.79 \[ \frac {\log \left (b x + a\right ) - \log \left (b x - a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.01, size = 33, normalized size = 2.36 \[ \frac {\log \left ({\left | b x + a \right |}\right )}{2 \, a b} - \frac {\log \left ({\left | b x - a \right |}\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.22, size = 31, normalized size = 2.21
method | result | size |
default | \(\frac {\ln \left (b x +a \right )}{2 a b}-\frac {\ln \left (-b x +a \right )}{2 a b}\) | \(31\) |
norman | \(\frac {\ln \left (b x +a \right )}{2 a b}-\frac {\ln \left (-b x +a \right )}{2 a b}\) | \(31\) |
risch | \(\frac {\ln \left (b x +a \right )}{2 a b}-\frac {\ln \left (-b x +a \right )}{2 a b}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 31, normalized size = 2.21 \[ \frac {\log \left (b x + a\right )}{2 \, a b} - \frac {\log \left (b x - a\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 14, normalized size = 1.00 \[ \frac {\mathrm {atanh}\left (\frac {b\,x}{a}\right )}{a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.14, size = 20, normalized size = 1.43 \[ - \frac {\frac {\log {\left (- \frac {a}{b} + x \right )}}{2} - \frac {\log {\left (\frac {a}{b} + x \right )}}{2}}{a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________