Optimal. Leaf size=30 \[ \frac {1}{2} x \log \left (\left (x^2-a^2\right )^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 30, 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 = {12, 2448, 321, 207} \[ \frac {1}{2} x \log \left (\left (x^2-a^2\right )^2\right )+2 a \tanh ^{-1}\left (\frac {x}{a}\right )-2 x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 207
Rule 321
Rule 2448
Rubi steps
\begin {align*} \int \frac {1}{2} \log \left (\left (-a^2+x^2\right )^2\right ) \, dx &=\frac {1}{2} \int \log \left (\left (-a^2+x^2\right )^2\right ) \, dx\\ &=\frac {1}{2} x \log \left (\left (-a^2+x^2\right )^2\right )-2 \int \frac {x^2}{-a^2+x^2} \, dx\\ &=-2 x+\frac {1}{2} x \log \left (\left (-a^2+x^2\right )^2\right )-\left (2 a^2\right ) \int \frac {1}{-a^2+x^2} \, dx\\ &=-2 x+2 a \tanh ^{-1}\left (\frac {x}{a}\right )+\frac {1}{2} x \log \left (\left (-a^2+x^2\right )^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 31, normalized size = 1.03 \[ \frac {1}{2} \left (x \log \left (\left (a^2-x^2\right )^2\right )+4 a \tanh ^{-1}\left (\frac {x}{a}\right )-4 x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.75, size = 38, normalized size = 1.27 \[ \frac {1}{2} \, x \log \left (a^{4} - 2 \, a^{2} x^{2} + x^{4}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.86, size = 36, normalized size = 1.20 \[ \frac {1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{2}\right ) + a \log \left ({\left | a + x \right |}\right ) - a \log \left ({\left | -a + x \right |}\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 35, normalized size = 1.17 \[ -a \ln \left (-a +x \right )+a \ln \left (a +x \right )+\frac {x \ln \left (\left (-a^{2}+x^{2}\right )^{2}\right )}{2}-2 x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 34, normalized size = 1.13 \[ \frac {1}{2} \, x \log \left ({\left (a^{2} - x^{2}\right )}^{2}\right ) + a \log \left (a + x\right ) - a \log \left (-a + x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.40, size = 28, normalized size = 0.93 \[ 2\,a\,\mathrm {atanh}\left (\frac {x}{a}\right )-2\,x+\frac {x\,\ln \left ({\left (a^2-x^2\right )}^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.18, size = 32, normalized size = 1.07 \[ - 2 a \left (\frac {\log {\left (- a + x \right )}}{2} - \frac {\log {\left (a + x \right )}}{2}\right ) + \frac {x \log {\left (\left (- a^{2} + x^{2}\right )^{2} \right )}}{2} - 2 x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________