Optimal. Leaf size=17 \[ -x-\frac {2 a \log (a-b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45}
\begin {gather*} -\frac {2 a \log (a-b x)}{b}-x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{a^2-b^2 x^2} \, dx &=\int \frac {a+b x}{a-b x} \, dx\\ &=\int \left (-1+\frac {2 a}{a-b x}\right ) \, dx\\ &=-x-\frac {2 a \log (a-b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -x-\frac {2 a \log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.45, size = 18, normalized size = 1.06
method | result | size |
default | \(-x -\frac {2 a \ln \left (-b x +a \right )}{b}\) | \(18\) |
norman | \(-x -\frac {2 a \ln \left (-b x +a \right )}{b}\) | \(18\) |
risch | \(-x -\frac {2 a \ln \left (-b x +a \right )}{b}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 18, normalized size = 1.06 \begin {gather*} -x - \frac {2 \, a \log \left (b x - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.39, size = 20, normalized size = 1.18 \begin {gather*} -\frac {b x + 2 \, a \log \left (b x - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.04, size = 14, normalized size = 0.82 \begin {gather*} - \frac {2 a \log {\left (- a + b x \right )}}{b} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.21, size = 19, normalized size = 1.12 \begin {gather*} -x - \frac {2 \, a \log \left ({\left | b x - a \right |}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.40, size = 18, normalized size = 1.06 \begin {gather*} -x-\frac {2\,a\,\ln \left (b\,x-a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________