Optimal. Leaf size=16 \[ \frac {2 \log (a+b x+1)}{b}-x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6161, 43} \[ \frac {2 \log (a+b x+1)}{b}-x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6161
Rubi steps
\begin {align*} \int e^{-2 \tanh ^{-1}(a+b x)} \, dx &=\int \frac {1-a-b x}{1+a+b x} \, dx\\ &=\int \left (-1+\frac {2}{1+a+b x}\right ) \, dx\\ &=-x+\frac {2 \log (1+a+b x)}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \frac {2 \log (a+b x+1)}{b}-x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.31, size = 18, normalized size = 1.12 \[ -\frac {b x - 2 \, \log \left (b x + a + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.53, size = 38, normalized size = 2.38 \[ -\frac {b x + a + 1}{b} - \frac {2 \, \log \left (\frac {{\left | b x + a + 1 \right |}}{{\left (b x + a + 1\right )}^{2} {\left | b \right |}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 17, normalized size = 1.06 \[ -x +\frac {2 \ln \left (b x +a +1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 16, normalized size = 1.00 \[ -x + \frac {2 \, \log \left (b x + a + 1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 16, normalized size = 1.00 \[ \frac {2\,\ln \left (a+b\,x+1\right )}{b}-x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.11, size = 12, normalized size = 0.75 \[ - x + \frac {2 \log {\left (a + b x + 1 \right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________