Optimal. Leaf size=17 \[ \frac {\log ^2\left (e^{a+b x}\right )}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2188, 30}
\begin {gather*} \frac {\log ^2\left (e^{a+b x}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 2188
Rubi steps
\begin {align*} \int \log \left (e^{a+b x}\right ) \, dx &=\frac {\text {Subst}\left (\int x \, dx,x,\log \left (e^{a+b x}\right )\right )}{b}\\ &=\frac {\log ^2\left (e^{a+b x}\right )}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log ^2\left (e^{a+b x}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 15, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {\ln \left ({\mathrm e}^{b x +a}\right )^{2}}{2 b}\) | \(15\) |
default | \(\frac {\ln \left ({\mathrm e}^{b x +a}\right )^{2}}{2 b}\) | \(15\) |
norman | \(\frac {\ln \left ({\mathrm e}^{b x +a}\right )^{2}}{2 b}\) | \(15\) |
risch | \(x \ln \left ({\mathrm e}^{b x +a}\right )-\frac {b \,x^{2}}{2}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.02, size = 8, normalized size = 0.47 \begin {gather*} a x + \frac {b x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 3.15, size = 10, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.07, size = 17, normalized size = 1.00 \begin {gather*} x\,\ln \left ({\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\right )-\frac {b\,x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________