Optimal. Leaf size=25 \[ \frac {\left (a+b e^x\right ) \log \left (a+b e^x\right )}{b}-e^x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2194, 2554, 12, 2248, 43} \[ e^x \log \left (a+b e^x\right )+\frac {a \log \left (a+b e^x\right )}{b}-e^x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 43
Rule 2194
Rule 2248
Rule 2554
Rubi steps
\begin {align*} \int e^x \log \left (a+b e^x\right ) \, dx &=e^x \log \left (a+b e^x\right )-\int \frac {b e^{2 x}}{a+b e^x} \, dx\\ &=e^x \log \left (a+b e^x\right )-b \int \frac {e^{2 x}}{a+b e^x} \, dx\\ &=e^x \log \left (a+b e^x\right )-b \operatorname {Subst}\left (\int \frac {x}{a+b x} \, dx,x,e^x\right )\\ &=e^x \log \left (a+b e^x\right )-b \operatorname {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,e^x\right )\\ &=-e^x+\frac {a \log \left (a+b e^x\right )}{b}+e^x \log \left (a+b e^x\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {\left (a+b e^x\right ) \log \left (a+b e^x\right )}{b}-e^x \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 25, normalized size = 1.00 \[ -\frac {b e^{x} - {\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 26, normalized size = 1.04 \[ -\frac {b e^{x} - {\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 34, normalized size = 1.36 \[ {\mathrm e}^{x} \ln \left (b \,{\mathrm e}^{x}+a \right )+\frac {a \ln \left (b \,{\mathrm e}^{x}+a \right )}{b}-{\mathrm e}^{x}-\frac {a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.74, size = 26, normalized size = 1.04 \[ -\frac {b e^{x} - {\left (b e^{x} + a\right )} \log \left (b e^{x} + a\right ) + a}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.66, size = 27, normalized size = 1.08 \[ {\mathrm {e}}^x\,\ln \left (a+b\,{\mathrm {e}}^x\right )-{\mathrm {e}}^x+\frac {a\,\ln \left (a+b\,{\mathrm {e}}^x\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________