Optimal. Leaf size=21 \[ \frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2248, 37} \[ \frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 2248
Rubi steps
\begin {align*} \int \frac {e^{2 x}}{\left (a+b e^x\right )^3} \, dx &=\operatorname {Subst}\left (\int \frac {x}{(a+b x)^3} \, dx,x,e^x\right )\\ &=\frac {e^{2 x}}{2 a \left (a+b e^x\right )^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 21, normalized size = 1.00 \[ \frac {e^{2 x}}{2 a \left (a+b e^x\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.39, size = 35, normalized size = 1.67 \[ -\frac {2 \, b e^{x} + a}{2 \, {\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 20, normalized size = 0.95 \[ -\frac {2 \, b e^{x} + a}{2 \, {\left (b e^{x} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 29, normalized size = 1.38 \[ \frac {a}{2 \left (b \,{\mathrm e}^{x}+a \right )^{2} b^{2}}-\frac {1}{\left (b \,{\mathrm e}^{x}+a \right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.48, size = 61, normalized size = 2.90 \[ -\frac {b e^{x}}{b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}} - \frac {a}{2 \, {\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.56, size = 29, normalized size = 1.38 \[ \frac {{\mathrm {e}}^{2\,x}}{2\,a\,\left (a^2+2\,{\mathrm {e}}^x\,a\,b+{\mathrm {e}}^{2\,x}\,b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.13, size = 37, normalized size = 1.76 \[ \frac {- a - 2 b e^{x}}{2 a^{2} b^{2} + 4 a b^{3} e^{x} + 2 b^{4} e^{2 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________