Optimal. Leaf size=23 \[ \frac{e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0267732, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2248, 37} \[ \frac{e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2248
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^3} \, dx,x,e^{2 x}\right )\\ &=\frac{e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2}\\ \end{align*}
Mathematica [A] time = 0.009315, size = 23, normalized size = 1. \[ \frac{e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 33, normalized size = 1.4 \begin{align*}{\frac{a}{4\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{2}}}-{\frac{1}{2\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.02851, size = 90, normalized size = 3.91 \begin{align*} -\frac{b e^{\left (2 \, x\right )}}{2 \,{\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} - \frac{a}{4 \,{\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.45707, size = 89, normalized size = 3.87 \begin{align*} -\frac{2 \, b e^{\left (2 \, x\right )} + a}{4 \,{\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 0.15944, size = 41, normalized size = 1.78 \begin{align*} \frac{- a - 2 b e^{2 x}}{4 a^{2} b^{2} + 8 a b^{3} e^{2 x} + 4 b^{4} e^{4 x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.25, size = 32, normalized size = 1.39 \begin{align*} -\frac{2 \, b e^{\left (2 \, x\right )} + a}{4 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]