Optimal. Leaf size=23 \[ \frac {e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 37
Rule 2248
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 = 0.01, size = 23, normalized size = 1.00 \[ \frac {e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 39, normalized size = 1.70 \[ -\frac {2 \, b e^{\left (2 \, x\right )} + a}{4 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 24, normalized size = 1.04 \[ -\frac {2 \, b e^{\left (2 \, x\right )} + a}{4 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 33, normalized size = 1.43 \[ \frac {a}{4 \left (b \,{\mathrm e}^{2 x}+a \right )^{2} b^{2}}-\frac {1}{2 \left (b \,{\mathrm e}^{2 x}+a \right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.46, size = 67, normalized size = 2.91 \[ -\frac {b e^{\left (2 \, x\right )}}{2 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} - \frac {a}{4 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.56, size = 31, normalized size = 1.35 \[ \frac {{\mathrm {e}}^{4\,x}}{4\,a\,\left (a^2+2\,{\mathrm {e}}^{2\,x}\,a\,b+{\mathrm {e}}^{4\,x}\,b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.14, size = 41, normalized size = 1.78 \[ \frac {- a - 2 b e^{2 x}}{4 a^{2} b^{2} + 8 a b^{3} e^{2 x} + 4 b^{4} e^{4 x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________