\(\int \frac {e^{4 x}}{(a+b e^{2 x})^3} \, dx\) [24]

Optimal result

Integrand size = 17, antiderivative size = 23 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=\frac {e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]

[Out]

1/4*exp(4*x)/a/(a+b*exp(2*x))^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2280, 37} \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=\frac {e^{4 x}}{4 a \left (a+b e^{2 x}\right )^2} \]

[In]

Int[E^(4*x)/(a + b*E^(2*x))^3,x]

[Out]

E^(4*x)/(4*a*(a + b*E^(2*x))^2)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 2280

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_.)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Wit
h[{m = FullSimplify[g*h*(Log[G]/(d*e*Log[F]))]}, Dist[Denominator[m]*(G^(f*h - c*g*(h/d))/(d*e*Log[F])), Subst
[Int[x^(Numerator[m] - 1)*(a + b*x^Denominator[m])^p, x], x, F^(e*((c + d*x)/Denominator[m]))], x] /; LeQ[m, -
1] || GeQ[m, 1]] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=\frac {-a-2 b e^{2 x}}{4 b^2 \left (a+b e^{2 x}\right )^2} \]

[In]

Integrate[E^(4*x)/(a + b*E^(2*x))^3,x]

[Out]

(-a - 2*b*E^(2*x))/(4*b^2*(a + b*E^(2*x))^2)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87

method	result	size
parallelrisch	\(\frac {{\mathrm e}^{4 x}}{4 a \left (a +b \,{\mathrm e}^{2 x}\right )^{2}}\)	\(20\)
risch	\(-\frac {2 b \,{\mathrm e}^{2 x}+a}{4 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{2}}\)	\(25\)
norman	\(\frac {-\frac {a}{4 b^{2}}-\frac {{\mathrm e}^{2 x}}{2 b}}{\left (a +b \,{\mathrm e}^{2 x}\right )^{2}}\)	\(28\)
default	\(\frac {a}{4 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )^{2}}-\frac {1}{2 b^{2} \left (a +b \,{\mathrm e}^{2 x}\right )}\)	\(33\)

[In]

int(exp(4*x)/(a+b*exp(2*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/4*exp(4*x)/a/(a+b*exp(2*x))^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.70 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=-\frac {2 \, b e^{\left (2 \, x\right )} + a}{4 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \]

[In]

integrate(exp(4*x)/(a+b*exp(2*x))^3,x, algorithm="fricas")

[Out]

-1/4*(2*b*e^(2*x) + a)/(b^4*e^(4*x) + 2*a*b^3*e^(2*x) + a^2*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).

Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=\frac {- a - 2 b e^{2 x}}{4 a^{2} b^{2} + 8 a b^{3} e^{2 x} + 4 b^{4} e^{4 x}} \]

[In]

integrate(exp(4*x)/(a+b*exp(2*x))**3,x)

[Out]

(-a - 2*b*exp(2*x))/(4*a**2*b**2 + 8*a*b**3*exp(2*x) + 4*b**4*exp(4*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (19) = 38\).

Time = 0.17 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.91 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=-\frac {b e^{\left (2 \, x\right )}}{2 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} - \frac {a}{4 \, {\left (b^{4} e^{\left (4 \, x\right )} + 2 \, a b^{3} e^{\left (2 \, x\right )} + a^{2} b^{2}\right )}} \]

[In]

integrate(exp(4*x)/(a+b*exp(2*x))^3,x, algorithm="maxima")

[Out]

-1/2*b*e^(2*x)/(b^4*e^(4*x) + 2*a*b^3*e^(2*x) + a^2*b^2) - 1/4*a/(b^4*e^(4*x) + 2*a*b^3*e^(2*x) + a^2*b^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=-\frac {2 \, b e^{\left (2 \, x\right )} + a}{4 \, {\left (b e^{\left (2 \, x\right )} + a\right )}^{2} b^{2}} \]

[In]

integrate(exp(4*x)/(a+b*exp(2*x))^3,x, algorithm="giac")

[Out]

-1/4*(2*b*e^(2*x) + a)/((b*e^(2*x) + a)^2*b^2)

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {e^{4 x}}{\left (a+b e^{2 x}\right )^3} \, dx=\frac {{\mathrm {e}}^{4\,x}}{4\,a\,\left (a^2+2\,{\mathrm {e}}^{2\,x}\,a\,b+{\mathrm {e}}^{4\,x}\,b^2\right )} \]

[In]

int(exp(4*x)/(a + b*exp(2*x))^3,x)

[Out]

exp(4*x)/(4*a*(b^2*exp(4*x) + a^2 + 2*a*b*exp(2*x)))