3.25 \(\int \frac{e^{4 x}}{(a+b e^{2 x})^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]

[Out]

a/(6*b^2*(a + b*E^(2*x))^3) - 1/(4*b^2*(a + b*E^(2*x))^2)

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Rubi [A]  time = 0.0366779, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2248, 43} \[ \frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*x)/(a + b*E^(2*x))^4,x]

[Out]

a/(6*b^2*(a + b*E^(2*x))^3) - 1/(4*b^2*(a + b*E^(2*x))^2)

Rule 2248

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{e^{4 x}}{\left (a+b e^{2 x}\right )^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^4} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^4}+\frac{1}{b (a+b x)^3}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{a}{6 b^2 \left (a+b e^{2 x}\right )^3}-\frac{1}{4 b^2 \left (a+b e^{2 x}\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.0181263, size = 28, normalized size = 0.74 \[ -\frac{a+3 b e^{2 x}}{12 b^2 \left (a+b e^{2 x}\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a + 3*b*E^(2*x))/(12*b^2*(a + b*E^(2*x))^3)

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Maple [A]  time = 0.004, size = 33, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{2}}}+{\frac{a}{6\,{b}^{2} \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/b^2/(a+b*exp(x)^2)^2+1/6*a/b^2/(a+b*exp(x)^2)^3

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Maxima [B]  time = 1.1199, size = 123, normalized size = 3.24 \begin{align*} -\frac{b e^{\left (2 \, x\right )}}{4 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} - \frac{a}{12 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*e^(2*x)/(b^5*e^(6*x) + 3*a*b^4*e^(4*x) + 3*a^2*b^3*e^(2*x) + a^3*b^2) - 1/12*a/(b^5*e^(6*x) + 3*a*b^4*e
^(4*x) + 3*a^2*b^3*e^(2*x) + a^3*b^2)

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Fricas [A]  time = 1.48252, size = 117, normalized size = 3.08 \begin{align*} -\frac{3 \, b e^{\left (2 \, x\right )} + a}{12 \,{\left (b^{5} e^{\left (6 \, x\right )} + 3 \, a b^{4} e^{\left (4 \, x\right )} + 3 \, a^{2} b^{3} e^{\left (2 \, x\right )} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b*e^(2*x) + a)/(b^5*e^(6*x) + 3*a*b^4*e^(4*x) + 3*a^2*b^3*e^(2*x) + a^3*b^2)

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Sympy [A]  time = 0.178177, size = 54, normalized size = 1.42 \begin{align*} \frac{- a - 3 b e^{2 x}}{12 a^{3} b^{2} + 36 a^{2} b^{3} e^{2 x} + 36 a b^{4} e^{4 x} + 12 b^{5} e^{6 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a - 3*b*exp(2*x))/(12*a**3*b**2 + 36*a**2*b**3*exp(2*x) + 36*a*b**4*exp(4*x) + 12*b**5*exp(6*x))

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Giac [A]  time = 1.34235, size = 32, normalized size = 0.84 \begin{align*} -\frac{3 \, b e^{\left (2 \, x\right )} + a}{12 \,{\left (b e^{\left (2 \, x\right )} + a\right )}^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*b*e^(2*x) + a)/((b*e^(2*x) + a)^3*b^2)