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

Optimal. Leaf size=31 \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]

[Out]

E^(2*x)/(2*b) - (a*Log[a + b*E^(2*x)])/(2*b^2)

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Rubi [A]  time = 0.0338813, antiderivative size = 31, 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} \[ \frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*x)/(a + b*E^(2*x)),x]

[Out]

E^(2*x)/(2*b) - (a*Log[a + b*E^(2*x)])/(2*b^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}}{a+b e^{2 x}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{a+b x} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx,x,e^{2 x}\right )\\ &=\frac{e^{2 x}}{2 b}-\frac{a \log \left (a+b e^{2 x}\right )}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0155778, size = 30, normalized size = 0.97 \[ \frac{1}{2} \left (\frac{e^{2 x}}{b}-\frac{a \log \left (a+b e^{2 x}\right )}{b^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*x)/(a + b*E^(2*x)),x]

[Out]

(E^(2*x)/b - (a*Log[a + b*E^(2*x)])/b^2)/2

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Maple [A]  time = 0.003, size = 26, normalized size = 0.8 \begin{align*}{\frac{ \left ({{\rm e}^{x}} \right ) ^{2}}{2\,b}}-{\frac{a\ln \left ( a+b \left ({{\rm e}^{x}} \right ) ^{2} \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13591, size = 34, normalized size = 1.1 \begin{align*} \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a \log \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]

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

[Out]

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

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Fricas [A]  time = 1.49142, size = 59, normalized size = 1.9 \begin{align*} \frac{b e^{\left (2 \, x\right )} - a \log \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]

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

[Out]

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

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Sympy [A]  time = 0.140633, size = 29, normalized size = 0.94 \begin{align*} - \frac{a \log{\left (\frac{a}{b} + e^{2 x} \right )}}{2 b^{2}} + \begin{cases} \frac{e^{2 x}}{2 b} & \text{for}\: 2 b \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(a/b + exp(2*x))/(2*b**2) + Piecewise((exp(2*x)/(2*b), Ne(2*b, 0)), (x/b, True))

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Giac [A]  time = 1.29887, size = 35, normalized size = 1.13 \begin{align*} \frac{e^{\left (2 \, x\right )}}{2 \, b} - \frac{a \log \left ({\left | b e^{\left (2 \, x\right )} + a \right |}\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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