3.74.32 \(\int \frac {e^4+432 e^{2 x}}{e^4+144 e^{2 x}} \, dx\)

Optimal. Leaf size=16 \[ x+\log \left (\frac {e^4}{144}+e^{2 x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2282, 72} \begin {gather*} x+\log \left (144 e^{2 x}+e^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^4 + 432*E^(2*x))/(E^4 + 144*E^(2*x)),x]

[Out]

x + Log[E^4 + 144*E^(2*x)]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^4+432 x}{x \left (e^4+144 x\right )} \, dx,x,e^{2 x}\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}+\frac {288}{e^4+144 x}\right ) \, dx,x,e^{2 x}\right )\\ &=x+\log \left (e^4+144 e^{2 x}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 14, normalized size = 0.88 \begin {gather*} x+\log \left (e^4+144 e^{2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^4 + 432*E^(2*x))/(E^4 + 144*E^(2*x)),x]

[Out]

x + Log[E^4 + 144*E^(2*x)]

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 12, normalized size = 0.75 \begin {gather*} x + \log \left (e^{4} + 144 \, e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((432*exp(2*x)+exp(2)^2)/(144*exp(2*x)+exp(2)^2),x, algorithm="fricas")

[Out]

x + log(e^4 + 144*e^(2*x))

________________________________________________________________________________________

giac [A]  time = 0.21, size = 12, normalized size = 0.75 \begin {gather*} x + \log \left (e^{4} + 144 \, e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((432*exp(2*x)+exp(2)^2)/(144*exp(2*x)+exp(2)^2),x, algorithm="giac")

[Out]

x + log(e^4 + 144*e^(2*x))

________________________________________________________________________________________

maple [A]  time = 0.04, size = 13, normalized size = 0.81

method	result	size
risch	\(x +\ln \left (\frac {{\mathrm e}^{4}}{144}+{\mathrm e}^{2 x}\right )\)	\(13\)
norman	\(x +\ln \left (144 \,{\mathrm e}^{2 x}+{\mathrm e}^{4}\right )\)	\(15\)
derivativedivides	\(\ln \left (144 \,{\mathrm e}^{2 x}+{\mathrm e}^{4}\right )+\frac {\ln \left ({\mathrm e}^{2 x}\right )}{2}\)	\(19\)
default	\(\ln \left (144 \,{\mathrm e}^{2 x}+{\mathrm e}^{4}\right )+\frac {\ln \left ({\mathrm e}^{2 x}\right )}{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((432*exp(2*x)+exp(2)^2)/(144*exp(2*x)+exp(2)^2),x,method=_RETURNVERBOSE)

[Out]

x+ln(1/144*exp(4)+exp(2*x))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 12, normalized size = 0.75 \begin {gather*} x + \log \left (e^{4} + 144 \, e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((432*exp(2*x)+exp(2)^2)/(144*exp(2*x)+exp(2)^2),x, algorithm="maxima")

[Out]

x + log(e^4 + 144*e^(2*x))

________________________________________________________________________________________

mupad [B]  time = 4.54, size = 12, normalized size = 0.75 \begin {gather*} x+\ln \left (144\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((432*exp(2*x) + exp(4))/(144*exp(2*x) + exp(4)),x)

[Out]

x + log(144*exp(2*x) + exp(4))

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} x + \log {\left (e^{2 x} + \frac {e^{4}}{144} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((432*exp(2*x)+exp(2)**2)/(144*exp(2*x)+exp(2)**2),x)

[Out]

x + log(exp(2*x) + exp(4)/144)

________________________________________________________________________________________