3.81.45 \(\int \frac {198 e+(-3-132 e) e^x+22 e^{1+2 x}}{9 e-6 e^{1+x}+e^{1+2 x}} \, dx\)

Optimal. Leaf size=25 \[ 1+\frac {e^{-1+x}}{-3+e^x}-25 (3-x)-3 x \]

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Rubi [A]  time = 0.07, antiderivative size = 18, normalized size of antiderivative = 0.72, number of steps used = 4, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2282, 12, 893} \begin {gather*} 22 x-\frac {3}{e \left (3-e^x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(198*E + (-3 - 132*E)*E^x + 22*E^(1 + 2*x))/(9*E - 6*E^(1 + x) + E^(1 + 2*x)),x]

[Out]

-3/(E*(3 - E^x)) + 22*x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

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} &=\operatorname {Subst}\left (\int \frac {198 e-3 (1+44 e) x+22 e x^2}{e (3-x)^2 x} \, dx,x,e^x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {198 e-3 (1+44 e) x+22 e x^2}{(3-x)^2 x} \, dx,x,e^x\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {3}{(-3+x)^2}+\frac {22 e}{x}\right ) \, dx,x,e^x\right )}{e}\\ &=-\frac {3}{e \left (3-e^x\right )}+22 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 20, normalized size = 0.80 \begin {gather*} \frac {-\frac {3}{3-e^x}+22 e x}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(198*E + (-3 - 132*E)*E^x + 22*E^(1 + 2*x))/(9*E - 6*E^(1 + x) + E^(1 + 2*x)),x]

[Out]

(-3/(3 - E^x) + 22*E*x)/E

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fricas [A]  time = 0.46, size = 28, normalized size = 1.12 \begin {gather*} \frac {66 \, x e - 22 \, x e^{\left (x + 1\right )} - 3}{3 \, e - e^{\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*exp(1)*exp(x)^2+(-132*exp(1)-3)*exp(x)+198*exp(1))/(exp(1)*exp(x)^2-6*exp(1)*exp(x)+9*exp(1)),x,
algorithm="fricas")

[Out]

(66*x*e - 22*x*e^(x + 1) - 3)/(3*e - e^(x + 1))

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giac [A]  time = 0.19, size = 14, normalized size = 0.56 \begin {gather*} 22 \, x + \frac {3 \, e^{\left (-1\right )}}{e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*exp(1)*exp(x)^2+(-132*exp(1)-3)*exp(x)+198*exp(1))/(exp(1)*exp(x)^2-6*exp(1)*exp(x)+9*exp(1)),x,
algorithm="giac")

[Out]

22*x + 3*e^(-1)/(e^x - 3)

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maple [A]  time = 0.05, size = 15, normalized size = 0.60

method	result	size
risch	\(22 x +\frac {3 \,{\mathrm e}^{-1}}{{\mathrm e}^{x}-3}\)	\(15\)
derivativedivides	\({\mathrm e}^{-1} \left (\frac {3}{{\mathrm e}^{x}-3}+22 \,{\mathrm e} \ln \left ({\mathrm e}^{x}\right )\right )\)	\(22\)
default	\({\mathrm e}^{-1} \left (\frac {3}{{\mathrm e}^{x}-3}+22 \,{\mathrm e} \ln \left ({\mathrm e}^{x}\right )\right )\)	\(22\)
norman	\(\frac {-66 x +22 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{-1}}{{\mathrm e}^{x}-3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22*exp(1)*exp(x)^2+(-132*exp(1)-3)*exp(x)+198*exp(1))/(exp(1)*exp(x)^2-6*exp(1)*exp(x)+9*exp(1)),x,method
=_RETURNVERBOSE)

[Out]

22*x+3/(exp(x)-3)*exp(-1)

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maxima [A]  time = 0.37, size = 19, normalized size = 0.76 \begin {gather*} 22 \, x - \frac {3}{3 \, e - e^{\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*exp(1)*exp(x)^2+(-132*exp(1)-3)*exp(x)+198*exp(1))/(exp(1)*exp(x)^2-6*exp(1)*exp(x)+9*exp(1)),x,
algorithm="maxima")

[Out]

22*x - 3/(3*e - e^(x + 1))

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mupad [B]  time = 6.55, size = 17, normalized size = 0.68 \begin {gather*} 22\,x+\frac {3}{{\mathrm {e}}^{x+1}-3\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((198*exp(1) + 22*exp(2*x)*exp(1) - exp(x)*(132*exp(1) + 3))/(9*exp(1) + exp(2*x)*exp(1) - 6*exp(1)*exp(x))
,x)

[Out]

22*x + 3/(exp(x + 1) - 3*exp(1))

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sympy [A]  time = 0.10, size = 15, normalized size = 0.60 \begin {gather*} 22 x + \frac {3}{e e^{x} - 3 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((22*exp(1)*exp(x)**2+(-132*exp(1)-3)*exp(x)+198*exp(1))/(exp(1)*exp(x)**2-6*exp(1)*exp(x)+9*exp(1)),
x)

[Out]

22*x + 3/(E*exp(x) - 3*E)

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