3.5.91 \(\int \frac {1+x+18 e^{1-6 e^{1+3 x}+3 x} x}{x} \, dx\)

Optimal. Leaf size=18 \[ 1-e^{-6 e^{1+3 x}}+x+\log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {14, 2282, 2194, 43} \begin {gather*} x-e^{-6 e^{3 x+1}}+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x + 18*E^(1 - 6*E^(1 + 3*x) + 3*x)*x)/x,x]

[Out]

-E^(-6*E^(1 + 3*x)) + x + Log[x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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])

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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} &=\int \left (18 e^{1-6 e^{1+3 x}+3 x}+\frac {1+x}{x}\right ) \, dx\\ &=18 \int e^{1-6 e^{1+3 x}+3 x} \, dx+\int \frac {1+x}{x} \, dx\\ &=6 \operatorname {Subst}\left (\int e^{1-6 e x} \, dx,x,e^{3 x}\right )+\int \left (1+\frac {1}{x}\right ) \, dx\\ &=-e^{-6 e^{1+3 x}}+x+\log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 17, normalized size = 0.94 \begin {gather*} -e^{-6 e^{1+3 x}}+x+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + 18*E^(1 - 6*E^(1 + 3*x) + 3*x)*x)/x,x]

[Out]

-E^(-6*E^(1 + 3*x)) + x + Log[x]

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fricas [B]  time = 0.72, size = 41, normalized size = 2.28 \begin {gather*} {\left (x e^{\left (3 \, x + 1\right )} + e^{\left (3 \, x + 1\right )} \log \relax (x) - e^{\left (3 \, x - 6 \, e^{\left (3 \, x + 1\right )} + 1\right )}\right )} e^{\left (-3 \, x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x*exp(3*x+1)*exp(-6*exp(3*x+1))+x+1)/x,x, algorithm="fricas")

[Out]

(x*e^(3*x + 1) + e^(3*x + 1)*log(x) - e^(3*x - 6*e^(3*x + 1) + 1))*e^(-3*x - 1)

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giac [B]  time = 0.43, size = 41, normalized size = 2.28 \begin {gather*} {\left (x e^{\left (3 \, x + 1\right )} + e^{\left (3 \, x + 1\right )} \log \relax (x) - e^{\left (3 \, x - 6 \, e^{\left (3 \, x + 1\right )} + 1\right )}\right )} e^{\left (-3 \, x - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x*exp(3*x+1)*exp(-6*exp(3*x+1))+x+1)/x,x, algorithm="giac")

[Out]

(x*e^(3*x + 1) + e^(3*x + 1)*log(x) - e^(3*x - 6*e^(3*x + 1) + 1))*e^(-3*x - 1)

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

method	result	size
norman	\(x -{\mathrm e}^{-6 \,{\mathrm e}^{3 x +1}}+\ln \relax (x )\)	\(16\)
risch	\(x -{\mathrm e}^{-6 \,{\mathrm e}^{3 x +1}}+\ln \relax (x )\)	\(16\)
default	\(x -{\mathrm e}^{-6 \,{\mathrm e}^{3 x +1}}+\ln \relax (x )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x*exp(3*x+1)*exp(-6*exp(3*x+1))+x+1)/x,x,method=_RETURNVERBOSE)

[Out]

x-exp(-6*exp(3*x+1))+ln(x)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.83 \begin {gather*} x - e^{\left (-6 \, e^{\left (3 \, x + 1\right )}\right )} + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x*exp(3*x+1)*exp(-6*exp(3*x+1))+x+1)/x,x, algorithm="maxima")

[Out]

x - e^(-6*e^(3*x + 1)) + log(x)

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mupad [B]  time = 0.45, size = 15, normalized size = 0.83 \begin {gather*} x-{\mathrm {e}}^{-6\,{\mathrm {e}}^{3\,x}\,\mathrm {e}}+\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 18*x*exp(-6*exp(3*x + 1))*exp(3*x + 1) + 1)/x,x)

[Out]

x - exp(-6*exp(3*x)*exp(1)) + log(x)

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sympy [A]  time = 0.14, size = 15, normalized size = 0.83 \begin {gather*} x + \log {\relax (x )} - e^{- 6 e^{3 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((18*x*exp(3*x+1)*exp(-6*exp(3*x+1))+x+1)/x,x)

[Out]

x + log(x) - exp(-6*exp(3*x + 1))

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