3.101.27 \(\int \frac {18 x-6 x^2+54 x^6+e^x (-12 x+2 x^2-x^3-36 x^6)+e^{2 x} (2 x+6 x^6)+(-18+e^x (12-12 x)+18 x+e^{2 x} (-2+2 x)) \log (x)}{9 x-6 e^x x+e^{2 x} x} \, dx\) [10027]

Optimal. Leaf size=30 \[ 3+\frac {x^2}{-3+e^x}+x \left (x+x^5\right )-(-x+\log (x))^2 \]

[Out]

x^2/(exp(x)-3)+3-(ln(x)-x)^2+(x^5+x)*x

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Rubi [A]
time = 0.79, antiderivative size = 29, normalized size of antiderivative = 0.97, number of steps used = 30, number of rules used = 14, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {6820, 6874, 2215, 2221, 2317, 2438, 2611, 2320, 6724, 2216, 2222, 2388, 2338, 2332} \begin {gather*} x^6-\frac {x^2}{3-e^x}-\log ^2(x)+2 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18*x - 6*x^2 + 54*x^6 + E^x*(-12*x + 2*x^2 - x^3 - 36*x^6) + E^(2*x)*(2*x + 6*x^6) + (-18 + E^x*(12 - 12*
x) + 18*x + E^(2*x)*(-2 + 2*x))*Log[x])/(9*x - 6*E^x*x + E^(2*x)*x),x]

[Out]

-(x^2/(3 - E^x)) + x^6 + 2*x*Log[x] - Log[x]^2

Rule 2215

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[(c
+ d*x)^(m + 1)/(a*d*(m + 1)), x] - Dist[b/a, Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n))
, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2222

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_.)*
((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1
)*Log[F])), x] - Dist[d*(m/(b*f*g*n*(p + 1)*Log[F])), Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1
), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

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]]]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {e^{2 x} \left (2+6 x^5\right )+6 \left (3-x+9 x^5\right )-e^x \left (12-2 x+x^2+36 x^5\right )}{\left (-3+e^x\right )^2}+\frac {2 (-1+x) \log (x)}{x}\right ) \, dx\\ &=2 \int \frac {(-1+x) \log (x)}{x} \, dx+\int \frac {e^{2 x} \left (2+6 x^5\right )+6 \left (3-x+9 x^5\right )-e^x \left (12-2 x+x^2+36 x^5\right )}{\left (-3+e^x\right )^2} \, dx\\ &=2 \int \log (x) \, dx-2 \int \frac {\log (x)}{x} \, dx+\int \left (-\frac {(-2+x) x}{-3+e^x}-\frac {3 x^2}{\left (-3+e^x\right )^2}+2 \left (1+3 x^5\right )\right ) \, dx\\ &=-2 x+2 x \log (x)-\log ^2(x)+2 \int \left (1+3 x^5\right ) \, dx-3 \int \frac {x^2}{\left (-3+e^x\right )^2} \, dx-\int \frac {(-2+x) x}{-3+e^x} \, dx\\ &=x^6+2 x \log (x)-\log ^2(x)-\int \frac {e^x x^2}{\left (-3+e^x\right )^2} \, dx+\int \frac {x^2}{-3+e^x} \, dx-\int \left (-\frac {2 x}{-3+e^x}+\frac {x^2}{-3+e^x}\right ) \, dx\\ &=-\frac {x^2}{3-e^x}-\frac {x^3}{9}+x^6+2 x \log (x)-\log ^2(x)+\frac {1}{3} \int \frac {e^x x^2}{-3+e^x} \, dx-\int \frac {x^2}{-3+e^x} \, dx\\ &=-\frac {x^2}{3-e^x}+x^6+\frac {1}{3} x^2 \log \left (1-\frac {e^x}{3}\right )+2 x \log (x)-\log ^2(x)-\frac {1}{3} \int \frac {e^x x^2}{-3+e^x} \, dx-\frac {2}{3} \int x \log \left (1-\frac {e^x}{3}\right ) \, dx\\ &=-\frac {x^2}{3-e^x}+x^6+2 x \log (x)-\log ^2(x)+\frac {2}{3} x \text {Li}_2\left (\frac {e^x}{3}\right )+\frac {2}{3} \int x \log \left (1-\frac {e^x}{3}\right ) \, dx-\frac {2}{3} \int \text {Li}_2\left (\frac {e^x}{3}\right ) \, dx\\ &=-\frac {x^2}{3-e^x}+x^6+2 x \log (x)-\log ^2(x)+\frac {2}{3} \int \text {Li}_2\left (\frac {e^x}{3}\right ) \, dx-\frac {2}{3} \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{3}\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {x^2}{3-e^x}+x^6+2 x \log (x)-\log ^2(x)-\frac {2 \text {Li}_3\left (\frac {e^x}{3}\right )}{3}+\frac {2}{3} \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{3}\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {x^2}{3-e^x}+x^6+2 x \log (x)-\log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 1.43, size = 26, normalized size = 0.87 \begin {gather*} \frac {x^2}{-3+e^x}+x^6+2 x \log (x)-\log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*x - 6*x^2 + 54*x^6 + E^x*(-12*x + 2*x^2 - x^3 - 36*x^6) + E^(2*x)*(2*x + 6*x^6) + (-18 + E^x*(12
- 12*x) + 18*x + E^(2*x)*(-2 + 2*x))*Log[x])/(9*x - 6*E^x*x + E^(2*x)*x),x]

[Out]

x^2/(-3 + E^x) + x^6 + 2*x*Log[x] - Log[x]^2

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Maple [A]
time = 0.07, size = 36, normalized size = 1.20

method	result	size
risch	\(-\ln \left (x \right )^{2}+2 x \ln \left (x \right )+\frac {x^{2} \left ({\mathrm e}^{x} x^{4}-3 x^{4}+1\right )}{{\mathrm e}^{x}-3}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x-2)*exp(x)^2+(-12*x+12)*exp(x)+18*x-18)*ln(x)+(6*x^6+2*x)*exp(x)^2+(-36*x^6-x^3+2*x^2-12*x)*exp(x)+5
4*x^6-6*x^2+18*x)/(x*exp(x)^2-6*exp(x)*x+9*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)^2+2*x*ln(x)+x^2*(exp(x)*x^4-3*x^4+1)/(exp(x)-3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (29) = 58\).
time = 0.30, size = 74, normalized size = 2.47 \begin {gather*} -\frac {3 \, x^{6} - x^{2} - {\left (x^{6} + 2 \, x \log \left (x\right ) - \log \left (x\right )^{2} - 2 \, x\right )} e^{x} + 6 \, x \log \left (x\right ) - 3 \, \log \left (x\right )^{2} - 6 \, x - 6}{e^{x} - 3} + \frac {2 \, {\left (x e^{x} - 3 \, x - 3\right )}}{e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+2*x)*exp(x)^2+(-12*x+12)*exp(x)+18*x-18)*log(x)+(6*x^6+2*x)*exp(x)^2+(-36*x^6-x^3+2*x^2-12*x)*
exp(x)+54*x^6-6*x^2+18*x)/(x*exp(x)^2-6*exp(x)*x+9*x),x, algorithm="maxima")

[Out]

-(3*x^6 - x^2 - (x^6 + 2*x*log(x) - log(x)^2 - 2*x)*e^x + 6*x*log(x) - 3*log(x)^2 - 6*x - 6)/(e^x - 3) + 2*(x*
e^x - 3*x - 3)/(e^x - 3)

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Fricas [A]
time = 0.45, size = 44, normalized size = 1.47 \begin {gather*} \frac {x^{6} e^{x} - 3 \, x^{6} - {\left (e^{x} - 3\right )} \log \left (x\right )^{2} + x^{2} + 2 \, {\left (x e^{x} - 3 \, x\right )} \log \left (x\right )}{e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+2*x)*exp(x)^2+(-12*x+12)*exp(x)+18*x-18)*log(x)+(6*x^6+2*x)*exp(x)^2+(-36*x^6-x^3+2*x^2-12*x)*
exp(x)+54*x^6-6*x^2+18*x)/(x*exp(x)^2-6*exp(x)*x+9*x),x, algorithm="fricas")

[Out]

(x^6*e^x - 3*x^6 - (e^x - 3)*log(x)^2 + x^2 + 2*(x*e^x - 3*x)*log(x))/(e^x - 3)

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Sympy [A]
time = 0.09, size = 22, normalized size = 0.73 \begin {gather*} x^{6} + \frac {x^{2}}{e^{x} - 3} + 2 x \log {\left (x \right )} - \log {\left (x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+2*x)*exp(x)**2+(-12*x+12)*exp(x)+18*x-18)*ln(x)+(6*x**6+2*x)*exp(x)**2+(-36*x**6-x**3+2*x**2-1
2*x)*exp(x)+54*x**6-6*x**2+18*x)/(x*exp(x)**2-6*exp(x)*x+9*x),x)

[Out]

x**6 + x**2/(exp(x) - 3) + 2*x*log(x) - log(x)**2

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Giac [A]
time = 0.42, size = 48, normalized size = 1.60 \begin {gather*} \frac {x^{6} e^{x} - 3 \, x^{6} + 2 \, x e^{x} \log \left (x\right ) - e^{x} \log \left (x\right )^{2} + x^{2} - 6 \, x \log \left (x\right ) + 3 \, \log \left (x\right )^{2}}{e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2+2*x)*exp(x)^2+(-12*x+12)*exp(x)+18*x-18)*log(x)+(6*x^6+2*x)*exp(x)^2+(-36*x^6-x^3+2*x^2-12*x)*
exp(x)+54*x^6-6*x^2+18*x)/(x*exp(x)^2-6*exp(x)*x+9*x),x, algorithm="giac")

[Out]

(x^6*e^x - 3*x^6 + 2*x*e^x*log(x) - e^x*log(x)^2 + x^2 - 6*x*log(x) + 3*log(x)^2)/(e^x - 3)

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Mupad [B]
time = 9.31, size = 25, normalized size = 0.83 \begin {gather*} 2\,x\,\ln \left (x\right )-{\ln \left (x\right )}^2+\frac {x^2}{{\mathrm {e}}^x-3}+x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + exp(2*x)*(2*x + 6*x^6) - exp(x)*(12*x - 2*x^2 + x^3 + 36*x^6) + log(x)*(18*x - exp(x)*(12*x - 12)
+ exp(2*x)*(2*x - 2) - 18) - 6*x^2 + 54*x^6)/(9*x + x*exp(2*x) - 6*x*exp(x)),x)

[Out]

2*x*log(x) - log(x)^2 + x^2/(exp(x) - 3) + x^6

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