3.87.30 \(\int \frac {6 x^2-4 x \log (2)+e^x (-4 x-2 x^2+(2+2 x) \log (2))+e^{7 x} (-4 x^2-2 x \log (2)+e^x (5 x-x^2+(1+x) \log (2)))+(-4 x+e^x (2+2 x)+e^{7 x} (-2 x+e^x (1+x))) \log (2+e^{7 x})}{2+e^{7 x}} \, dx\)

Optimal. Leaf size=23 \[ \left (e^x-x\right ) x \left (-x+\log (2)+\log \left (2+e^{7 x}\right )\right ) \]

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Rubi [B]  time = 3.50, antiderivative size = 1004, normalized size of antiderivative = 43.65, number of steps used = 100, number of rules used = 24, integrand size = 117, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.205, Rules used = {6742, 2254, 2184, 2190, 2279, 2391, 2531, 2282, 6589, 14, 43, 2532, 2176, 2194, 2554, 12, 2248, 321, 201, 634, 618, 204, 628, 31} \begin {gather*} x^3-e^x x^2-\log \left (2+e^{7 x}\right ) x^2-\log (2) x^2-\frac {(-1)^{5/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {x^2}{2^{6/7}}+\left (-\frac {1}{2}\right )^{6/7} x^2+2 e^x x-(-1)^{6/7} \sqrt [7]{2} \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right ) x-\sqrt [7]{2} \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right ) x+(-1)^{5/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{4/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right ) x+(-1)^{3/7} \sqrt [7]{2} \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right ) x-(-1)^{2/7} \sqrt [7]{2} \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right ) x+\sqrt [7]{-2} \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right ) x+e^x \log \left (2+e^{7 x}\right ) x+e^x (5+\log (2)) x-2 e^x+\frac {(-1)^{5/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{4/7} (1-x)^2}{2^{6/7}}+\frac {(-1)^{3/7} (1-x)^2}{2^{6/7}}-\frac {(-1)^{2/7} (1-x)^2}{2^{6/7}}+\frac {\sqrt [7]{-1} (1-x)^2}{2^{6/7}}-\frac {(1-x)^2}{2^{6/7}}-\left (-\frac {1}{2}\right )^{6/7} (1-x)^2+7 e^x (1-x)+\sqrt [7]{2} \log \left (\sqrt [7]{2}+e^x\right )-(-1)^{6/7} \sqrt [7]{2} (1-x) \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right )-\sqrt [7]{2} (1-x) \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right )+(-1)^{5/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{4/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right )+(-1)^{3/7} \sqrt [7]{2} (1-x) \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{2/7} \sqrt [7]{2} (1-x) \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right )+\sqrt [7]{-2} (1-x) \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right )-e^x \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\sqrt [7]{2} \cos \left (\frac {\pi }{7}\right ) \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \cos \left (\frac {\pi }{7}\right )\right )+\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}+2 \sqrt [7]{2} e^x \sin \left (\frac {3 \pi }{14}\right )\right ) \sin \left (\frac {3 \pi }{14}\right )+2 \sqrt [7]{2} \tan ^{-1}\left (\frac {\left (e^x-\sqrt [7]{2} \cos \left (\frac {\pi }{7}\right )\right ) \csc \left (\frac {\pi }{7}\right )}{\sqrt [7]{2}}\right ) \sin \left (\frac {\pi }{7}\right )-\sqrt [7]{2} \log \left (2^{2/7}+e^{2 x}-2 \sqrt [7]{2} e^x \sin \left (\frac {\pi }{14}\right )\right ) \sin \left (\frac {\pi }{14}\right )-e^x (5+\log (2))+2 \sqrt [7]{2} \tan ^{-1}\left (\frac {\sec \left (\frac {3 \pi }{14}\right ) \left (e^x+\sqrt [7]{2} \sin \left (\frac {3 \pi }{14}\right )\right )}{\sqrt [7]{2}}\right ) \cos \left (\frac {3 \pi }{14}\right )+2 \sqrt [7]{2} \tan ^{-1}\left (\frac {\sec \left (\frac {\pi }{14}\right ) \left (e^x-\sqrt [7]{2} \sin \left (\frac {\pi }{14}\right )\right )}{\sqrt [7]{2}}\right ) \cos \left (\frac {\pi }{14}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(6*x^2 - 4*x*Log[2] + E^x*(-4*x - 2*x^2 + (2 + 2*x)*Log[2]) + E^(7*x)*(-4*x^2 - 2*x*Log[2] + E^x*(5*x - x^
2 + (1 + x)*Log[2])) + (-4*x + E^x*(2 + 2*x) + E^(7*x)*(-2*x + E^x*(1 + x)))*Log[2 + E^(7*x)])/(2 + E^(7*x)),x
]

[Out]

-2*E^x + 7*E^x*(1 - x) - (-1/2)^(6/7)*(1 - x)^2 - (1 - x)^2/2^(6/7) + ((-1)^(1/7)*(1 - x)^2)/2^(6/7) - ((-1)^(
2/7)*(1 - x)^2)/2^(6/7) + ((-1)^(3/7)*(1 - x)^2)/2^(6/7) - ((-1)^(4/7)*(1 - x)^2)/2^(6/7) + ((-1)^(5/7)*(1 - x
)^2)/2^(6/7) + 2*E^x*x + (-1/2)^(6/7)*x^2 + x^2/2^(6/7) - ((-1)^(1/7)*x^2)/2^(6/7) + ((-1)^(2/7)*x^2)/2^(6/7)
- ((-1)^(3/7)*x^2)/2^(6/7) + ((-1)^(4/7)*x^2)/2^(6/7) - ((-1)^(5/7)*x^2)/2^(6/7) - E^x*x^2 + x^3 + 2*2^(1/7)*A
rcTan[(Sec[Pi/14]*(E^x - 2^(1/7)*Sin[Pi/14]))/2^(1/7)]*Cos[Pi/14] + 2*2^(1/7)*ArcTan[(Sec[(3*Pi)/14]*(E^x + 2^
(1/7)*Sin[(3*Pi)/14]))/2^(1/7)]*Cos[(3*Pi)/14] - x^2*Log[2] - E^x*(5 + Log[2]) + E^x*x*(5 + Log[2]) + 2^(1/7)*
Log[2^(1/7) + E^x] - (-1)^(6/7)*2^(1/7)*(1 - x)*Log[1 - (-1/2)^(1/7)*E^x] - (-1)^(6/7)*2^(1/7)*x*Log[1 - (-1/2
)^(1/7)*E^x] - 2^(1/7)*(1 - x)*Log[1 + E^x/2^(1/7)] - 2^(1/7)*x*Log[1 + E^x/2^(1/7)] + (-1)^(5/7)*2^(1/7)*(1 -
x)*Log[1 + ((-1)^(2/7)*E^x)/2^(1/7)] + (-1)^(5/7)*2^(1/7)*x*Log[1 + ((-1)^(2/7)*E^x)/2^(1/7)] - (-1)^(4/7)*2^
(1/7)*(1 - x)*Log[1 - ((-1)^(3/7)*E^x)/2^(1/7)] - (-1)^(4/7)*2^(1/7)*x*Log[1 - ((-1)^(3/7)*E^x)/2^(1/7)] + (-1
)^(3/7)*2^(1/7)*(1 - x)*Log[1 + ((-1)^(4/7)*E^x)/2^(1/7)] + (-1)^(3/7)*2^(1/7)*x*Log[1 + ((-1)^(4/7)*E^x)/2^(1
/7)] - (-1)^(2/7)*2^(1/7)*(1 - x)*Log[1 - ((-1)^(5/7)*E^x)/2^(1/7)] - (-1)^(2/7)*2^(1/7)*x*Log[1 - ((-1)^(5/7)
*E^x)/2^(1/7)] + (-2)^(1/7)*(1 - x)*Log[1 + ((-1)^(6/7)*E^x)/2^(1/7)] + (-2)^(1/7)*x*Log[1 + ((-1)^(6/7)*E^x)/
2^(1/7)] - E^x*Log[2 + E^(7*x)] + E^x*x*Log[2 + E^(7*x)] - x^2*Log[2 + E^(7*x)] + E^x*Log[2*(2 + E^(7*x))] - 2
^(1/7)*Cos[Pi/7]*Log[2^(2/7) + E^(2*x) - 2*2^(1/7)*E^x*Cos[Pi/7]] - 2^(1/7)*Log[2^(2/7) + E^(2*x) - 2*2^(1/7)*
E^x*Sin[Pi/14]]*Sin[Pi/14] + 2*2^(1/7)*ArcTan[((E^x - 2^(1/7)*Cos[Pi/7])*Csc[Pi/7])/2^(1/7)]*Sin[Pi/7] + 2^(1/
7)*Log[2^(2/7) + E^(2*x) + 2*2^(1/7)*E^x*Sin[(3*Pi)/14]]*Sin[(3*Pi)/14]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, 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])

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (r*
Int[1/(r + s*x), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[
(n - 3)/2, 0] && PosQ[a/b]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 2176

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

Rule 2184

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 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 2254

Int[((a_.) + (b_.)*(F_)^(u_))^(p_.)*((c_.) + (d_.)*(F_)^(v_))^(q_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> W
ith[{w = ExpandIntegrand[(e + f*x)^m, (a + b*F^u)^p*(c + d*F^v)^q, x]}, Int[w, x] /; SumQ[w]] /; FreeQ[{F, a,
b, c, d, e, f, m}, x] && IntegersQ[p, q] && LinearQ[{u, v}, x] && RationalQ[Simplify[u/v]]

Rule 2279

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 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]]]

Rule 2391

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 2531

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 2532

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

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6589

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 6742

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 {14 \left (e^x-x\right ) x}{2+e^{7 x}}-2 x \left (2 x+\log (2)+\log \left (2+e^{7 x}\right )\right )+e^x \left (-x^2+5 x \left (1+\frac {\log (2)}{5}\right )+x \log \left (2+e^{7 x}\right )+\log \left (2 \left (2+e^{7 x}\right )\right )\right )\right ) \, dx\\ &=-\left (2 \int x \left (2 x+\log (2)+\log \left (2+e^{7 x}\right )\right ) \, dx\right )-14 \int \frac {\left (e^x-x\right ) x}{2+e^{7 x}} \, dx+\int e^x \left (-x^2+5 x \left (1+\frac {\log (2)}{5}\right )+x \log \left (2+e^{7 x}\right )+\log \left (2 \left (2+e^{7 x}\right )\right )\right ) \, dx\\ &=-\left (2 \int \left (x (2 x+\log (2))+x \log \left (2+e^{7 x}\right )\right ) \, dx\right )-14 \int \left (\frac {e^x x}{2+e^{7 x}}-\frac {x^2}{2+e^{7 x}}\right ) \, dx+\int \left (-e^x x^2+e^x x (5+\log (2))+e^x x \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )\right ) \, dx\\ &=-(2 \int x (2 x+\log (2)) \, dx)-2 \int x \log \left (2+e^{7 x}\right ) \, dx-14 \int \frac {e^x x}{2+e^{7 x}} \, dx+14 \int \frac {x^2}{2+e^{7 x}} \, dx+(5+\log (2)) \int e^x x \, dx-\int e^x x^2 \, dx+\int e^x x \log \left (2+e^{7 x}\right ) \, dx+\int e^x \log \left (2 \left (2+e^{7 x}\right )\right ) \, dx\\ &=-e^x x^2+\frac {7 x^3}{3}+e^x x (5+\log (2))+x^2 \log \left (1+\frac {e^{7 x}}{2}\right )-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )+2 \int e^x x \, dx-2 \int \left (2 x^2+x \log (2)\right ) \, dx-2 \int x \log \left (1+\frac {e^{7 x}}{2}\right ) \, dx-7 \int \frac {e^{7 x} x^2}{2+e^{7 x}} \, dx-14 \int \left (-\frac {x}{7\ 2^{5/7} \left (\sqrt [7]{2}+e^x\right )}-\frac {(-1)^{6/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-\sqrt [7]{-1} e^x\right )}+\frac {\left (-\frac {1}{2}\right )^{5/7} x}{7 \left (\sqrt [7]{2}+(-1)^{2/7} e^x\right )}-\frac {(-1)^{4/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-(-1)^{3/7} e^x\right )}+\frac {(-1)^{3/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}+(-1)^{4/7} e^x\right )}-\frac {(-1)^{2/7} x}{7\ 2^{5/7} \left (\sqrt [7]{2}-(-1)^{5/7} e^x\right )}+\frac {\sqrt [7]{-1} x}{7\ 2^{5/7} \left (\sqrt [7]{2}+(-1)^{6/7} e^x\right )}\right ) \, dx+(-5-\log (2)) \int e^x \, dx-\int \frac {7 e^{8 x}}{2+e^{7 x}} \, dx-\int \frac {7 e^{8 x} (-1+x)}{2+e^{7 x}} \, dx\\ &=2 e^x x-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )+\frac {2}{7} x \text {Li}_2\left (-\frac {e^{7 x}}{2}\right )-\frac {2}{7} \int \text {Li}_2\left (-\frac {e^{7 x}}{2}\right ) \, dx-2 \int e^x \, dx+2 \int x \log \left (1+\frac {e^{7 x}}{2}\right ) \, dx-7 \int \frac {e^{8 x}}{2+e^{7 x}} \, dx-7 \int \frac {e^{8 x} (-1+x)}{2+e^{7 x}} \, dx+(-2)^{2/7} \int \frac {x}{\sqrt [7]{2}-(-1)^{5/7} e^x} \, dx+2^{2/7} \int \frac {x}{\sqrt [7]{2}+e^x} \, dx-\left (\sqrt [7]{-1} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{6/7} e^x} \, dx-\left ((-1)^{3/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{4/7} e^x} \, dx+\left ((-1)^{4/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}-(-1)^{3/7} e^x} \, dx-\left ((-1)^{5/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}+(-1)^{2/7} e^x} \, dx+\left ((-1)^{6/7} 2^{2/7}\right ) \int \frac {x}{\sqrt [7]{2}-\sqrt [7]{-1} e^x} \, dx\\ &=-2 e^x+2 e^x x+\left (-\frac {1}{2}\right )^{6/7} x^2+\frac {x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{5/7} x^2}{2^{6/7}}-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\frac {2}{49} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,e^{7 x}\right )+\frac {2}{7} \int \text {Li}_2\left (-\frac {e^{7 x}}{2}\right ) \, dx-7 \int \left (e^x (-1+x)-\frac {2 e^x (-1+x)}{2+e^{7 x}}\right ) \, dx-7 \operatorname {Subst}\left (\int \frac {x^7}{2+x^7} \, dx,x,e^x\right )-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-\sqrt [7]{-1} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{2/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-(-1)^{3/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{4/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}-(-1)^{5/7} e^x} \, dx-\sqrt [7]{2} \int \frac {e^x x}{\sqrt [7]{2}+(-1)^{6/7} e^x} \, dx\\ &=-9 e^x+2 e^x x+\left (-\frac {1}{2}\right )^{6/7} x^2+\frac {x^2}{2^{6/7}}-\frac {\sqrt [7]{-1} x^2}{2^{6/7}}+\frac {(-1)^{2/7} x^2}{2^{6/7}}-\frac {(-1)^{3/7} x^2}{2^{6/7}}+\frac {(-1)^{4/7} x^2}{2^{6/7}}-\frac {(-1)^{5/7} x^2}{2^{6/7}}-e^x x^2+x^3-x^2 \log (2)-e^x (5+\log (2))+e^x x (5+\log (2))-(-1)^{6/7} \sqrt [7]{2} x \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right )-\sqrt [7]{2} x \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right )+(-1)^{5/7} \sqrt [7]{2} x \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{4/7} \sqrt [7]{2} x \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right )+(-1)^{3/7} \sqrt [7]{2} x \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right )-(-1)^{2/7} \sqrt [7]{2} x \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right )+\sqrt [7]{-2} x \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right )-e^x \log \left (2+e^{7 x}\right )+e^x x \log \left (2+e^{7 x}\right )-x^2 \log \left (2+e^{7 x}\right )+e^x \log \left (2 \left (2+e^{7 x}\right )\right )-\frac {2}{49} \text {Li}_3\left (-\frac {e^{7 x}}{2}\right )+\frac {2}{49} \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {x}{2}\right )}{x} \, dx,x,e^{7 x}\right )-7 \int e^x (-1+x) \, dx+14 \int \frac {e^x (-1+x)}{2+e^{7 x}} \, dx+14 \operatorname {Subst}\left (\int \frac {1}{2+x^7} \, dx,x,e^x\right )-\sqrt [7]{-2} \int \log \left (1+\frac {(-1)^{6/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\sqrt [7]{2} \int \log \left (1+\frac {e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{2/7} \sqrt [7]{2}\right ) \int \log \left (1-\frac {(-1)^{5/7} e^x}{\sqrt [7]{2}}\right ) \, dx-\left ((-1)^{3/7} \sqrt [7]{2}\right ) \int \log \left (1+\frac {(-1)^{4/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{4/7} \sqrt [7]{2}\right ) \int \log \left (1-\frac {(-1)^{3/7} e^x}{\sqrt [7]{2}}\right ) \, dx-\left ((-1)^{5/7} \sqrt [7]{2}\right ) \int \log \left (1+\frac {(-1)^{2/7} e^x}{\sqrt [7]{2}}\right ) \, dx+\left ((-1)^{6/7} \sqrt [7]{2}\right ) \int \log \left (1-\sqrt [7]{-\frac {1}{2}} e^x\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 24, normalized size = 1.04 \begin {gather*} -\left (\left (e^x-x\right ) x \left (x-\log \left (4+2 e^{7 x}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6*x^2 - 4*x*Log[2] + E^x*(-4*x - 2*x^2 + (2 + 2*x)*Log[2]) + E^(7*x)*(-4*x^2 - 2*x*Log[2] + E^x*(5*
x - x^2 + (1 + x)*Log[2])) + (-4*x + E^x*(2 + 2*x) + E^(7*x)*(-2*x + E^x*(1 + x)))*Log[2 + E^(7*x)])/(2 + E^(7
*x)),x]

[Out]

-((E^x - x)*x*(x - Log[4 + 2*E^(7*x)]))

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fricas [A]  time = 0.59, size = 42, normalized size = 1.83 \begin {gather*} x^{3} - x^{2} \log \relax (2) - {\left (x^{2} - x \log \relax (2)\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(x)-2*x)*exp(7*x)+(2*x+2)*exp(x)-4*x)*log(exp(7*x)+2)+((log(2)*(x+1)-x^2+5*x)*exp(x)-2*x
*log(2)-4*x^2)*exp(7*x)+((2*x+2)*log(2)-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="fricas"
)

[Out]

x^3 - x^2*log(2) - (x^2 - x*log(2))*e^x - (x^2 - x*e^x)*log(e^(7*x) + 2)

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giac [B]  time = 0.19, size = 47, normalized size = 2.04 \begin {gather*} x^{3} - x^{2} e^{x} - x^{2} \log \relax (2) + x e^{x} \log \relax (2) - x^{2} \log \left (e^{\left (7 \, x\right )} + 2\right ) + x e^{x} \log \left (e^{\left (7 \, x\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(x)-2*x)*exp(7*x)+(2*x+2)*exp(x)-4*x)*log(exp(7*x)+2)+((log(2)*(x+1)-x^2+5*x)*exp(x)-2*x
*log(2)-4*x^2)*exp(7*x)+((2*x+2)*log(2)-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="giac")

[Out]

x^3 - x^2*e^x - x^2*log(2) + x*e^x*log(2) - x^2*log(e^(7*x) + 2) + x*e^x*log(e^(7*x) + 2)

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maple [A]  time = 0.14, size = 43, normalized size = 1.87

method	result	size
risch	\(\left ({\mathrm e}^{x} x -x^{2}\right ) \ln \left ({\mathrm e}^{7 x}+2\right )-x^{2} \ln \relax (2)+x \ln \relax (2) {\mathrm e}^{x}+x^{3}-{\mathrm e}^{x} x^{2}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x+1)*exp(x)-2*x)*exp(7*x)+(2*x+2)*exp(x)-4*x)*ln(exp(7*x)+2)+((ln(2)*(x+1)-x^2+5*x)*exp(x)-2*x*ln(2)-4
*x^2)*exp(7*x)+((2*x+2)*ln(2)-2*x^2-4*x)*exp(x)-4*x*ln(2)+6*x^2)/(exp(7*x)+2),x,method=_RETURNVERBOSE)

[Out]

(exp(x)*x-x^2)*ln(exp(7*x)+2)-x^2*ln(2)+x*ln(2)*exp(x)+x^3-exp(x)*x^2

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maxima [A]  time = 0.49, size = 42, normalized size = 1.83 \begin {gather*} x^{3} - x^{2} \log \relax (2) - {\left (x^{2} - x \log \relax (2)\right )} e^{x} - {\left (x^{2} - x e^{x}\right )} \log \left (e^{\left (7 \, x\right )} + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(x)-2*x)*exp(7*x)+(2*x+2)*exp(x)-4*x)*log(exp(7*x)+2)+((log(2)*(x+1)-x^2+5*x)*exp(x)-2*x
*log(2)-4*x^2)*exp(7*x)+((2*x+2)*log(2)-2*x^2-4*x)*exp(x)-4*x*log(2)+6*x^2)/(exp(7*x)+2),x, algorithm="maxima"
)

[Out]

x^3 - x^2*log(2) - (x^2 - x*log(2))*e^x - (x^2 - x*e^x)*log(e^(7*x) + 2)

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mupad [B]  time = 5.49, size = 42, normalized size = 1.83 \begin {gather*} {\mathrm {e}}^x\,\left (x\,\ln \relax (2)-x^2\right )-\frac {x^2\,\ln \relax (4)}{2}+\ln \left ({\mathrm {e}}^{7\,x}+2\right )\,\left (x\,{\mathrm {e}}^x-x^2\right )+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x*log(2) + exp(x)*(4*x - log(2)*(2*x + 2) + 2*x^2) + exp(7*x)*(2*x*log(2) - exp(x)*(5*x + log(2)*(x +
1) - x^2) + 4*x^2) - 6*x^2 + log(exp(7*x) + 2)*(4*x - exp(x)*(2*x + 2) + exp(7*x)*(2*x - exp(x)*(x + 1))))/(ex
p(7*x) + 2),x)

[Out]

exp(x)*(x*log(2) - x^2) - (x^2*log(4))/2 + log(exp(7*x) + 2)*(x*exp(x) - x^2) + x^3

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x+1)*exp(x)-2*x)*exp(7*x)+(2*x+2)*exp(x)-4*x)*ln(exp(7*x)+2)+((ln(2)*(x+1)-x**2+5*x)*exp(x)-2*x*
ln(2)-4*x**2)*exp(7*x)+((2*x+2)*ln(2)-2*x**2-4*x)*exp(x)-4*x*ln(2)+6*x**2)/(exp(7*x)+2),x)

[Out]

Timed out

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