\(\int \frac {-8 e^{e^6}-32 x^2+(64 e x-64 x^2) \log (-e+x)}{e-x} \, dx\) [6766]

Optimal result

Integrand size = 38, antiderivative size = 19 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=8 \left (e^{e^6}+4 x^2\right ) \log (-e+x) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(42\) vs. \(2(19)=38\).

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.21, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6874, 711, 2442, 45} \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=32 x^2 \log (x-e)+8 \left (4 e^2+e^{e^6}\right ) \log (e-x)-32 e^2 \log (e-x) \]

[In]

Int[(-8*E^E^6 - 32*x^2 + (64*E*x - 64*x^2)*Log[-E + x])/(E - x),x]

[Out]

-32*E^2*Log[E - x] + 8*(4*E^2 + E^E^6)*Log[E - x] + 32*x^2*Log[-E + x]

Rule 45

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 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2442

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {8 \left (e^{e^6}+4 x^2\right )}{e-x}+64 x \log (-e+x)\right ) \, dx \\ & = -\left (8 \int \frac {e^{e^6}+4 x^2}{e-x} \, dx\right )+64 \int x \log (-e+x) \, dx \\ & = 32 x^2 \log (-e+x)-8 \int \left (-4 e+\frac {4 e^2+e^{e^6}}{e-x}-4 x\right ) \, dx-32 \int \frac {x^2}{-e+x} \, dx \\ & = 32 e x+16 x^2+8 \left (4 e^2+e^{e^6}\right ) \log (e-x)+32 x^2 \log (-e+x)-32 \int \left (e-\frac {e^2}{e-x}+x\right ) \, dx \\ & = -32 e^2 \log (e-x)+8 \left (4 e^2+e^{e^6}\right ) \log (e-x)+32 x^2 \log (-e+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=-8 \left (-e^{e^6} \log (e-x)-4 x^2 \log (-e+x)\right ) \]

[In]

Integrate[(-8*E^E^6 - 32*x^2 + (64*E*x - 64*x^2)*Log[-E + x])/(E - x),x]

[Out]

-8*(-(E^E^6*Log[E - x]) - 4*x^2*Log[-E + x])

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.37

method	result	size
risch	\(32 \ln \left (x -{\mathrm e}\right ) x^{2}+8 \,{\mathrm e}^{{\mathrm e}^{6}} \ln \left (x -{\mathrm e}\right )\)	\(26\)
norman	\(32 \ln \left (x -{\mathrm e}\right ) x^{2}+8 \,{\mathrm e}^{{\mathrm e}^{6}} \ln \left (x -{\mathrm e}\right )\)	\(28\)
parallelrisch	\(32 \ln \left (x -{\mathrm e}\right ) x^{2}+8 \,{\mathrm e}^{{\mathrm e}^{6}} \ln \left (x -{\mathrm e}\right )\)	\(28\)
derivativedivides	\(64 \,{\mathrm e} \left (\left (x -{\mathrm e}\right ) \ln \left (x -{\mathrm e}\right )-x +{\mathrm e}\right )+32 \ln \left (x -{\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}+32 \,{\mathrm e}^{2} \ln \left (x -{\mathrm e}\right )+64 \,{\mathrm e} \left (x -{\mathrm e}\right )+8 \,{\mathrm e}^{{\mathrm e}^{6}} \ln \left (x -{\mathrm e}\right )\)	\(80\)
default	\(64 \,{\mathrm e} \left (\left (x -{\mathrm e}\right ) \ln \left (x -{\mathrm e}\right )-x +{\mathrm e}\right )+32 \ln \left (x -{\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}+32 \,{\mathrm e}^{2} \ln \left (x -{\mathrm e}\right )+64 \,{\mathrm e} \left (x -{\mathrm e}\right )+8 \,{\mathrm e}^{{\mathrm e}^{6}} \ln \left (x -{\mathrm e}\right )\)	\(80\)
parts	\(64 \,{\mathrm e} \left (\left (x -{\mathrm e}\right ) \ln \left (x -{\mathrm e}\right )-x +{\mathrm e}\right )+32 \ln \left (x -{\mathrm e}\right ) \left (x -{\mathrm e}\right )^{2}-16 \left (x -{\mathrm e}\right )^{2}+16 x^{2}+32 x \,{\mathrm e}-8 \left (-4 \,{\mathrm e}^{2}-{\mathrm e}^{{\mathrm e}^{6}}\right ) \ln \left (x -{\mathrm e}\right )\)	\(82\)

[In]

int(((64*x*exp(1)-64*x^2)*ln(x-exp(1))-8*exp(exp(3)^2)-32*x^2)/(exp(1)-x),x,method=_RETURNVERBOSE)

[Out]

32*ln(x-exp(1))*x^2+8*exp(exp(6))*ln(x-exp(1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=8 \, {\left (4 \, x^{2} + e^{\left (e^{6}\right )}\right )} \log \left (x - e\right ) \]

[In]

integrate(((64*x*exp(1)-64*x^2)*log(x-exp(1))-8*exp(exp(3)^2)-32*x^2)/(exp(1)-x),x, algorithm="fricas")

[Out]

8*(4*x^2 + e^(e^6))*log(x - e)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.26 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=32 x^{2} \log {\left (x - e \right )} + 8 e^{e^{6}} \log {\left (x - e \right )} \]

[In]

integrate(((64*x*exp(1)-64*x**2)*ln(x-exp(1))-8*exp(exp(3)**2)-32*x**2)/(exp(1)-x),x)

[Out]

32*x**2*log(x - E) + 8*exp(exp(6))*log(x - E)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (18) = 36\).

Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 6.58 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=-64 \, {\left (e \log \left (x - e\right ) + x\right )} e \log \left (x - e\right ) - 32 \, e^{2} \log \left (x - e\right )^{2} + 32 \, {\left (e \log \left (x - e\right )^{2} + 2 \, e \log \left (x - e\right ) + 2 \, x\right )} e - 64 \, x e + 32 \, {\left (x^{2} + 2 \, x e + 2 \, e^{2} \log \left (x - e\right )\right )} \log \left (x - e\right ) - 64 \, e^{2} \log \left (x - e\right ) + 8 \, e^{\left (e^{6}\right )} \log \left (x - e\right ) \]

[In]

integrate(((64*x*exp(1)-64*x^2)*log(x-exp(1))-8*exp(exp(3)^2)-32*x^2)/(exp(1)-x),x, algorithm="maxima")

[Out]

-64*(e*log(x - e) + x)*e*log(x - e) - 32*e^2*log(x - e)^2 + 32*(e*log(x - e)^2 + 2*e*log(x - e) + 2*x)*e - 64*
x*e + 32*(x^2 + 2*x*e + 2*e^2*log(x - e))*log(x - e) - 64*e^2*log(x - e) + 8*e^(e^6)*log(x - e)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=32 \, x^{2} \log \left (x - e\right ) + 8 \, e^{\left (e^{6}\right )} \log \left (x - e\right ) \]

[In]

integrate(((64*x*exp(1)-64*x^2)*log(x-exp(1))-8*exp(exp(3)^2)-32*x^2)/(exp(1)-x),x, algorithm="giac")

[Out]

32*x^2*log(x - e) + 8*e^(e^6)*log(x - e)

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \frac {-8 e^{e^6}-32 x^2+\left (64 e x-64 x^2\right ) \log (-e+x)}{e-x} \, dx=8\,\ln \left (x-\mathrm {e}\right )\,\left (4\,x^2+{\mathrm {e}}^{{\mathrm {e}}^6}\right ) \]

[In]

int((8*exp(exp(6)) - log(x - exp(1))*(64*x*exp(1) - 64*x^2) + 32*x^2)/(x - exp(1)),x)

[Out]

8*log(x - exp(1))*(exp(exp(6)) + 4*x^2)