\(\int \frac {-189-198 x+e^4 (22 x+23 x^2)+(-9-9 x+e^4 (x+x^2)) \log (\frac {9 x+9 x^2+e^4 (-x^2-x^3)}{e^4})}{-180-180 x+e^4 (20 x+20 x^2)} \, dx\) [3861]

Optimal result

Integrand size = 83, antiderivative size = 24 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=3+x+\frac {1}{20} x \log \left (\left (\frac {9}{e^4}-x\right ) \left (x+x^2\right )\right ) \]

[Out]

3+1/20*x*ln((x^2+x)*(9/exp(2)^2-x))+x

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.072, Rules used = {6873, 6820, 12, 6874, 907, 2603} \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {4 x}{5}+\frac {1}{20} x \log \left (x (x+1) \left (9-e^4 x\right )\right ) \]

[In]

Int[(-189 - 198*x + E^4*(22*x + 23*x^2) + (-9 - 9*x + E^4*(x + x^2))*Log[(9*x + 9*x^2 + E^4*(-x^2 - x^3))/E^4]
)/(-180 - 180*x + E^4*(20*x + 20*x^2)),x]

[Out]

(4*x)/5 + (x*Log[x*(1 + x)*(9 - E^4*x)])/20

Rule 12

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

Rule 907

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 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6820

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

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {189+198 x-e^4 \left (22 x+23 x^2\right )-\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{180+20 \left (9-e^4\right ) x-20 e^4 x^2} \, dx \\ & = \int \frac {153-18 \left (-9+e^4\right ) x-19 e^4 x^2-(1+x) \left (-9+e^4 x\right ) \log \left (-x (1+x) \left (-9+e^4 x\right )\right )}{20 (1+x) \left (9-e^4 x\right )} \, dx \\ & = \frac {1}{20} \int \frac {153-18 \left (-9+e^4\right ) x-19 e^4 x^2-(1+x) \left (-9+e^4 x\right ) \log \left (-x (1+x) \left (-9+e^4 x\right )\right )}{(1+x) \left (9-e^4 x\right )} \, dx \\ & = \frac {1}{20} \int \left (\frac {153+18 \left (9-e^4\right ) x-19 e^4 x^2}{(1+x) \left (9-e^4 x\right )}+\log \left (-x (1+x) \left (-9+e^4 x\right )\right )\right ) \, dx \\ & = \frac {1}{20} \int \frac {153+18 \left (9-e^4\right ) x-19 e^4 x^2}{(1+x) \left (9-e^4 x\right )} \, dx+\frac {1}{20} \int \log \left (-x (1+x) \left (-9+e^4 x\right )\right ) \, dx \\ & = \frac {1}{20} x \log \left (x (1+x) \left (9-e^4 x\right )\right )-\frac {1}{20} \int \frac {9+2 \left (9-e^4\right ) x-3 e^4 x^2}{(1+x) \left (9-e^4 x\right )} \, dx+\frac {1}{20} \int \left (19+\frac {1}{-1-x}+\frac {9}{-9+e^4 x}\right ) \, dx \\ & = \frac {19 x}{20}-\frac {1}{20} \log (1+x)+\frac {9 \log \left (9-e^4 x\right )}{20 e^4}+\frac {1}{20} x \log \left (x (1+x) \left (9-e^4 x\right )\right )-\frac {1}{20} \int \left (3+\frac {1}{-1-x}+\frac {9}{-9+e^4 x}\right ) \, dx \\ & = \frac {4 x}{5}+\frac {1}{20} x \log \left (x (1+x) \left (9-e^4 x\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {1}{20} \left (16 x+x \log \left (-x (1+x) \left (-9+e^4 x\right )\right )\right ) \]

[In]

Integrate[(-189 - 198*x + E^4*(22*x + 23*x^2) + (-9 - 9*x + E^4*(x + x^2))*Log[(9*x + 9*x^2 + E^4*(-x^2 - x^3)
)/E^4])/(-180 - 180*x + E^4*(20*x + 20*x^2)),x]

[Out]

(16*x + x*Log[-(x*(1 + x)*(-9 + E^4*x))])/20

Maple [A] (verified)

Time = 4.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33

method	result	size
default	\(\frac {4 x}{5}+\frac {x \ln \left (x \left (-x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{4}+9 x +9\right )\right )}{20}\)	\(32\)
parts	\(\frac {4 x}{5}+\frac {x \ln \left (x \left (-x^{2} {\mathrm e}^{4}-x \,{\mathrm e}^{4}+9 x +9\right )\right )}{20}\)	\(32\)
risch	\(x +\frac {x \ln \left (\left (\left (-x^{3}-x^{2}\right ) {\mathrm e}^{4}+9 x^{2}+9 x \right ) {\mathrm e}^{-4}\right )}{20}\)	\(33\)
norman	\(x +\frac {x \ln \left (\left (\left (-x^{3}-x^{2}\right ) {\mathrm e}^{4}+9 x^{2}+9 x \right ) {\mathrm e}^{-4}\right )}{20}\)	\(37\)
parallelrisch	\(\frac {\left (\ln \left (\left (\left (-x^{3}-x^{2}\right ) {\mathrm e}^{4}+9 x^{2}+9 x \right ) {\mathrm e}^{-4}\right ) x \,{\mathrm e}^{8}+20 x \,{\mathrm e}^{8}-40 \,{\mathrm e}^{8}+360 \,{\mathrm e}^{4}\right ) {\mathrm e}^{-8}}{20}\)	\(64\)

[In]

int((((x^2+x)*exp(2)^2-9*x-9)*ln(((-x^3-x^2)*exp(2)^2+9*x^2+9*x)/exp(2)^2)+(23*x^2+22*x)*exp(2)^2-198*x-189)/(
(20*x^2+20*x)*exp(2)^2-180*x-180),x,method=_RETURNVERBOSE)

[Out]

4/5*x+1/20*x*ln(x*(-x^2*exp(2)^2-x*exp(2)^2+9*x+9))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {1}{20} \, x \log \left ({\left (9 \, x^{2} - {\left (x^{3} + x^{2}\right )} e^{4} + 9 \, x\right )} e^{\left (-4\right )}\right ) + x \]

[In]

integrate((((x^2+x)*exp(2)^2-9*x-9)*log(((-x^3-x^2)*exp(2)^2+9*x^2+9*x)/exp(2)^2)+(23*x^2+22*x)*exp(2)^2-198*x
-189)/((20*x^2+20*x)*exp(2)^2-180*x-180),x, algorithm="fricas")

[Out]

1/20*x*log((9*x^2 - (x^3 + x^2)*e^4 + 9*x)*e^(-4)) + x

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {x \log {\left (\frac {9 x^{2} + 9 x + \left (- x^{3} - x^{2}\right ) e^{4}}{e^{4}} \right )}}{20} + x \]

[In]

integrate((((x**2+x)*exp(2)**2-9*x-9)*ln(((-x**3-x**2)*exp(2)**2+9*x**2+9*x)/exp(2)**2)+(23*x**2+22*x)*exp(2)*
*2-198*x-189)/((20*x**2+20*x)*exp(2)**2-180*x-180),x)

[Out]

x*log((9*x**2 + 9*x + (-x**3 - x**2)*exp(4))*exp(-4))/20 + x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 162, normalized size of antiderivative = 6.75 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {23}{20} \, {\left (x e^{\left (-4\right )} + \frac {81 \, \log \left (x e^{4} - 9\right )}{e^{12} + 9 \, e^{8}} - \frac {\log \left (x + 1\right )}{e^{4} + 9}\right )} e^{4} + \frac {11}{10} \, {\left (\frac {9 \, \log \left (x e^{4} - 9\right )}{e^{8} + 9 \, e^{4}} + \frac {\log \left (x + 1\right )}{e^{4} + 9}\right )} e^{4} + \frac {1}{20} \, {\left (x e^{4} \log \left (x\right ) - 7 \, x e^{4} + {\left (x e^{4} - 9\right )} \log \left (-x e^{4} + 9\right ) + {\left (x e^{4} + e^{4}\right )} \log \left (x + 1\right )\right )} e^{\left (-4\right )} - \frac {891 \, \log \left (x e^{4} - 9\right )}{10 \, {\left (e^{8} + 9 \, e^{4}\right )}} - \frac {189 \, \log \left (x e^{4} - 9\right )}{20 \, {\left (e^{4} + 9\right )}} - \frac {9 \, \log \left (x + 1\right )}{20 \, {\left (e^{4} + 9\right )}} \]

[In]

integrate((((x^2+x)*exp(2)^2-9*x-9)*log(((-x^3-x^2)*exp(2)^2+9*x^2+9*x)/exp(2)^2)+(23*x^2+22*x)*exp(2)^2-198*x
-189)/((20*x^2+20*x)*exp(2)^2-180*x-180),x, algorithm="maxima")

[Out]

23/20*(x*e^(-4) + 81*log(x*e^4 - 9)/(e^12 + 9*e^8) - log(x + 1)/(e^4 + 9))*e^4 + 11/10*(9*log(x*e^4 - 9)/(e^8
+ 9*e^4) + log(x + 1)/(e^4 + 9))*e^4 + 1/20*(x*e^4*log(x) - 7*x*e^4 + (x*e^4 - 9)*log(-x*e^4 + 9) + (x*e^4 + e
^4)*log(x + 1))*e^(-4) - 891/10*log(x*e^4 - 9)/(e^8 + 9*e^4) - 189/20*log(x*e^4 - 9)/(e^4 + 9) - 9/20*log(x +
1)/(e^4 + 9)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {1}{20} \, x \log \left (-x^{3} e^{4} - x^{2} e^{4} + 9 \, x^{2} + 9 \, x\right ) + \frac {4}{5} \, x \]

[In]

integrate((((x^2+x)*exp(2)^2-9*x-9)*log(((-x^3-x^2)*exp(2)^2+9*x^2+9*x)/exp(2)^2)+(23*x^2+22*x)*exp(2)^2-198*x
-189)/((20*x^2+20*x)*exp(2)^2-180*x-180),x, algorithm="giac")

[Out]

1/20*x*log(-x^3*e^4 - x^2*e^4 + 9*x^2 + 9*x) + 4/5*x

Mupad [B] (verification not implemented)

Time = 10.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-189-198 x+e^4 \left (22 x+23 x^2\right )+\left (-9-9 x+e^4 \left (x+x^2\right )\right ) \log \left (\frac {9 x+9 x^2+e^4 \left (-x^2-x^3\right )}{e^4}\right )}{-180-180 x+e^4 \left (20 x+20 x^2\right )} \, dx=\frac {x\,\left (\ln \left ({\mathrm {e}}^{-4}\,\left (9\,x-{\mathrm {e}}^4\,\left (x^3+x^2\right )+9\,x^2\right )\right )+20\right )}{20} \]

[In]

int((198*x - exp(4)*(22*x + 23*x^2) + log(exp(-4)*(9*x - exp(4)*(x^2 + x^3) + 9*x^2))*(9*x - exp(4)*(x + x^2)
+ 9) + 189)/(180*x - exp(4)*(20*x + 20*x^2) + 180),x)

[Out]

(x*(log(exp(-4)*(9*x - exp(4)*(x^2 + x^3) + 9*x^2)) + 20))/20