\(\int \frac {2+4 x+4 x^2+2 x^3+e^3 (-2-2 x-2 x^2)+e^4 (2+2 x+2 x^2)+(-2-2 x-2 x^2) \log (\frac {3+3 x}{x})}{x+x^2} \, dx\) [9078]

Optimal result

Integrand size = 72, antiderivative size = 23 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (-1+e^3-e^4-x+\log \left (3+\frac {3}{x}\right )\right )^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1607, 6820, 12, 6818} \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (x-\log \left (\frac {3}{x}+3\right )+e^4-e^3+1\right )^2 \]

[In]

Int[(2 + 4*x + 4*x^2 + 2*x^3 + E^3*(-2 - 2*x - 2*x^2) + E^4*(2 + 2*x + 2*x^2) + (-2 - 2*x - 2*x^2)*Log[(3 + 3*
x)/x])/(x + x^2),x]

[Out]

(1 - E^3 + E^4 + x - Log[3 + 3/x])^2

Rule 12

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

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6818

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*(y^(m + 1)/(m + 1)), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x (1+x)} \, dx \\ & = \int \frac {2 \left (1+x+x^2\right ) \left (1+(-1+e) e^3+x-\log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx \\ & = 2 \int \frac {\left (1+x+x^2\right ) \left (1+(-1+e) e^3+x-\log \left (3+\frac {3}{x}\right )\right )}{x (1+x)} \, dx \\ & = \left (1-e^3+e^4+x-\log \left (3+\frac {3}{x}\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (1-e^3+e^4+x-\log \left (3+\frac {3}{x}\right )\right )^2 \]

[In]

Integrate[(2 + 4*x + 4*x^2 + 2*x^3 + E^3*(-2 - 2*x - 2*x^2) + E^4*(2 + 2*x + 2*x^2) + (-2 - 2*x - 2*x^2)*Log[(
3 + 3*x)/x])/(x + x^2),x]

[Out]

(1 - E^3 + E^4 + x - Log[3 + 3/x])^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(62\) vs. \(2(21)=42\).

Time = 0.66 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.74

method	result	size
derivativedivides	\(-2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}\right ) \ln \left (\frac {3}{x}+3\right )+\frac {2 \left (3 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{3}+3\right ) x}{3}+x^{2}+\ln \left (\frac {3}{x}+3\right )^{2}-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\)	\(63\)
default	\(-2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}\right ) \ln \left (\frac {3}{x}+3\right )+\frac {2 \left (3 \,{\mathrm e}^{4}-3 \,{\mathrm e}^{3}+3\right ) x}{3}+x^{2}+\ln \left (\frac {3}{x}+3\right )^{2}-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\)	\(63\)
norman	\(x^{2}+\ln \left (\frac {3 x +3}{x}\right )^{2}+\left (2+2 \,{\mathrm e}^{4}-2 \,{\mathrm e}^{3}\right ) x +\left (-2 \,{\mathrm e}^{4}+2 \,{\mathrm e}^{3}-2\right ) \ln \left (\frac {3 x +3}{x}\right )-2 x \ln \left (\frac {3 x +3}{x}\right )\)	\(63\)
parts	\(x^{2}+2 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{3}+2 x +2 \left ({\mathrm e}^{4}-{\mathrm e}^{3}+1\right ) \ln \left (x \right )+2 \left (-{\mathrm e}^{4}+{\mathrm e}^{3}\right ) \ln \left (1+x \right )+\ln \left (\frac {3}{x}+3\right )^{2}+2 \ln \left (\frac {3}{x}\right )-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\)	\(79\)
parallelrisch	\(2 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{4} \ln \left (\frac {3 x +3}{x}\right )-2 x \,{\mathrm e}^{3}+2 \ln \left (\frac {3 x +3}{x}\right ) {\mathrm e}^{3}-5+x^{2}-2 x \ln \left (\frac {3 x +3}{x}\right )+\ln \left (\frac {3 x +3}{x}\right )^{2}-4 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{3}+2 x -2 \ln \left (\frac {3 x +3}{x}\right )\)	\(87\)
risch	\(2 x \,{\mathrm e}^{4}-2 x \,{\mathrm e}^{3}+x^{2}+2 x +2 \,{\mathrm e}^{4} \ln \left (x \right )-2 \ln \left (x \right ) {\mathrm e}^{3}+2 \ln \left (x \right )-2 \,{\mathrm e}^{3} \ln \left (1+x \right ) {\mathrm e}+2 \,{\mathrm e}^{3} \ln \left (1+x \right )+\ln \left (\frac {3}{x}+3\right )^{2}+2 \ln \left (\frac {3}{x}\right )-\frac {2 \ln \left (\frac {3}{x}+3\right ) \left (\frac {3}{x}+3\right ) x}{3}\)	\(88\)

[In]

int(((-2*x^2-2*x-2)*ln((3*x+3)/x)+(2*x^2+2*x+2)*exp(4)+(-2*x^2-2*x-2)*exp(3)+2*x^3+4*x^2+4*x+2)/(x^2+x),x,meth
od=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} + 2 \, x e^{4} - 2 \, x e^{3} - 2 \, {\left (x + e^{4} - e^{3} + 1\right )} \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right ) + \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right )^{2} + 2 \, x \]

[In]

integrate(((-2*x^2-2*x-2)*log((3*x+3)/x)+(2*x^2+2*x+2)*exp(4)+(-2*x^2-2*x-2)*exp(3)+2*x^3+4*x^2+4*x+2)/(x^2+x)
,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} - 2 x \log {\left (\frac {3 x + 3}{x} \right )} + x \left (- 2 e^{3} + 2 + 2 e^{4}\right ) + \left (- 2 e^{3} + 2 + 2 e^{4}\right ) \log {\left (x \right )} + \log {\left (\frac {3 x + 3}{x} \right )}^{2} + \left (- 2 e^{4} - 2 + 2 e^{3}\right ) \log {\left (x + 1 \right )} \]

[In]

integrate(((-2*x**2-2*x-2)*ln((3*x+3)/x)+(2*x**2+2*x+2)*exp(4)+(-2*x**2-2*x-2)*exp(3)+2*x**3+4*x**2+4*x+2)/(x*
*2+x),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.74 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=x^{2} + 2 \, x {\left (e^{4} - e^{3} - \log \left (3\right ) + 1\right )} - 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} e^{4} + 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} e^{3} - 2 \, {\left (x + e^{4} - e^{3} + 2\right )} \log \left (x + 1\right ) + 2 \, e^{4} \log \left (x + 1\right ) - 2 \, e^{3} \log \left (x + 1\right ) - \log \left (x + 1\right )^{2} + 2 \, x \log \left (x\right ) + 2 \, \log \left (x + 1\right ) \log \left (x\right ) - \log \left (x\right )^{2} + 2 \, {\left (\log \left (x + 1\right ) - \log \left (x\right )\right )} \log \left (\frac {3}{x} + 3\right ) + 2 \, \log \left (x + 1\right ) + 2 \, \log \left (x\right ) \]

[In]

integrate(((-2*x^2-2*x-2)*log((3*x+3)/x)+(2*x^2+2*x+2)*exp(4)+(-2*x^2-2*x-2)*exp(3)+2*x^3+4*x^2+4*x+2)/(x^2+x)
,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\int { \frac {2 \, {\left (x^{3} + 2 \, x^{2} + {\left (x^{2} + x + 1\right )} e^{4} - {\left (x^{2} + x + 1\right )} e^{3} - {\left (x^{2} + x + 1\right )} \log \left (\frac {3 \, {\left (x + 1\right )}}{x}\right ) + 2 \, x + 1\right )}}{x^{2} + x} \,d x } \]

[In]

integrate(((-2*x^2-2*x-2)*log((3*x+3)/x)+(2*x^2+2*x+2)*exp(4)+(-2*x^2-2*x-2)*exp(3)+2*x^3+4*x^2+4*x+2)/(x^2+x)
,x, algorithm="giac")

[Out]

integrate(2*(x^3 + 2*x^2 + (x^2 + x + 1)*e^4 - (x^2 + x + 1)*e^3 - (x^2 + x + 1)*log(3*(x + 1)/x) + 2*x + 1)/(
x^2 + x), x)

Mupad [B] (verification not implemented)

Time = 13.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {2+4 x+4 x^2+2 x^3+e^3 \left (-2-2 x-2 x^2\right )+e^4 \left (2+2 x+2 x^2\right )+\left (-2-2 x-2 x^2\right ) \log \left (\frac {3+3 x}{x}\right )}{x+x^2} \, dx=\left (x-\ln \left (\frac {3\,\left (x+1\right )}{x}\right )\right )\,\left (x-2\,{\mathrm {e}}^3+2\,{\mathrm {e}}^4-\ln \left (\frac {3\,\left (x+1\right )}{x}\right )+2\right ) \]

[In]

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

[Out]

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