\(\int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+(4+2 x+2 \log ^2(2)) \log (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}})}{x+\log ^2(2)} \, dx\) [2910]

Optimal result

Integrand size = 65, antiderivative size = 19 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=\left (1+x+\log \left (\frac {25 \left (x+\log ^2(2)\right )^2}{e^{10}}\right )\right )^2 \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6820, 12, 6818} \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=\left (-x-\log \left (\left (x+\log ^2(2)\right )^2\right )+9-\log (25)\right )^2 \]

[In]

Int[(4 + 6*x + 2*x^2 + (2 + 2*x)*Log[2]^2 + (4 + 2*x + 2*Log[2]^2)*Log[(25*x^2 + 50*x*Log[2]^2 + 25*Log[2]^4)/
E^10])/(x + Log[2]^2),x]

[Out]

(9 - x - Log[25] - Log[(x + Log[2]^2)^2])^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 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 \left (2+x+\log ^2(2)\right ) \left (x-9 \left (1-\frac {2 \log (5)}{9}\right )+\log \left (\left (x+\log ^2(2)\right )^2\right )\right )}{x+\log ^2(2)} \, dx \\ & = 2 \int \frac {\left (2+x+\log ^2(2)\right ) \left (x-9 \left (1-\frac {2 \log (5)}{9}\right )+\log \left (\left (x+\log ^2(2)\right )^2\right )\right )}{x+\log ^2(2)} \, dx \\ & = \left (9-x-\log (25)-\log \left (\left (x+\log ^2(2)\right )^2\right )\right )^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=\left (-9+x+\log (25)+\log \left (\left (x+\log ^2(2)\right )^2\right )\right )^2 \]

[In]

Integrate[(4 + 6*x + 2*x^2 + (2 + 2*x)*Log[2]^2 + (4 + 2*x + 2*Log[2]^2)*Log[(25*x^2 + 50*x*Log[2]^2 + 25*Log[
2]^4)/E^10])/(x + Log[2]^2),x]

[Out]

(-9 + x + Log[25] + Log[(x + Log[2]^2)^2])^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs. \(2(23)=46\).

Time = 0.52 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74

method	result	size
norman	\(x^{2}+{\ln \left (\left (25 \ln \left (2\right )^{4}+50 x \ln \left (2\right )^{2}+25 x^{2}\right ) {\mathrm e}^{-10}\right )}^{2}+2 \ln \left (\left (25 \ln \left (2\right )^{4}+50 x \ln \left (2\right )^{2}+25 x^{2}\right ) {\mathrm e}^{-10}\right )+2 x +2 x \ln \left (\left (25 \ln \left (2\right )^{4}+50 x \ln \left (2\right )^{2}+25 x^{2}\right ) {\mathrm e}^{-10}\right )\)	\(90\)
parallelrisch	\(-\ln \left (2\right )^{4}-4 \ln \left (2\right )^{2}+x^{2}+2 \ln \left (25 \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right ) {\mathrm e}^{-10}\right ) x +{\ln \left (25 \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right ) {\mathrm e}^{-10}\right )}^{2}+2 x +2 \ln \left (25 \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right ) {\mathrm e}^{-10}\right )\)	\(93\)
risch	\(8 \ln \left (5\right ) \ln \left (\ln \left (2\right )^{2}+x \right )+4 x \ln \left (5\right )-4 \ln \left (\ln \left (2\right )^{2}+x \right )^{2}+4 \ln \left (\ln \left (2\right )^{2}+x \right ) \ln \left (\left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right ) {\mathrm e}^{-10}\right )+2 \ln \left (\left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right ) {\mathrm e}^{-10}\right ) x +x^{2}+4 \ln \left (\ln \left (2\right )^{2}+x \right )+2 x\)	\(94\)
default	\(x^{2}-18 x +2 \left (-2 \ln \left (2\right )^{2}-18\right ) \ln \left (\ln \left (2\right )^{2}+x \right )+4 \ln \left (5\right ) \left (x +2 \ln \left (\ln \left (2\right )^{2}+x \right )\right )+2 x \ln \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right )+4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )^{2}+x \right )+4 \ln \left (\ln \left (2\right )^{2}+x \right ) \ln \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right )-4 \ln \left (\ln \left (2\right )^{2}+x \right )^{2}\)	\(108\)
parts	\(-18 x +x^{2}+2 \left (-2 \ln \left (2\right )^{2}+2\right ) \ln \left (\ln \left (2\right )^{2}+x \right )+2 x \ln \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right )+4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )^{2}+x \right )+4 \ln \left (\ln \left (2\right )^{2}+x \right ) \ln \left (\ln \left (2\right )^{4}+2 x \ln \left (2\right )^{2}+x^{2}\right )-4 \ln \left (\ln \left (2\right )^{2}+x \right )^{2}-40 \ln \left (\ln \left (2\right )^{2}+x \right )+4 \ln \left (5\right ) \left (x +2 \ln \left (\ln \left (2\right )^{2}+x \right )\right )\)	\(117\)

[In]

int(((2*ln(2)^2+2*x+4)*ln((25*ln(2)^4+50*x*ln(2)^2+25*x^2)/exp(5)^2)+(2+2*x)*ln(2)^2+2*x^2+6*x+4)/(ln(2)^2+x),
x,method=_RETURNVERBOSE)

[Out]

x^2+ln((25*ln(2)^4+50*x*ln(2)^2+25*x^2)/exp(5)^2)^2+2*ln((25*ln(2)^4+50*x*ln(2)^2+25*x^2)/exp(5)^2)+2*x+2*x*ln
((25*ln(2)^4+50*x*ln(2)^2+25*x^2)/exp(5)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (18) = 36\).

Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.84 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=x^{2} + 2 \, {\left (x + 1\right )} \log \left (25 \, {\left (\log \left (2\right )^{4} + 2 \, x \log \left (2\right )^{2} + x^{2}\right )} e^{\left (-10\right )}\right ) + \log \left (25 \, {\left (\log \left (2\right )^{4} + 2 \, x \log \left (2\right )^{2} + x^{2}\right )} e^{\left (-10\right )}\right )^{2} + 2 \, x \]

[In]

integrate(((2*log(2)^2+2*x+4)*log((25*log(2)^4+50*x*log(2)^2+25*x^2)/exp(5)^2)+(2+2*x)*log(2)^2+2*x^2+6*x+4)/(
log(2)^2+x),x, algorithm="fricas")

[Out]

x^2 + 2*(x + 1)*log(25*(log(2)^4 + 2*x*log(2)^2 + x^2)*e^(-10)) + log(25*(log(2)^4 + 2*x*log(2)^2 + x^2)*e^(-1
0))^2 + 2*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).

Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=x^{2} + 2 x \log {\left (\frac {25 x^{2} + 50 x \log {\left (2 \right )}^{2} + 25 \log {\left (2 \right )}^{4}}{e^{10}} \right )} + 2 x + \log {\left (\frac {25 x^{2} + 50 x \log {\left (2 \right )}^{2} + 25 \log {\left (2 \right )}^{4}}{e^{10}} \right )}^{2} + 4 \log {\left (x + \log {\left (2 \right )}^{2} \right )} \]

[In]

integrate(((2*ln(2)**2+2*x+4)*ln((25*ln(2)**4+50*x*ln(2)**2+25*x**2)/exp(5)**2)+(2+2*x)*ln(2)**2+2*x**2+6*x+4)
/(ln(2)**2+x),x)

[Out]

x**2 + 2*x*log((25*x**2 + 50*x*log(2)**2 + 25*log(2)**4)*exp(-10)) + 2*x + log((25*x**2 + 50*x*log(2)**2 + 25*
log(2)**4)*exp(-10))**2 + 4*log(x + log(2)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (18) = 36\).

Time = 0.30 (sec) , antiderivative size = 244, normalized size of antiderivative = 12.84 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=2 \, \log \left (2\right )^{4} \log \left (\log \left (2\right )^{2} + x\right ) + 2 \, \log \left (2\right )^{2} \log \left (25 \, e^{\left (-10\right )} \log \left (2\right )^{4} + 50 \, x e^{\left (-10\right )} \log \left (2\right )^{2} + 25 \, x^{2} e^{\left (-10\right )}\right ) \log \left (\log \left (2\right )^{2} + x\right ) + 2 \, \log \left (2\right )^{2} \log \left (\log \left (2\right )^{2} + x\right )^{2} - 2 \, {\left (\log \left (2\right )^{2} \log \left (\log \left (2\right )^{2} + x\right ) - x\right )} \log \left (2\right )^{2} + 2 \, {\left (2 \, {\left (\log \left (5\right ) - 5\right )} \log \left (\log \left (2\right )^{2} + x\right ) - \log \left (25 \, e^{\left (-10\right )} \log \left (2\right )^{4} + 50 \, x e^{\left (-10\right )} \log \left (2\right )^{2} + 25 \, x^{2} e^{\left (-10\right )}\right ) \log \left (\log \left (2\right )^{2} + x\right ) + \log \left (\log \left (2\right )^{2} + x\right )^{2}\right )} \log \left (2\right )^{2} - 2 \, x \log \left (2\right )^{2} + x^{2} - 2 \, {\left (\log \left (2\right )^{2} \log \left (\log \left (2\right )^{2} + x\right ) - x\right )} \log \left (25 \, e^{\left (-10\right )} \log \left (2\right )^{4} + 50 \, x e^{\left (-10\right )} \log \left (2\right )^{2} + 25 \, x^{2} e^{\left (-10\right )}\right ) + 8 \, {\left (\log \left (5\right ) - 5\right )} \log \left (\log \left (2\right )^{2} + x\right ) + 4 \, \log \left (\log \left (2\right )^{2} + x\right )^{2} + 2 \, x + 4 \, \log \left (\log \left (2\right )^{2} + x\right ) \]

[In]

integrate(((2*log(2)^2+2*x+4)*log((25*log(2)^4+50*x*log(2)^2+25*x^2)/exp(5)^2)+(2+2*x)*log(2)^2+2*x^2+6*x+4)/(
log(2)^2+x),x, algorithm="maxima")

[Out]

2*log(2)^4*log(log(2)^2 + x) + 2*log(2)^2*log(25*e^(-10)*log(2)^4 + 50*x*e^(-10)*log(2)^2 + 25*x^2*e^(-10))*lo
g(log(2)^2 + x) + 2*log(2)^2*log(log(2)^2 + x)^2 - 2*(log(2)^2*log(log(2)^2 + x) - x)*log(2)^2 + 2*(2*(log(5)
- 5)*log(log(2)^2 + x) - log(25*e^(-10)*log(2)^4 + 50*x*e^(-10)*log(2)^2 + 25*x^2*e^(-10))*log(log(2)^2 + x) +
log(log(2)^2 + x)^2)*log(2)^2 - 2*x*log(2)^2 + x^2 - 2*(log(2)^2*log(log(2)^2 + x) - x)*log(25*e^(-10)*log(2)
^4 + 50*x*e^(-10)*log(2)^2 + 25*x^2*e^(-10)) + 8*(log(5) - 5)*log(log(2)^2 + x) + 4*log(log(2)^2 + x)^2 + 2*x
+ 4*log(log(2)^2 + x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (18) = 36\).

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.21 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=x^{2} + 2 \, x \log \left (25 \, \log \left (2\right )^{4} + 50 \, x \log \left (2\right )^{2} + 25 \, x^{2}\right ) + \log \left (25 \, \log \left (2\right )^{4} + 50 \, x \log \left (2\right )^{2} + 25 \, x^{2}\right )^{2} - 18 \, x - 36 \, \log \left (\log \left (2\right )^{2} + x\right ) \]

[In]

integrate(((2*log(2)^2+2*x+4)*log((25*log(2)^4+50*x*log(2)^2+25*x^2)/exp(5)^2)+(2+2*x)*log(2)^2+2*x^2+6*x+4)/(
log(2)^2+x),x, algorithm="giac")

[Out]

x^2 + 2*x*log(25*log(2)^4 + 50*x*log(2)^2 + 25*x^2) + log(25*log(2)^4 + 50*x*log(2)^2 + 25*x^2)^2 - 18*x - 36*
log(log(2)^2 + x)

Mupad [B] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.32 \[ \int \frac {4+6 x+2 x^2+(2+2 x) \log ^2(2)+\left (4+2 x+2 \log ^2(2)\right ) \log \left (\frac {25 x^2+50 x \log ^2(2)+25 \log ^4(2)}{e^{10}}\right )}{x+\log ^2(2)} \, dx=2\,x+2\,\ln \left ({\left (x+{\ln \left (2\right )}^2\right )}^2\right )+{\ln \left (25\,{\mathrm {e}}^{-10}\,\left (x^2+2\,{\ln \left (2\right )}^2\,x+{\ln \left (2\right )}^4\right )\right )}^2+2\,x\,\ln \left (25\,{\mathrm {e}}^{-10}\,\left (x^2+2\,{\ln \left (2\right )}^2\,x+{\ln \left (2\right )}^4\right )\right )+x^2 \]

[In]

int((6*x + log(exp(-10)*(50*x*log(2)^2 + 25*log(2)^4 + 25*x^2))*(2*x + 2*log(2)^2 + 4) + log(2)^2*(2*x + 2) +
2*x^2 + 4)/(x + log(2)^2),x)

[Out]

2*x + 2*log((x + log(2)^2)^2) + log(25*exp(-10)*(2*x*log(2)^2 + log(2)^4 + x^2))^2 + 2*x*log(25*exp(-10)*(2*x*
log(2)^2 + log(2)^4 + x^2)) + x^2