Integrand size = 200, antiderivative size = 24 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=-x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \]
Time = 0.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=-x+x^4 \log ^2\left (4 \log (5) \log \left ((7+x+e x)^2\right )\right ) \]
Integrate[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2) ] + (4*x^4 + 4*E*x^4)*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 + E *(14*x + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2* x^2)]]^2)/((7 + x + E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] ),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(-e x-x-7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )+\left (4 e x^4+4 x^4\right ) \log \left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )+\left (4 e x^4+4 x^4+28 x^3\right ) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right ) \log ^2\left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )}{(e x+x+7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {(-e x-x-7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )+\left (4 e x^4+4 x^4\right ) \log \left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )+\left (4 e x^4+4 x^4+28 x^3\right ) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right ) \log ^2\left (4 \log (5) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )\right )}{((1+e) x+7) \log \left (e^2 x^2+x^2+e \left (2 x^2+14 x\right )+14 x+49\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (\frac {4 (1+e) x^4 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{((1+e) x+7) \log \left (((1+e) x+7)^2\right )}+4 x^3 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1372 \text {Subst}\left (\int \frac {\log \left (4 \log (5) \log \left (x^2\right )\right )}{\log \left (x^2\right )}dx,x,(1+e) x+7\right )}{(1+e)^4}+4 \int x^3 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )dx+4 \int \frac {x^3 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx-\frac {28 \int \frac {x^2 \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx}{1+e}+\frac {196 \int \frac {x \log \left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{\log \left (((1+e) x+7)^2\right )}dx}{(1+e)^2}-x+\frac {2401 \log ^2\left (4 \log (5) \log \left (((1+e) x+7)^2\right )\right )}{(1+e)^4}\) |
Int[((-7 - x - E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)] + (4 *x^4 + 4*E*x^4)*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x ^2)]] + (28*x^3 + 4*x^4 + 4*E*x^4)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]*Log[4*Log[5]*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]] ^2)/((7 + x + E*x)*Log[49 + 14*x + x^2 + E^2*x^2 + E*(14*x + 2*x^2)]),x]
3.26.88.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 3.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88
method | result | size |
parallelrisch | \({\ln \left (4 \ln \left (5\right ) \ln \left (x^{2} {\mathrm e}^{2}+\left (2 x^{2}+14 x \right ) {\mathrm e}+x^{2}+14 x +49\right )\right )}^{2} x^{4}-x\) | \(45\) |
int(((4*x^4*exp(1)+4*x^4+28*x^3)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+1 4*x+49)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))^2+(4* x^4*exp(1)+4*x^4)*ln(4*ln(5)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 49))+(-x*exp(1)-x-7)*ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49))/(x* exp(1)+x+7)/ln(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49),x,method=_RET URNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log \left (4 \, \log \left (5\right ) \log \left (x^{2} e^{2} + x^{2} + 2 \, {\left (x^{2} + 7 \, x\right )} e + 14 \, x + 49\right )\right )^{2} - x \]
integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 )+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp (1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 +14*x+49))/(x*exp(1)+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 ),x, algorithm=\
Time = 0.71 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log {\left (4 \log {\left (5 \right )} \log {\left (x^{2} + x^{2} e^{2} + 14 x + e \left (2 x^{2} + 14 x\right ) + 49 \right )} \right )}^{2} - x \]
integrate(((4*x**4*exp(1)+4*x**4+28*x**3)*ln(x**2*exp(1)**2+(2*x**2+14*x)* exp(1)+x**2+14*x+49)*ln(4*ln(5)*ln(x**2*exp(1)**2+(2*x**2+14*x)*exp(1)+x** 2+14*x+49))**2+(4*x**4*exp(1)+4*x**4)*ln(4*ln(5)*ln(x**2*exp(1)**2+(2*x**2 +14*x)*exp(1)+x**2+14*x+49))+(-x*exp(1)-x-7)*ln(x**2*exp(1)**2+(2*x**2+14* x)*exp(1)+x**2+14*x+49))/(x*exp(1)+x+7)/ln(x**2*exp(1)**2+(2*x**2+14*x)*ex p(1)+x**2+14*x+49),x)
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (25) = 50\).
Time = 0.42 (sec) , antiderivative size = 267, normalized size of antiderivative = 11.12 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=2 \, x^{4} {\left (3 \, \log \left (2\right ) + \log \left (\log \left (5\right )\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right ) + x^{4} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )^{2} + {\left (9 \, \log \left (2\right )^{2} + 6 \, \log \left (2\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} - {\left (\frac {x}{e + 1} - \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1}\right )} e - \frac {7 \, \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{2 \, {\left (e + 1\right )}} - \frac {x}{e + 1} + \frac {7 \, {\left (\frac {{\left (e \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right ) + \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )} \log \left (\log \left (x {\left (e + 1\right )} + 7\right )\right )}{e + 1} - 2 \, \log \left (x {\left (e + 1\right )} + 7\right )\right )}}{2 \, {\left (e + 1\right )}} + \frac {7 \, \log \left (x {\left (e + 1\right )} + 7\right )}{e^{2} + 2 \, e + 1} \]
integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 )+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp (1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 +14*x+49))/(x*exp(1)+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 ),x, algorithm=\
2*x^4*(3*log(2) + log(log(5)))*log(log(x*(e + 1) + 7)) + x^4*log(log(x*(e + 1) + 7))^2 + (9*log(2)^2 + 6*log(2)*log(log(5)) + log(log(5))^2)*x^4 - ( x/(e + 1) - 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1))*e - 7/2*log(x^2*e^2 + 2* x^2*e + x^2 + 14*x*e + 14*x + 49)*log(log(x*(e + 1) + 7))/(e + 1) - x/(e + 1) + 7/2*((e*log(x^2*e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49) + log(x^2* e^2 + 2*x^2*e + x^2 + 14*x*e + 14*x + 49))*log(log(x*(e + 1) + 7))/(e + 1) - 2*log(x*(e + 1) + 7))/(e + 1) + 7*log(x*(e + 1) + 7)/(e^2 + 2*e + 1)
Time = 2.60 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x^{4} \log \left (4 \, \log \left (5\right ) \log \left (x^{2} e^{2} + 2 \, x^{2} e + x^{2} + 14 \, x e + 14 \, x + 49\right )\right )^{2} - x \]
integrate(((4*x^4*exp(1)+4*x^4+28*x^3)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1 )+x^2+14*x+49)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+ 49))^2+(4*x^4*exp(1)+4*x^4)*log(4*log(5)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp (1)+x^2+14*x+49))+(-x*exp(1)-x-7)*log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2 +14*x+49))/(x*exp(1)+x+7)/log(x^2*exp(1)^2+(2*x^2+14*x)*exp(1)+x^2+14*x+49 ),x, algorithm=\
Time = 8.84 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.75 \[ \int \frac {(-7-x-e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )+\left (4 x^4+4 e x^4\right ) \log \left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )+\left (28 x^3+4 x^4+4 e x^4\right ) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right ) \log ^2\left (4 \log (5) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )\right )}{(7+x+e x) \log \left (49+14 x+x^2+e^2 x^2+e \left (14 x+2 x^2\right )\right )} \, dx=x\,\left (x^3\,{\ln \left (4\,\ln \left (14\,x+\mathrm {e}\,\left (2\,x^2+14\,x\right )+x^2\,{\mathrm {e}}^2+x^2+49\right )\,\ln \left (5\right )\right )}^2-1\right ) \]
int((log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5 ))*(4*x^4*exp(1) + 4*x^4) - log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*(x + x*exp(1) + 7) + log(14*x + exp(1)*(14*x + 2*x^2) + x^2*ex p(2) + x^2 + 49)*log(4*log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*log(5))^2*(4*x^4*exp(1) + 28*x^3 + 4*x^4))/(log(14*x + exp(1)*(14*x + 2*x^2) + x^2*exp(2) + x^2 + 49)*(x + x*exp(1) + 7)),x)