Integrand size = 131, antiderivative size = 28 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\frac {\left (-1+e-\frac {x}{2}\right ) (x+\log (2 x))}{4 \left (-5+x+x^2+\log (4)\right )} \]
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\frac {-5+(-1+2 e) x+\log (4)+(-2+2 e-x) \log (2 x)}{8 \left (-5+x+x^2+\log (4)\right )} \]
Integrate[(10 + 13*x + 7*x^2 + E*(-10 - 8*x + 2*x^2 - 2*x^3) + (-2 - 3*x - 2*x^2 + E*(2 + 2*x))*Log[4] + (7*x + 4*x^2 + x^3 + E*(-2*x - 4*x^2) - x*L og[4])*Log[2*x])/(200*x - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5 + (-80*x + 16*x ^2 + 16*x^3)*Log[4] + 8*x*Log[4]^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.
| \(\displaystyle \int \frac {7 x^2+\left (-2 x^2-3 x+e (2 x+2)-2\right ) \log (4)+e \left (-2 x^3+2 x^2-8 x-10\right )+\left (x^3+4 x^2+e \left (-4 x^2-2 x\right )+7 x-x \log (4)\right ) \log (2 x)+13 x+10}{8 x^5+16 x^4-72 x^3-80 x^2+\left (16 x^3+16 x^2-80 x\right ) \log (4)+200 x+8 x \log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {7 x^2+\left (-2 x^2-3 x+e (2 x+2)-2\right ) \log (4)+e \left (-2 x^3+2 x^2-8 x-10\right )+\left (x^3+4 x^2+e \left (-4 x^2-2 x\right )+7 x-x \log (4)\right ) \log (2 x)+13 x+10}{8 x^5+16 x^4-72 x^3-80 x^2+\left (16 x^3+16 x^2-80 x\right ) \log (4)+x \left (200+8 \log ^2(4)\right )}dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {7 x^2+\left (-2 x^2-3 x+e (2 x+2)-2\right ) \log (4)+e \left (-2 x^3+2 x^2-8 x-10\right )+\left (x^3+4 x^2+e \left (-4 x^2-2 x\right )+7 x-x \log (4)\right ) \log (2 x)+13 x+10}{x \left (8 x^4+16 x^3-8 x^2 (9-\log (16))-16 x (5-\log (4))+8 (5-\log (4))^2\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {7 x}{8 \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}+\frac {\left (x^2+4 (1-e) x-2 e+7-\log (4)\right ) \log (2 x)}{8 \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}+\frac {13}{8 \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}+\frac {e \left (-x^3+x^2-4 x-5\right )}{4 x \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}+\frac {\left (-2 x^2-(3-2 e) x-2 (1-e)\right ) \log (4)}{8 x \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}+\frac {5}{4 x \left (x^4+2 x^3-x^2 (9-\log (16))-2 x (5-\log (4))+(\log (4)-5)^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 e (2 x+1)}{\left (-(2 x+1)^2-4 \log (4)+21\right ) (21-\log (256))}+\frac {13 (2 x+1)}{2 \left (-(2 x+1)^2-4 \log (4)+21\right ) (21-\log (256))}-\frac {(3-2 e) \log (4) (2 x+1)}{2 \left (-(2 x+1)^2-4 \log (4)+21\right ) (21-\log (256))}-\frac {\log (4) (-x-\log (16)+10)}{\left (-(2 x+1)^2-4 \log (4)+21\right ) (21-2 \log (16))}-\frac {(1-e) \log (4) \log (x)}{4 (5-\log (4))^2}-\frac {5 e \log (x)}{4 (5-\log (4))^2}+\frac {5 \log (x)}{4 (5-\log (4))^2}+\frac {(1-e) \log (4) \log \left (-x^2-x-\log (4)+5\right )}{8 (5-\log (4))^2}+\frac {5 e \log \left (-x^2-x-\log (4)+5\right )}{8 (5-\log (4))^2}-\frac {5 \log \left (-x^2-x-\log (4)+5\right )}{8 (5-\log (4))^2}+\frac {1}{8} (7-2 (e+\log (2))) \int \frac {\log (2 x)}{x^4+2 x^3-(9-\log (16)) x^2-2 (5-\log (4)) x+(-5+\log (4))^2}dx+\frac {1}{2} (1-e) \int \frac {x \log (2 x)}{x^4+2 x^3-(9-\log (16)) x^2-2 (5-\log (4)) x+(-5+\log (4))^2}dx+\frac {1}{8} \int \frac {x^2 \log (2 x)}{x^4+2 x^3-(9-\log (16)) x^2-2 (5-\log (4)) x+(-5+\log (4))^2}dx-\frac {(1-e) \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) \log (4) (31-\log (4096))}{4 (21-4 \log (4))^{3/2} (5-\log (4))^2}-\frac {5 e \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) (31-\log (4096))}{4 (21-4 \log (4))^{3/2} (5-\log (4))^2}+\frac {5 \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) (31-\log (4096))}{4 (21-4 \log (4))^{3/2} (5-\log (4))^2}+\frac {e (-x-\log (16)+10)}{\left (-(2 x+1)^2-4 \log (4)+21\right ) (21-\log (256))}+\frac {7 (-x-\log (16)+10)}{2 \left (-(2 x+1)^2-4 \log (4)+21\right ) (21-\log (256))}-\frac {5 e (x-\log (16)+11)}{4 (5-\log (4)) \left (-x^2-x-\log (4)+5\right ) (21-\log (256))}+\frac {5 (x-\log (16)+11)}{4 (5-\log (4)) \left (-x^2-x-\log (4)+5\right ) (21-\log (256))}-\frac {(1-e) \log (4) (x-\log (16)+11)}{4 (5-\log (4)) \left (-x^2-x-\log (4)+5\right ) (21-\log (256))}-\frac {9 e \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right )}{2 (21-\log (256))^{3/2}}+\frac {19 \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right )}{4 (21-\log (256))^{3/2}}-\frac {(3-2 e) \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) \log (4)}{2 (21-\log (256))^{3/2}}+\frac {\text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) \log (4)}{2 (21-\log (256))^{3/2}}+\frac {e (-((11-\log (16)) (2 x+1))-4 \log (4)+21)}{2 \left (-(2 x+1)^2-4 \log (4)+21\right ) (21-2 \log (16))}+\frac {e \text {arctanh}\left (\frac {2 x+1}{\sqrt {21-4 \log (4)}}\right ) (5-\log (4))}{(21-4 \log (4))^{3/2}}\) |
Int[(10 + 13*x + 7*x^2 + E*(-10 - 8*x + 2*x^2 - 2*x^3) + (-2 - 3*x - 2*x^2 + E*(2 + 2*x))*Log[4] + (7*x + 4*x^2 + x^3 + E*(-2*x - 4*x^2) - x*Log[4]) *Log[2*x])/(200*x - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5 + (-80*x + 16*x^2 + 1 6*x^3)*Log[4] + 8*x*Log[4]^2),x]
3.13.15.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 0.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64
| method | result | size |
| norman | \(\frac {\left (-\frac {1}{8}+\frac {{\mathrm e}}{4}\right ) x +\left (\frac {{\mathrm e}}{4}-\frac {1}{4}\right ) \ln \left (2 x \right )-\frac {x \ln \left (2 x \right )}{8}-\frac {5}{8}+\frac {\ln \left (2\right )}{4}}{x^{2}+2 \ln \left (2\right )+x -5}\) | \(46\) |
| parallelrisch | \(\frac {2 \,{\mathrm e} \ln \left (2 x \right )-x \ln \left (2 x \right )+2 x \,{\mathrm e}-2 \ln \left (2 x \right )+2 \ln \left (2\right )-x -5}{8 x^{2}+16 \ln \left (2\right )+8 x -40}\) | \(50\) |
| risch | \(\frac {\left (-x +2 \,{\mathrm e}-2\right ) \ln \left (2 x \right )}{8 x^{2}+16 \ln \left (2\right )+8 x -40}+\frac {2 x \,{\mathrm e}+2 \ln \left (2\right )-x -5}{8 x^{2}+16 \ln \left (2\right )+8 x -40}\) | \(57\) |
| parts | \(\text {Expression too large to display}\) | \(813\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(817\) |
| default | \(\text {Expression too large to display}\) | \(817\) |
int(((-2*x*ln(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*ln(2*x)+2*((2+2*x)*exp (1)-2*x^2-3*x-2)*ln(2)+(-2*x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10)/(32*x*l n(2)^2+2*(16*x^3+16*x^2-80*x)*ln(2)+8*x^5+16*x^4-72*x^3-80*x^2+200*x),x,me thod=_RETURNVERBOSE)
((-1/8+1/4*exp(1))*x+(1/4*exp(1)-1/4)*ln(2*x)-1/8*x*ln(2*x)-5/8+1/4*ln(2)) /(x^2+2*ln(2)+x-5)
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\frac {2 \, x e - {\left (x - 2 \, e + 2\right )} \log \left (2 \, x\right ) - x + 2 \, \log \left (2\right ) - 5}{8 \, {\left (x^{2} + x + 2 \, \log \left (2\right ) - 5\right )}} \]
integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2+ 2*x)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10 )/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^2+ 200*x),x, algorithm=\
Time = 1.13 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\frac {\left (- x - 2 + 2 e\right ) \log {\left (2 x \right )}}{8 x^{2} + 8 x - 40 + 16 \log {\left (2 \right )}} + \frac {x \left (-1 + 2 e\right ) - 5 + 2 \log {\left (2 \right )}}{8 x^{2} + 8 x - 40 + 16 \log {\left (2 \right )}} \]
integrate(((-2*x*ln(2)+(-4*x**2-2*x)*exp(1)+x**3+4*x**2+7*x)*ln(2*x)+2*((2 +2*x)*exp(1)-2*x**2-3*x-2)*ln(2)+(-2*x**3+2*x**2-8*x-10)*exp(1)+7*x**2+13* x+10)/(32*x*ln(2)**2+2*(16*x**3+16*x**2-80*x)*ln(2)+8*x**5+16*x**4-72*x**3 -80*x**2+200*x),x)
(-x - 2 + 2*E)*log(2*x)/(8*x**2 + 8*x - 40 + 16*log(2)) + (x*(-1 + 2*E) - 5 + 2*log(2))/(8*x**2 + 8*x - 40 + 16*log(2))
Leaf count of result is larger than twice the leaf count of optimal. 1589 vs. \(2 (27) = 54\).
Time = 0.35 (sec) , antiderivative size = 1589, normalized size of antiderivative = 56.75 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\text {Too large to display} \]
integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2+ 2*x)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10 )/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^2+ 200*x),x, algorithm=\
-1/4*((12*log(2) - 31)*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8 *log(2) + 21) + 1))/((32*log(2)^3 - 244*log(2)^2 + 620*log(2) - 525)*sqrt( -8*log(2) + 21)) + 2*(x - 4*log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105)* x^2 + 32*log(2)^3 + (16*log(2)^2 - 82*log(2) + 105)*x - 244*log(2)^2 + 620 *log(2) - 525) + log(x^2 + x + 2*log(2) - 5)/(4*log(2)^2 - 20*log(2) + 25) - 2*log(x)/(4*log(2)^2 - 20*log(2) + 25))*e*log(2) + 1/2*((2*x + 1)/(x^2* (8*log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105) + 2*l og((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((8* log(2) - 21)*sqrt(-8*log(2) + 21)))*e*log(2) + 5/8*((12*log(2) - 31)*log(( 2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((32*log (2)^3 - 244*log(2)^2 + 620*log(2) - 525)*sqrt(-8*log(2) + 21)) + 2*(x - 4* log(2) + 11)/((16*log(2)^2 - 82*log(2) + 105)*x^2 + 32*log(2)^3 + (16*log( 2)^2 - 82*log(2) + 105)*x - 244*log(2)^2 + 620*log(2) - 525) + log(x^2 + x + 2*log(2) - 5)/(4*log(2)^2 - 20*log(2) + 25) - 2*log(x)/(4*log(2)^2 - 20 *log(2) + 25))*e - 1/4*(2*(2*log(2) - 5)*log((2*x - sqrt(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 21)*sqrt(-8*log(2) + 21 )) - (x*(4*log(2) - 11) - 2*log(2) + 5)/(x^2*(8*log(2) - 21) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105))*e - ((2*x + 1)/(x^2*(8*log(2) - 2 1) + x*(8*log(2) - 21) + 16*log(2)^2 - 82*log(2) + 105) + 2*log((2*x - sqr t(-8*log(2) + 21) + 1)/(2*x + sqrt(-8*log(2) + 21) + 1))/((8*log(2) - 2...
Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\frac {2 \, x e - x \log \left (2\right ) + 2 \, e \log \left (2\right ) - x \log \left (x\right ) + 2 \, e \log \left (x\right ) - x - 2 \, \log \left (x\right ) - 5}{8 \, {\left (x^{2} + x + 2 \, \log \left (2\right ) - 5\right )}} \]
integrate(((-2*x*log(2)+(-4*x^2-2*x)*exp(1)+x^3+4*x^2+7*x)*log(2*x)+2*((2+ 2*x)*exp(1)-2*x^2-3*x-2)*log(2)+(-2*x^3+2*x^2-8*x-10)*exp(1)+7*x^2+13*x+10 )/(32*x*log(2)^2+2*(16*x^3+16*x^2-80*x)*log(2)+8*x^5+16*x^4-72*x^3-80*x^2+ 200*x),x, algorithm=\
1/8*(2*x*e - x*log(2) + 2*e*log(2) - x*log(x) + 2*e*log(x) - x - 2*log(x) - 5)/(x^2 + x + 2*log(2) - 5)
Timed out. \[ \int \frac {10+13 x+7 x^2+e \left (-10-8 x+2 x^2-2 x^3\right )+\left (-2-3 x-2 x^2+e (2+2 x)\right ) \log (4)+\left (7 x+4 x^2+x^3+e \left (-2 x-4 x^2\right )-x \log (4)\right ) \log (2 x)}{200 x-80 x^2-72 x^3+16 x^4+8 x^5+\left (-80 x+16 x^2+16 x^3\right ) \log (4)+8 x \log ^2(4)} \, dx=\int \frac {13\,x-2\,\ln \left (2\right )\,\left (3\,x+2\,x^2-\mathrm {e}\,\left (2\,x+2\right )+2\right )+\ln \left (2\,x\right )\,\left (7\,x-\mathrm {e}\,\left (4\,x^2+2\,x\right )-2\,x\,\ln \left (2\right )+4\,x^2+x^3\right )-\mathrm {e}\,\left (2\,x^3-2\,x^2+8\,x+10\right )+7\,x^2+10}{200\,x+2\,\ln \left (2\right )\,\left (16\,x^3+16\,x^2-80\,x\right )+32\,x\,{\ln \left (2\right )}^2-80\,x^2-72\,x^3+16\,x^4+8\,x^5} \,d x \]
int((13*x - 2*log(2)*(3*x + 2*x^2 - exp(1)*(2*x + 2) + 2) + log(2*x)*(7*x - exp(1)*(2*x + 4*x^2) - 2*x*log(2) + 4*x^2 + x^3) - exp(1)*(8*x - 2*x^2 + 2*x^3 + 10) + 7*x^2 + 10)/(200*x + 2*log(2)*(16*x^2 - 80*x + 16*x^3) + 32 *x*log(2)^2 - 80*x^2 - 72*x^3 + 16*x^4 + 8*x^5),x)