Integrand size = 112, antiderivative size = 23 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {-2+e^x}{x+\frac {9}{2-2 x-\log (4)}} \] Output:
(exp(x)-2)/(9/(-2*x+2-2*ln(2))+x)
Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {\left (-2+e^x\right ) (-2+2 x+\log (4))}{-9+2 x^2+x (-2+\log (4))} \] Input:
Integrate[(44 - 16*x + 8*x^2 + (-8 + 8*x)*Log[4] + 2*Log[4]^2 + E^x*(-4 - 6*x - 12*x^2 + 4*x^3 + (-5 - 8*x + 4*x^2)*Log[4] + (-1 + x)*Log[4]^2))/(81 + 36*x - 32*x^2 - 8*x^3 + 4*x^4 + (-18*x - 4*x^2 + 4*x^3)*Log[4] + x^2*Lo g[4]^2),x]
Output:
((-2 + E^x)*(-2 + 2*x + Log[4]))/(-9 + 2*x^2 + x*(-2 + Log[4]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 3.82 (sec) , antiderivative size = 1435, normalized size of antiderivative = 62.39, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6, 2463, 6, 7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {8 x^2+e^x \left (4 x^3-12 x^2+\left (4 x^2-8 x-5\right ) \log (4)-6 x+(x-1) \log ^2(4)-4\right )-16 x+(8 x-8) \log (4)+44+2 \log ^2(4)}{4 x^4-8 x^3-32 x^2+x^2 \log ^2(4)+\left (4 x^3-4 x^2-18 x\right ) \log (4)+36 x+81} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^2+e^x \left (4 x^3-12 x^2+\left (4 x^2-8 x-5\right ) \log (4)-6 x+(x-1) \log ^2(4)-4\right )-16 x+(8 x-8) \log (4)+44+2 \log ^2(4)}{4 x^4-8 x^3+x^2 \left (\log ^2(4)-32\right )+\left (4 x^3-4 x^2-18 x\right ) \log (4)+36 x+81}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {8 x^2+e^x \left (4 x^3-12 x^2+\left (4 x^2-8 x-5\right ) \log (4)-6 x+(x-1) \log ^2(4)-4\right )-16 x+(8 x-8) \log (4)+44+2 \log ^2(4)}{\left (2 x^2-2 x+x \log (4)-9\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {8 x^2+e^x \left (4 x^3-12 x^2+\left (4 x^2-8 x-5\right ) \log (4)-6 x+(x-1) \log ^2(4)-4\right )-16 x+(8 x-8) \log (4)+44+2 \log ^2(4)}{\left (2 x^2+x (\log (4)-2)-9\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 x^2+e^x \left (4 x^3-12 x^2+\left (4 x^2-8 x-5\right ) \log (4)-6 x+(x-1) \log ^2(4)-4\right )-16 x+(8 x-8) \log (4)+44 \left (1+\frac {\log ^2(4)}{22}\right )}{\left (-2 x^2+x (2-\log (4))+9\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (22+\log ^2(4)\right )}{\left (-2 x^2+x (2-\log (4))+9\right )^2}+\frac {8 x^2}{\left (-2 x^2+x (2-\log (4))+9\right )^2}-\frac {16 x}{\left (-2 x^2+x (2-\log (4))+9\right )^2}+\frac {8 (x-1) \log (4)}{\left (-2 x^2+x (2-\log (4))+9\right )^2}+\frac {e^x \left (4 x^3-4 x^2 (3-\log (4))-x \left (6-\log ^2(4)+8 \log (4)\right )-(1+\log (4)) (4+\log (4))\right )}{\left (-2 x^2+x (2-\log (4))+9\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16 \left (22+\log ^2(4)\right ) \text {arctanh}\left (\frac {-4 x-\log (4)+2}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {16 \log (4) (2+\log (4)) \text {arctanh}\left (\frac {-4 x-\log (4)+2}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {32 (2-\log (4)) \text {arctanh}\left (\frac {-4 x-\log (4)+2}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {288 \text {arctanh}\left (\frac {-4 x-\log (4)+2}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {8 \log (4) (-((2+\log (4)) x)-\log (4)+20)}{\left (-2 x^2+(2-\log (4)) x+9\right ) \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {2 e^x (2-\log (4)) \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x-\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}-\frac {4 e^x \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x-\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}+\frac {2 e^x (2-\log (4)) \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x+\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}-\frac {4 e^x \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right ) \left (-4 x+\sqrt {76-4 \log (4)+\log ^2(4)}-\log (4)+2\right )}+\frac {e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4)) \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{2 \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {e^{\frac {1}{4} \left (2-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4)) \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )}{2 \sqrt {2} \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (1+\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+\frac {1}{2} e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (1-\frac {6-\log (4)}{\sqrt {76-4 \log (4)+\log ^2(4)}}\right )+\frac {8 x ((2-\log (4)) x+18)}{\left (-2 x^2+(2-\log (4)) x+9\right ) \left (76-4 \log (4)+\log ^2(4)\right )}-\frac {16 ((2-\log (4)) x+18)}{\left (-2 x^2+(2-\log (4)) x+9\right ) \left (76-4 \log (4)+\log ^2(4)\right )}-\frac {e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{76-4 \log (4)+\log ^2(4)}-\frac {e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{76-4 \log (4)+\log ^2(4)}-\frac {2 (-4 x-\log (4)+2) \left (22+\log ^2(4)\right )}{\left (-2 x^2+(2-\log (4)) x+9\right ) \left (76-4 \log (4)+\log ^2(4)\right )}+\frac {4 e^{\frac {1}{4} \left (2-\log (4)+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {4 e^{\frac {1}{4} \left (2-\log (4)-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) \left (40-8 \log (2)+\log ^2(4)\right )}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}-\frac {\sqrt {2} e^{\frac {1}{4} \left (2+\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x-\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4))^2}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}+\frac {\sqrt {2} e^{\frac {1}{4} \left (2-\sqrt {76-4 \log (4)+\log ^2(4)}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{4} \left (4 x+\sqrt {76-4 \log (4)+\log ^2(4)}+\log (4)-2\right )\right ) (2-\log (4))^2}{\left (76-4 \log (4)+\log ^2(4)\right )^{3/2}}\) |
Input:
Int[(44 - 16*x + 8*x^2 + (-8 + 8*x)*Log[4] + 2*Log[4]^2 + E^x*(-4 - 6*x - 12*x^2 + 4*x^3 + (-5 - 8*x + 4*x^2)*Log[4] + (-1 + x)*Log[4]^2))/(81 + 36* x - 32*x^2 - 8*x^3 + 4*x^4 + (-18*x - 4*x^2 + 4*x^3)*Log[4] + x^2*Log[4]^2 ),x]
Output:
(288*ArcTanh[(2 - 4*x - Log[4])/Sqrt[76 - 4*Log[4] + Log[4]^2]])/(76 - 4*L og[4] + Log[4]^2)^(3/2) + (32*ArcTanh[(2 - 4*x - Log[4])/Sqrt[76 - 4*Log[4 ] + Log[4]^2]]*(2 - Log[4]))/(76 - 4*Log[4] + Log[4]^2)^(3/2) - (Sqrt[2]*E ^((2 + Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(2 - Log[4])^2)/(76 - 4*Log[4] + Log [4]^2)^(3/2) + (Sqrt[2]*E^((2 - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*ExpInte gralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(2 - Log[4] )^2)/(76 - 4*Log[4] + Log[4]^2)^(3/2) + (16*ArcTanh[(2 - 4*x - Log[4])/Sqr t[76 - 4*Log[4] + Log[4]^2]]*Log[4]*(2 + Log[4]))/(76 - 4*Log[4] + Log[4]^ 2)^(3/2) - (16*ArcTanh[(2 - 4*x - Log[4])/Sqrt[76 - 4*Log[4] + Log[4]^2]]* (22 + Log[4]^2))/(76 - 4*Log[4] + Log[4]^2)^(3/2) + (4*E^((2 - Log[4] + Sq rt[76 - 4*Log[4] + Log[4]^2])/4)*ExpIntegralEi[(-2 + 4*x + Log[4] - Sqrt[7 6 - 4*Log[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log[4]^2))/(76 - 4*Log[4] + Log[4]^2)^(3/2) - (4*E^((2 - Log[4] - Sqrt[76 - 4*Log[4] + Log[4]^2])/4)*E xpIntegralEi[(-2 + 4*x + Log[4] + Sqrt[76 - 4*Log[4] + Log[4]^2])/4]*(40 - 8*Log[2] + Log[4]^2))/(76 - 4*Log[4] + Log[4]^2)^(3/2) - (16*(18 + x*(2 - Log[4])))/((9 - 2*x^2 + x*(2 - Log[4]))*(76 - 4*Log[4] + Log[4]^2)) + (8* x*(18 + x*(2 - Log[4])))/((9 - 2*x^2 + x*(2 - Log[4]))*(76 - 4*Log[4] + Lo g[4]^2)) - (2*(2 - 4*x - Log[4])*(22 + Log[4]^2))/((9 - 2*x^2 + x*(2 - Log [4]))*(76 - 4*Log[4] + Log[4]^2)) - (E^((2 - Log[4] + Sqrt[76 - 4*Log[4...
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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83
| method | result | size |
| norman | \(\frac {-4 x +\left (2 \ln \left (2\right )-2\right ) {\mathrm e}^{x}+2 \,{\mathrm e}^{x} x +4-4 \ln \left (2\right )}{2 x \ln \left (2\right )+2 x^{2}-2 x -9}\) | \(42\) |
| parallelrisch | \(\frac {8+4 \,{\mathrm e}^{x} \ln \left (2\right )+4 \,{\mathrm e}^{x} x -8 \ln \left (2\right )-8 x -4 \,{\mathrm e}^{x}}{4 x \ln \left (2\right )+4 x^{2}-4 x -18}\) | \(44\) |
| risch | \(\frac {-2 x +2-2 \ln \left (2\right )}{x \ln \left (2\right )+x^{2}-x -\frac {9}{2}}+\frac {2 \left (x +\ln \left (2\right )-1\right ) {\mathrm e}^{x}}{2 x \ln \left (2\right )+2 x^{2}-2 x -9}\) | \(52\) |
| parts | \(\frac {-2 x +2-2 \ln \left (2\right )}{x \ln \left (2\right )+x^{2}-x -\frac {9}{2}}+\frac {3 \,{\mathrm e}^{x} \left (2 x \ln \left (2\right )^{2}-4 x \ln \left (2\right )-9 \ln \left (2\right )+20 x +9\right )}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}-\frac {3 \,{\mathrm e}^{x} \left (x \ln \left (2\right )-x -9\right )}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {2 \,{\mathrm e}^{x} \left (\ln \left (2\right )+2 x -1\right )}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}-\frac {22 \ln \left (2\right ) {\mathrm e}^{x} x}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {2 \ln \left (2\right )^{2} {\mathrm e}^{x} x}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {3 \,{\mathrm e}^{x} \ln \left (2\right )^{2}}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {49 \,{\mathrm e}^{x} \ln \left (2\right )}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {{\mathrm e}^{x} \left (2 x \ln \left (2\right )^{3}-6 x \ln \left (2\right )^{2}-9 \ln \left (2\right )^{2}+33 x \ln \left (2\right )+18 \ln \left (2\right )-29 x -90\right )}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}-\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{x} x}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}+\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{x}}{\left (\ln \left (2\right )^{2}-2 \ln \left (2\right )+19\right ) \left (2 x \ln \left (2\right )+2 x^{2}-2 x -9\right )}\) | \(450\) |
| default | \(\text {Expression too large to display}\) | \(1035\) |
Input:
int(((4*(-1+x)*ln(2)^2+2*(4*x^2-8*x-5)*ln(2)+4*x^3-12*x^2-6*x-4)*exp(x)+8* ln(2)^2+2*(8*x-8)*ln(2)+8*x^2-16*x+44)/(4*x^2*ln(2)^2+2*(4*x^3-4*x^2-18*x) *ln(2)+4*x^4-8*x^3-32*x^2+36*x+81),x,method=_RETURNVERBOSE)
Output:
(-4*x+(2*ln(2)-2)*exp(x)+2*exp(x)*x+4-4*ln(2))/(2*x*ln(2)+2*x^2-2*x-9)
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {2 \, {\left ({\left (x + \log \left (2\right ) - 1\right )} e^{x} - 2 \, x - 2 \, \log \left (2\right ) + 2\right )}}{2 \, x^{2} + 2 \, x \log \left (2\right ) - 2 \, x - 9} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*e xp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3- 4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="fricas")
Output:
2*((x + log(2) - 1)*e^x - 2*x - 2*log(2) + 2)/(2*x^2 + 2*x*log(2) - 2*x - 9)
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (17) = 34\).
Time = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.35 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {- 4 x - 4 \log {\left (2 \right )} + 4}{2 x^{2} + x \left (-2 + 2 \log {\left (2 \right )}\right ) - 9} + \frac {\left (2 x - 2 + 2 \log {\left (2 \right )}\right ) e^{x}}{2 x^{2} - 2 x + 2 x \log {\left (2 \right )} - 9} \] Input:
integrate(((4*(-1+x)*ln(2)**2+2*(4*x**2-8*x-5)*ln(2)+4*x**3-12*x**2-6*x-4) *exp(x)+8*ln(2)**2+2*(8*x-8)*ln(2)+8*x**2-16*x+44)/(4*x**2*ln(2)**2+2*(4*x **3-4*x**2-18*x)*ln(2)+4*x**4-8*x**3-32*x**2+36*x+81),x)
Output:
(-4*x - 4*log(2) + 4)/(2*x**2 + x*(-2 + 2*log(2)) - 9) + (2*x - 2 + 2*log( 2))*exp(x)/(2*x**2 - 2*x + 2*x*log(2) - 9)
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (21) = 42\).
Time = 0.17 (sec) , antiderivative size = 692, normalized size of antiderivative = 30.09 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\text {Too large to display} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*e xp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3- 4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="maxima")
Output:
-4*((2*x + log(2) - 1)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - 3 *log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) + log((2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log (2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2))*log(2)^2 + 4*(( log(2) - 1)*log((2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^ (3/2) + 2*(x*(log(2) - 1) - 9)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log( 2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171))*log (2) + 8*((2*x + log(2) - 1)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^ 3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) + log(( 2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2))*log(2) + 2 *(x + log(2) - 1)*e^x/(2*x^2 + 2*x*(log(2) - 1) - 9) - 4*(log(2) - 1)*log( (2*x - sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1)/(2*x + sqrt(log(2)^2 - 2*log(2) + 19) + log(2) - 1))/(log(2)^2 - 2*log(2) + 19)^(3/2) - 2*(2*(lo g(2)^2 - 2*log(2) + 10)*x - 9*log(2) + 9)/(2*(log(2)^2 - 2*log(2) + 19)*x^ 2 + 2*(log(2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) - 8*(x*(log(2) - 1) - 9)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log (2)^3 - 3*log(2)^2 + 21*log(2) - 19)*x - 9*log(2)^2 + 18*log(2) - 171) - 2 2*(2*x + log(2) - 1)/(2*(log(2)^2 - 2*log(2) + 19)*x^2 + 2*(log(2)^3 - ...
Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {2 \, {\left (x e^{x} + e^{x} \log \left (2\right ) - 2 \, x - e^{x} - 2 \, \log \left (2\right ) + 2\right )}}{2 \, x^{2} + 2 \, x \log \left (2\right ) - 2 \, x - 9} \] Input:
integrate(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*e xp(x)+8*log(2)^2+2*(8*x-8)*log(2)+8*x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3- 4*x^2-18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x, algorithm="giac")
Output:
2*(x*e^x + e^x*log(2) - 2*x - e^x - 2*log(2) + 2)/(2*x^2 + 2*x*log(2) - 2* x - 9)
Timed out. \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\int \frac {2\,\ln \left (2\right )\,\left (8\,x-8\right )-16\,x-{\mathrm {e}}^x\,\left (6\,x-4\,{\ln \left (2\right )}^2\,\left (x-1\right )+2\,\ln \left (2\right )\,\left (-4\,x^2+8\,x+5\right )+12\,x^2-4\,x^3+4\right )+8\,{\ln \left (2\right )}^2+8\,x^2+44}{36\,x+4\,x^2\,{\ln \left (2\right )}^2-2\,\ln \left (2\right )\,\left (-4\,x^3+4\,x^2+18\,x\right )-32\,x^2-8\,x^3+4\,x^4+81} \,d x \] Input:
int((2*log(2)*(8*x - 8) - 16*x - exp(x)*(6*x - 4*log(2)^2*(x - 1) + 2*log( 2)*(8*x - 4*x^2 + 5) + 12*x^2 - 4*x^3 + 4) + 8*log(2)^2 + 8*x^2 + 44)/(36* x + 4*x^2*log(2)^2 - 2*log(2)*(18*x + 4*x^2 - 4*x^3) - 32*x^2 - 8*x^3 + 4* x^4 + 81),x)
Output:
int((2*log(2)*(8*x - 8) - 16*x - exp(x)*(6*x - 4*log(2)^2*(x - 1) + 2*log( 2)*(8*x - 4*x^2 + 5) + 12*x^2 - 4*x^3 + 4) + 8*log(2)^2 + 8*x^2 + 44)/(36* x + 4*x^2*log(2)^2 - 2*log(2)*(18*x + 4*x^2 - 4*x^3) - 32*x^2 - 8*x^3 + 4* x^4 + 81), x)
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int \frac {44-16 x+8 x^2+(-8+8 x) \log (4)+2 \log ^2(4)+e^x \left (-4-6 x-12 x^2+4 x^3+\left (-5-8 x+4 x^2\right ) \log (4)+(-1+x) \log ^2(4)\right )}{81+36 x-32 x^2-8 x^3+4 x^4+\left (-18 x-4 x^2+4 x^3\right ) \log (4)+x^2 \log ^2(4)} \, dx=\frac {2 e^{x} \mathrm {log}\left (2\right )^{2}+2 e^{x} \mathrm {log}\left (2\right ) x -4 e^{x} \mathrm {log}\left (2\right )-2 e^{x} x +2 e^{x}-4 \mathrm {log}\left (2\right )^{2}+8 \,\mathrm {log}\left (2\right )+4 x^{2}-22}{2 \mathrm {log}\left (2\right )^{2} x +2 \,\mathrm {log}\left (2\right ) x^{2}-4 \,\mathrm {log}\left (2\right ) x -9 \,\mathrm {log}\left (2\right )-2 x^{2}+2 x +9} \] Input:
int(((4*(-1+x)*log(2)^2+2*(4*x^2-8*x-5)*log(2)+4*x^3-12*x^2-6*x-4)*exp(x)+ 8*log(2)^2+2*(8*x-8)*log(2)+8*x^2-16*x+44)/(4*x^2*log(2)^2+2*(4*x^3-4*x^2- 18*x)*log(2)+4*x^4-8*x^3-32*x^2+36*x+81),x)
Output:
(2*(e**x*log(2)**2 + e**x*log(2)*x - 2*e**x*log(2) - e**x*x + e**x - 2*log (2)**2 + 4*log(2) + 2*x**2 - 11))/(2*log(2)**2*x + 2*log(2)*x**2 - 4*log(2 )*x - 9*log(2) - 2*x**2 + 2*x + 9)