Integrand size = 91, antiderivative size = 26 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=e \left (x+\frac {(-1+2 x)^2 (2+\log (2))}{4 (1+x)}\right )^2 \] Output:
(x+1/4*(ln(2)+2)/(1+x)*(-1+2*x)^2)^2*exp(1)
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(26)=52\).
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=\frac {e \left (324+81 \log ^2(2)+16 (1+x)^4 (3+\log (2))^2+49 \log (16)+16 \log (256)-24 (1+x) \left (42-25 \log (2)+9 \log ^2(2)+8 \log (16)+4 \log (256)\right )-32 (1+x)^3 \left (21+3 \log ^2(2)+\log (65536)\right )\right )}{16 (1+x)^2} \] Input:
Integrate[(E*(-12 + 72*x - 144*x^2 + 240*x^3 + 144*x^4) + E*(-16 + 84*x - 192*x^2 + 128*x^3 + 96*x^4)*Log[2] + E*(-5 + 28*x - 48*x^2 + 16*x^3 + 16*x ^4)*Log[2]^2)/(8 + 24*x + 24*x^2 + 8*x^3),x]
Output:
(E*(324 + 81*Log[2]^2 + 16*(1 + x)^4*(3 + Log[2])^2 + 49*Log[16] + 16*Log[ 256] - 24*(1 + x)*(42 - 25*Log[2] + 9*Log[2]^2 + 8*Log[16] + 4*Log[256]) - 32*(1 + x)^3*(21 + 3*Log[2]^2 + Log[65536])))/(16*(1 + x)^2)
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(26)=52\).
Time = 0.52 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2007, 2389, 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.
\(\displaystyle \int \frac {e \left (144 x^4+240 x^3-144 x^2+72 x-12\right )+e \left (16 x^4+16 x^3-48 x^2+28 x-5\right ) \log ^2(2)+e \left (96 x^4+128 x^3-192 x^2+84 x-16\right ) \log (2)}{8 x^3+24 x^2+24 x+8} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {e \left (144 x^4+240 x^3-144 x^2+72 x-12\right )+e \left (16 x^4+16 x^3-48 x^2+28 x-5\right ) \log ^2(2)+e \left (96 x^4+128 x^3-192 x^2+84 x-16\right ) \log (2)}{(2 x+2)^3}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (x+1)^2}+2 e x (3+\log (2))^2-\frac {81 e (2+\log (2))^2}{8 (x+1)^3}-4 e \left (6+\log ^2(2)+\log (32)\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e x^2 (3+\log (2))^2-4 e x \left (6+\log ^2(2)+\log (32)\right )-\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (x+1)}+\frac {81 e (2+\log (2))^2}{16 (x+1)^2}\) |
Input:
Int[(E*(-12 + 72*x - 144*x^2 + 240*x^3 + 144*x^4) + E*(-16 + 84*x - 192*x^ 2 + 128*x^3 + 96*x^4)*Log[2] + E*(-5 + 28*x - 48*x^2 + 16*x^3 + 16*x^4)*Lo g[2]^2)/(8 + 24*x + 24*x^2 + 8*x^3),x]
Output:
(81*E*(2 + Log[2])^2)/(16*(1 + x)^2) + E*x^2*(3 + Log[2])^2 - 4*E*x*(6 + L og[2]^2 + Log[32]) - (9*E*(14 + 3*Log[2]^2 + Log[8192]))/(2*(1 + x))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(25)=50\).
Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08
method | result | size |
default | \(\frac {{\mathrm e} \left (8 x^{2} \ln \left (2\right )^{2}-32 x \ln \left (2\right )^{2}+48 x^{2} \ln \left (2\right )-160 x \ln \left (2\right )+72 x^{2}-192 x -\frac {108 \ln \left (2\right )^{2}+468 \ln \left (2\right )+504}{1+x}-\frac {-81 \ln \left (2\right )^{2}-324 \ln \left (2\right )-324}{2 \left (1+x \right )^{2}}\right )}{8}\) | \(80\) |
gosper | \(\frac {\left (16 x^{4} \ln \left (2\right )^{2}-32 x^{3} \ln \left (2\right )^{2}+96 x^{4} \ln \left (2\right )-128 x^{3} \ln \left (2\right )+144 x^{4}-56 x \ln \left (2\right )^{2}-96 x^{3}-23 \ln \left (2\right )^{2}-168 x \ln \left (2\right )-68 \ln \left (2\right )-144 x -60\right ) {\mathrm e}}{16 x^{2}+32 x +16}\) | \(84\) |
norman | \(\frac {\left (-2 \,{\mathrm e} \ln \left (2\right )^{2}-8 \,{\mathrm e} \ln \left (2\right )-6 \,{\mathrm e}\right ) x^{3}+\left ({\mathrm e} \ln \left (2\right )^{2}+6 \,{\mathrm e} \ln \left (2\right )+9 \,{\mathrm e}\right ) x^{4}+\left (-\frac {7 \,{\mathrm e} \ln \left (2\right )^{2}}{2}-\frac {21 \,{\mathrm e} \ln \left (2\right )}{2}-9 \,{\mathrm e}\right ) x -\frac {23 \,{\mathrm e} \ln \left (2\right )^{2}}{16}-\frac {17 \,{\mathrm e} \ln \left (2\right )}{4}-\frac {15 \,{\mathrm e}}{4}}{\left (1+x \right )^{2}}\) | \(92\) |
risch | \(x^{2} {\mathrm e} \ln \left (2\right )^{2}-4 \,{\mathrm e} \ln \left (2\right )^{2} x +6 \,{\mathrm e} \ln \left (2\right ) x^{2}-20 x \,{\mathrm e} \ln \left (2\right )+9 x^{2} {\mathrm e}-24 x \,{\mathrm e}+\frac {\left (-\frac {27 \,{\mathrm e} \ln \left (2\right )^{2}}{2}-\frac {117 \,{\mathrm e} \ln \left (2\right )}{2}-63 \,{\mathrm e}\right ) x -\frac {135 \,{\mathrm e} \ln \left (2\right )^{2}}{16}-\frac {153 \,{\mathrm e} \ln \left (2\right )}{4}-\frac {171 \,{\mathrm e}}{4}}{x^{2}+2 x +1}\) | \(100\) |
parallelrisch | \(\frac {16 \,{\mathrm e} \ln \left (2\right )^{2} x^{4}-32 \,{\mathrm e} \ln \left (2\right )^{2} x^{3}+96 \,{\mathrm e} \ln \left (2\right ) x^{4}-128 \,{\mathrm e} \ln \left (2\right ) x^{3}+144 x^{4} {\mathrm e}-56 \,{\mathrm e} \ln \left (2\right )^{2} x -96 x^{3} {\mathrm e}-23 \,{\mathrm e} \ln \left (2\right )^{2}-168 x \,{\mathrm e} \ln \left (2\right )-68 \,{\mathrm e} \ln \left (2\right )-144 x \,{\mathrm e}-60 \,{\mathrm e}}{16 x^{2}+32 x +16}\) | \(107\) |
Input:
int(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*ln(2)^2+(96*x^4+128*x^3-192*x^2+ 84*x-16)*exp(1)*ln(2)+(144*x^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8*x^3+24* x^2+24*x+8),x,method=_RETURNVERBOSE)
Output:
1/8*exp(1)*(8*x^2*ln(2)^2-32*x*ln(2)^2+48*x^2*ln(2)-160*x*ln(2)+72*x^2-192 *x-(108*ln(2)^2+468*ln(2)+504)/(1+x)-1/2*(-81*ln(2)^2-324*ln(2)-324)/(1+x) ^2)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (27) = 54\).
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.46 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=\frac {{\left (16 \, x^{4} - 32 \, x^{3} - 112 \, x^{2} - 280 \, x - 135\right )} e \log \left (2\right )^{2} + 4 \, {\left (24 \, x^{4} - 32 \, x^{3} - 136 \, x^{2} - 314 \, x - 153\right )} e \log \left (2\right ) + 12 \, {\left (12 \, x^{4} - 8 \, x^{3} - 52 \, x^{2} - 116 \, x - 57\right )} e}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \] Input:
integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-1 92*x^2+84*x-16)*exp(1)*log(2)+(144*x^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8 *x^3+24*x^2+24*x+8),x, algorithm="fricas")
Output:
1/16*((16*x^4 - 32*x^3 - 112*x^2 - 280*x - 135)*e*log(2)^2 + 4*(24*x^4 - 3 2*x^3 - 136*x^2 - 314*x - 153)*e*log(2) + 12*(12*x^4 - 8*x^3 - 52*x^2 - 11 6*x - 57)*e)/(x^2 + 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (24) = 48\).
Time = 0.36 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.31 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=x^{2} \left (e \log {\left (2 \right )}^{2} + 6 e \log {\left (2 \right )} + 9 e\right ) + x \left (- 24 e - 20 e \log {\left (2 \right )} - 4 e \log {\left (2 \right )}^{2}\right ) + \frac {x \left (- 1008 e - 936 e \log {\left (2 \right )} - 216 e \log {\left (2 \right )}^{2}\right ) - 684 e - 612 e \log {\left (2 \right )} - 135 e \log {\left (2 \right )}^{2}}{16 x^{2} + 32 x + 16} \] Input:
integrate(((16*x**4+16*x**3-48*x**2+28*x-5)*exp(1)*ln(2)**2+(96*x**4+128*x **3-192*x**2+84*x-16)*exp(1)*ln(2)+(144*x**4+240*x**3-144*x**2+72*x-12)*ex p(1))/(8*x**3+24*x**2+24*x+8),x)
Output:
x**2*(E*log(2)**2 + 6*E*log(2) + 9*E) + x*(-24*E - 20*E*log(2) - 4*E*log(2 )**2) + (x*(-1008*E - 936*E*log(2) - 216*E*log(2)**2) - 684*E - 612*E*log( 2) - 135*E*log(2)**2)/(16*x**2 + 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.73 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx={\left (e \log \left (2\right )^{2} + 6 \, e \log \left (2\right ) + 9 \, e\right )} x^{2} - 4 \, {\left (e \log \left (2\right )^{2} + 5 \, e \log \left (2\right ) + 6 \, e\right )} x - \frac {9 \, {\left (15 \, e \log \left (2\right )^{2} + 8 \, {\left (3 \, e \log \left (2\right )^{2} + 13 \, e \log \left (2\right ) + 14 \, e\right )} x + 68 \, e \log \left (2\right ) + 76 \, e\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \] Input:
integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-1 92*x^2+84*x-16)*exp(1)*log(2)+(144*x^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8 *x^3+24*x^2+24*x+8),x, algorithm="maxima")
Output:
(e*log(2)^2 + 6*e*log(2) + 9*e)*x^2 - 4*(e*log(2)^2 + 5*e*log(2) + 6*e)*x - 9/16*(15*e*log(2)^2 + 8*(3*e*log(2)^2 + 13*e*log(2) + 14*e)*x + 68*e*log (2) + 76*e)/(x^2 + 2*x + 1)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (27) = 54\).
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=x^{2} e \log \left (2\right )^{2} + 6 \, x^{2} e \log \left (2\right ) - 4 \, x e \log \left (2\right )^{2} + 9 \, x^{2} e - 20 \, x e \log \left (2\right ) - 24 \, x e - \frac {9 \, {\left (24 \, x e \log \left (2\right )^{2} + 104 \, x e \log \left (2\right ) + 15 \, e \log \left (2\right )^{2} + 112 \, x e + 68 \, e \log \left (2\right ) + 76 \, e\right )}}{16 \, {\left (x + 1\right )}^{2}} \] Input:
integrate(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-1 92*x^2+84*x-16)*exp(1)*log(2)+(144*x^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8 *x^3+24*x^2+24*x+8),x, algorithm="giac")
Output:
x^2*e*log(2)^2 + 6*x^2*e*log(2) - 4*x*e*log(2)^2 + 9*x^2*e - 20*x*e*log(2) - 24*x*e - 9/16*(24*x*e*log(2)^2 + 104*x*e*log(2) + 15*e*log(2)^2 + 112*x *e + 68*e*log(2) + 76*e)/(x + 1)^2
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=x^2\,\left (9\,\mathrm {e}+6\,\mathrm {e}\,\ln \left (2\right )+\mathrm {e}\,{\ln \left (2\right )}^2\right )-x\,\left (24\,\mathrm {e}+20\,\mathrm {e}\,\ln \left (2\right )+4\,\mathrm {e}\,{\ln \left (2\right )}^2\right )-\frac {342\,\mathrm {e}+306\,\mathrm {e}\,\ln \left (2\right )+\frac {135\,\mathrm {e}\,{\ln \left (2\right )}^2}{2}+x\,\left (504\,\mathrm {e}+468\,\mathrm {e}\,\ln \left (2\right )+108\,\mathrm {e}\,{\ln \left (2\right )}^2\right )}{8\,x^2+16\,x+8} \] Input:
int((exp(1)*(72*x - 144*x^2 + 240*x^3 + 144*x^4 - 12) + exp(1)*log(2)*(84* x - 192*x^2 + 128*x^3 + 96*x^4 - 16) + exp(1)*log(2)^2*(28*x - 48*x^2 + 16 *x^3 + 16*x^4 - 5))/(24*x + 24*x^2 + 8*x^3 + 8),x)
Output:
x^2*(9*exp(1) + 6*exp(1)*log(2) + exp(1)*log(2)^2) - x*(24*exp(1) + 20*exp (1)*log(2) + 4*exp(1)*log(2)^2) - (342*exp(1) + 306*exp(1)*log(2) + (135*e xp(1)*log(2)^2)/2 + x*(504*exp(1) + 468*exp(1)*log(2) + 108*exp(1)*log(2)^ 2))/(16*x + 8*x^2 + 8)
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int \frac {e \left (-12+72 x-144 x^2+240 x^3+144 x^4\right )+e \left (-16+84 x-192 x^2+128 x^3+96 x^4\right ) \log (2)+e \left (-5+28 x-48 x^2+16 x^3+16 x^4\right ) \log ^2(2)}{8+24 x+24 x^2+8 x^3} \, dx=\frac {e \left (16 \mathrm {log}\left (2\right )^{2} x^{4}-32 \mathrm {log}\left (2\right )^{2} x^{3}+28 \mathrm {log}\left (2\right )^{2} x^{2}+5 \mathrm {log}\left (2\right )^{2}+96 \,\mathrm {log}\left (2\right ) x^{4}-128 \,\mathrm {log}\left (2\right ) x^{3}+84 \,\mathrm {log}\left (2\right ) x^{2}+16 \,\mathrm {log}\left (2\right )+144 x^{4}-96 x^{3}+72 x^{2}+12\right )}{16 x^{2}+32 x +16} \] Input:
int(((16*x^4+16*x^3-48*x^2+28*x-5)*exp(1)*log(2)^2+(96*x^4+128*x^3-192*x^2 +84*x-16)*exp(1)*log(2)+(144*x^4+240*x^3-144*x^2+72*x-12)*exp(1))/(8*x^3+2 4*x^2+24*x+8),x)
Output:
(e*(16*log(2)**2*x**4 - 32*log(2)**2*x**3 + 28*log(2)**2*x**2 + 5*log(2)** 2 + 96*log(2)*x**4 - 128*log(2)*x**3 + 84*log(2)*x**2 + 16*log(2) + 144*x* *4 - 96*x**3 + 72*x**2 + 12))/(16*(x**2 + 2*x + 1))