Optimal. Leaf size=26 \[ e \left (x+\frac {(-1+2 x)^2 (2+\log (2))}{4 (1+x)}\right )^2 \]
________________________________________________________________________________________
Rubi [B] time = 0.13, antiderivative size = 60, normalized size of antiderivative = 2.31, number of steps used = 2, number of rules used = 1, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2074} \begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {81 e (2+\log (2))^2}{8 (1+x)^3}+2 e x (3+\log (2))^2-4 e \left (6+\log ^2(2)+\log (32)\right )+\frac {9 e \left (14+3 \log ^2(2)+\log (8192)\right )}{2 (1+x)^2}\right ) \, dx\\ &=\frac {81 e (2+\log (2))^2}{16 (1+x)^2}+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 (1+x)}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.07, size = 81, normalized size = 3.12 \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.65, size = 90, normalized size = 3.46 \begin {gather*} \frac {{\left (16 \, x^{4} - 32 \, x^{3} - 112 \, x^{2} - 280 \, x - 135\right )} e \log \relax (2)^{2} + 4 \, {\left (24 \, x^{4} - 32 \, x^{3} - 136 \, x^{2} - 314 \, x - 153\right )} e \log \relax (2) + 12 \, {\left (12 \, x^{4} - 8 \, x^{3} - 52 \, x^{2} - 116 \, x - 57\right )} e}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.33, size = 95, normalized size = 3.65 \begin {gather*} x^{2} e \log \relax (2)^{2} + 6 \, x^{2} e \log \relax (2) - 4 \, x e \log \relax (2)^{2} + 9 \, x^{2} e - 20 \, x e \log \relax (2) - 24 \, x e - \frac {9 \, {\left (24 \, x e \log \relax (2)^{2} + 104 \, x e \log \relax (2) + 15 \, e \log \relax (2)^{2} + 112 \, x e + 68 \, e \log \relax (2) + 76 \, e\right )}}{16 \, {\left (x + 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.07, size = 80, normalized size = 3.08
method | result | size |
default | \(\frac {{\mathrm e} \left (8 x^{2} \ln \relax (2)^{2}-32 x \ln \relax (2)^{2}+48 x^{2} \ln \relax (2)-160 x \ln \relax (2)+72 x^{2}-192 x -\frac {-81 \ln \relax (2)^{2}-324 \ln \relax (2)-324}{2 \left (x +1\right )^{2}}-\frac {108 \ln \relax (2)^{2}+468 \ln \relax (2)+504}{x +1}\right )}{8}\) | \(80\) |
gosper | \(\frac {\left (16 x^{4} \ln \relax (2)^{2}-32 x^{3} \ln \relax (2)^{2}+96 x^{4} \ln \relax (2)-128 x^{3} \ln \relax (2)+144 x^{4}-56 x \ln \relax (2)^{2}-96 x^{3}-23 \ln \relax (2)^{2}-168 x \ln \relax (2)-68 \ln \relax (2)-144 x -60\right ) {\mathrm e}}{16 x^{2}+32 x +16}\) | \(84\) |
norman | \(\frac {\left (-2 \,{\mathrm e} \ln \relax (2)^{2}-8 \,{\mathrm e} \ln \relax (2)-6 \,{\mathrm e}\right ) x^{3}+\left ({\mathrm e} \ln \relax (2)^{2}+6 \,{\mathrm e} \ln \relax (2)+9 \,{\mathrm e}\right ) x^{4}+\left (-\frac {7 \,{\mathrm e} \ln \relax (2)^{2}}{2}-\frac {21 \,{\mathrm e} \ln \relax (2)}{2}-9 \,{\mathrm e}\right ) x -\frac {23 \,{\mathrm e} \ln \relax (2)^{2}}{16}-\frac {17 \,{\mathrm e} \ln \relax (2)}{4}-\frac {15 \,{\mathrm e}}{4}}{\left (x +1\right )^{2}}\) | \(92\) |
risch | \(x^{2} {\mathrm e} \ln \relax (2)^{2}-4 \ln \relax (2)^{2} {\mathrm e} x +6 \ln \relax (2) {\mathrm e} x^{2}-20 x \,{\mathrm e} \ln \relax (2)+9 x^{2} {\mathrm e}-24 x \,{\mathrm e}+\frac {\left (-\frac {27 \,{\mathrm e} \ln \relax (2)^{2}}{2}-\frac {117 \,{\mathrm e} \ln \relax (2)}{2}-63 \,{\mathrm e}\right ) x -\frac {135 \,{\mathrm e} \ln \relax (2)^{2}}{16}-\frac {153 \,{\mathrm e} \ln \relax (2)}{4}-\frac {171 \,{\mathrm e}}{4}}{x^{2}+2 x +1}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.41, size = 97, normalized size = 3.73 \begin {gather*} {\left (e \log \relax (2)^{2} + 6 \, e \log \relax (2) + 9 \, e\right )} x^{2} - 4 \, {\left (e \log \relax (2)^{2} + 5 \, e \log \relax (2) + 6 \, e\right )} x - \frac {9 \, {\left (15 \, e \log \relax (2)^{2} + 8 \, {\left (3 \, e \log \relax (2)^{2} + 13 \, e \log \relax (2) + 14 \, e\right )} x + 68 \, e \log \relax (2) + 76 \, e\right )}}{16 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.20, size = 99, normalized size = 3.81 \begin {gather*} x^2\,\left (9\,\mathrm {e}+6\,\mathrm {e}\,\ln \relax (2)+\mathrm {e}\,{\ln \relax (2)}^2\right )-x\,\left (24\,\mathrm {e}+20\,\mathrm {e}\,\ln \relax (2)+4\,\mathrm {e}\,{\ln \relax (2)}^2\right )-\frac {342\,\mathrm {e}+306\,\mathrm {e}\,\ln \relax (2)+\frac {135\,\mathrm {e}\,{\ln \relax (2)}^2}{2}+x\,\left (504\,\mathrm {e}+468\,\mathrm {e}\,\ln \relax (2)+108\,\mathrm {e}\,{\ln \relax (2)}^2\right )}{8\,x^2+16\,x+8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.50, size = 112, normalized size = 4.31 \begin {gather*} x^{2} \left (e \log {\relax (2 )}^{2} + 6 e \log {\relax (2 )} + 9 e\right ) + x \left (- 24 e - 20 e \log {\relax (2 )} - 4 e \log {\relax (2 )}^{2}\right ) + \frac {x \left (- 1008 e - 936 e \log {\relax (2 )} - 216 e \log {\relax (2 )}^{2}\right ) - 684 e - 612 e \log {\relax (2 )} - 135 e \log {\relax (2 )}^{2}}{16 x^{2} + 32 x + 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________