Optimal. Leaf size=26 \[ \log \left (4+2 e^{2 x}+\frac {1}{x}+\frac {4}{e^x+256 x^4}\right ) \]
________________________________________________________________________________________
Rubi [F] time = 4.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-e^{2 x}-4096 x^5-65536 x^8+e^x \left (-4 x^2-512 x^4\right )+2 e^{2 x} \left (2 e^{2 x} x^2+1024 e^x x^6+131072 x^{10}\right )}{1024 x^6+65536 x^9+262144 x^{10}+e^{2 x} \left (x+4 x^2\right )+e^x \left (4 x^2+512 x^5+2048 x^6\right )+2 e^{2 x} \left (e^{2 x} x^2+512 e^x x^6+65536 x^{10}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {256 (-4+x) x^3}{e^x+256 x^4}-\frac {e^x+2 e^x x+12 x^2+8 e^x x^2-768 x^4-3328 x^5-2048 e^{2 x} x^5+3072 x^6+512 e^{2 x} x^6}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )}\right ) \, dx\\ &=2 x+256 \int \frac {(-4+x) x^3}{e^x+256 x^4} \, dx-\int \frac {e^x+2 e^x x+12 x^2+8 e^x x^2-768 x^4-3328 x^5-2048 e^{2 x} x^5+3072 x^6+512 e^{2 x} x^6}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )} \, dx\\ &=2 x+256 \int \left (-\frac {4 x^3}{e^x+256 x^4}+\frac {x^4}{e^x+256 x^4}\right ) \, dx-\int \left (\frac {2 e^x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {e^x}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )}+\frac {12 x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {8 e^x x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {768 x^3}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {3328 x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}-\frac {2048 e^{2 x} x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {3072 x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}+\frac {512 e^{2 x} x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5}\right ) \, dx\\ &=2 x-2 \int \frac {e^x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-8 \int \frac {e^x x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-12 \int \frac {x}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+256 \int \frac {x^4}{e^x+256 x^4} \, dx-512 \int \frac {e^{2 x} x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+768 \int \frac {x^3}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-1024 \int \frac {x^3}{e^x+256 x^4} \, dx+2048 \int \frac {e^{2 x} x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-3072 \int \frac {x^5}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx+3328 \int \frac {x^4}{e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5} \, dx-\int \frac {e^x}{x \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right )} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [B] time = 0.15, size = 59, normalized size = 2.27 \begin {gather*} -\log (x)-\log \left (e^x+256 x^4\right )+\log \left (e^x+4 x+4 e^x x+2 e^{3 x} x+256 x^4+1024 x^5+512 e^{2 x} x^5\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.70, size = 55, normalized size = 2.12 \begin {gather*} -\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.85, size = 54, normalized size = 2.08 \begin {gather*} \log \left (512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + 4 \, x e^{x} + 4 \, x + e^{x}\right ) - \log \left (256 \, x^{4} + e^{x}\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 51, normalized size = 1.96
method | result | size |
risch | \(-\ln \left ({\mathrm e}^{x}+256 x^{4}\right )+\ln \left ({\mathrm e}^{3 x}+256 \,{\mathrm e}^{2 x} x^{4}+\frac {\left (4 x +1\right ) {\mathrm e}^{x}}{2 x}+512 x^{4}+128 x^{3}+2\right )\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.44, size = 56, normalized size = 2.15 \begin {gather*} -\log \left (256 \, x^{4} + e^{x}\right ) + \log \left (\frac {512 \, x^{5} e^{\left (2 \, x\right )} + 1024 \, x^{5} + 256 \, x^{4} + 2 \, x e^{\left (3 \, x\right )} + {\left (4 \, x + 1\right )} e^{x} + 4 \, x}{2 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (512\,x^4+4\,x^2\right )-{\mathrm {e}}^{2\,x+\ln \relax (2)}\,\left (1024\,x^6\,{\mathrm {e}}^x+2\,x^2\,{\mathrm {e}}^{2\,x}+131072\,x^{10}\right )+4096\,x^5+65536\,x^8}{{\mathrm {e}}^x\,\left (2048\,x^6+512\,x^5+4\,x^2\right )+{\mathrm {e}}^{2\,x+\ln \relax (2)}\,\left (512\,x^6\,{\mathrm {e}}^x+x^2\,{\mathrm {e}}^{2\,x}+65536\,x^{10}\right )+{\mathrm {e}}^{2\,x}\,\left (4\,x^2+x\right )+1024\,x^6+65536\,x^9+262144\,x^{10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 1.70, size = 49, normalized size = 1.88 \begin {gather*} - \log {\left (256 x^{4} + e^{x} \right )} + \log {\left (256 x^{4} e^{2 x} + 512 x^{4} + 128 x^{3} + e^{3 x} + 2 + \frac {\left (4 x + 1\right ) e^{x}}{2 x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________