Optimal. Leaf size=25 \[ -\frac {2 \left (e^5-x\right )}{\left (4+\frac {e^{4 x}}{x^8}\right ) x} \]
________________________________________________________________________________________
Rubi [F] time = 8.82, 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 {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{\left (e^{4 x}+4 x^8\right )^2} \, dx\\ &=\int \left (-\frac {\left (e^5-x\right ) (-2+x) x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {\left (e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{4 \left (e^{2 x}-2 e^x x^2+2 x^4\right )}-\frac {\left (e^5-x\right ) (-2+x) x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {\left (-e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{4 \left (e^{2 x}+2 e^x x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{4} \int \frac {\left (-e^x-2 x^2\right ) \left (e^5-\left (2+e^5\right ) x+x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\int \frac {\left (e^5-x\right ) (-2+x) x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\int \frac {\left (e^5-x\right ) (-2+x) x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {e^5 \left (e^x-2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {\left (2+e^5\right ) x \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}-\frac {x^2 \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} \int \left (-\frac {e^5 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {\left (2+e^5\right ) x \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}-\frac {x^2 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx-\int \left (-\frac {2 e^5 x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {\left (2+e^5\right ) x^5 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}-\frac {x^6 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx-\int \left (-\frac {2 e^5 x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {\left (2+e^5\right ) x^5 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}-\frac {x^6 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {x^2 \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx\right )-\frac {1}{4} \int \frac {x^2 \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{4} e^5 \int \frac {e^x-2 x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} e^5 \int \frac {e^x+2 x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\left (2 e^5\right ) \int \frac {x^4 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^4 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {x \left (-e^x+2 x^2\right )}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {x \left (e^x+2 x^2\right )}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2+e^5\right ) \int \frac {x^5 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {x^5 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^6 \left (-e^x+x^2\right )}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^6 \left (e^x+x^2\right )}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ &=-\left (\frac {1}{4} \int \left (-\frac {e^x x^2}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {2 x^4}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx\right )-\frac {1}{4} \int \left (\frac {e^x x^2}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^4}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} e^5 \int \left (\frac {e^x}{e^{2 x}-2 e^x x^2+2 x^4}-\frac {2 x^2}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx-\frac {1}{4} e^5 \int \left (\frac {e^x}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^2}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx+\left (2 e^5\right ) \int \left (-\frac {e^x x^4}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\left (2 e^5\right ) \int \left (\frac {e^x x^4}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\frac {1}{4} \left (2+e^5\right ) \int \left (-\frac {e^x x}{e^{2 x}-2 e^x x^2+2 x^4}+\frac {2 x^3}{e^{2 x}-2 e^x x^2+2 x^4}\right ) \, dx+\frac {1}{4} \left (2+e^5\right ) \int \left (\frac {e^x x}{e^{2 x}+2 e^x x^2+2 x^4}+\frac {2 x^3}{e^{2 x}+2 e^x x^2+2 x^4}\right ) \, dx-\left (2+e^5\right ) \int \left (-\frac {e^x x^5}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^7}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx-\left (2+e^5\right ) \int \left (\frac {e^x x^5}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^7}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\int \left (-\frac {e^x x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}+\frac {x^8}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2}\right ) \, dx+\int \left (\frac {e^x x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}+\frac {x^8}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {e^x x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} \int \frac {e^x x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\frac {1}{2} \int \frac {x^4}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{2} \int \frac {x^4}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{4} e^5 \int \frac {e^x}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{4} e^5 \int \frac {e^x}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\frac {1}{2} e^5 \int \frac {x^2}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\frac {1}{2} e^5 \int \frac {x^2}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2 e^5\right ) \int \frac {e^x x^4}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {e^x x^4}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\left (2 e^5\right ) \int \frac {x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (-2-e^5\right ) \int \frac {e^x x}{e^{2 x}-2 e^x x^2+2 x^4} \, dx-\left (-2-e^5\right ) \int \frac {e^x x^5}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {e^x x}{e^{2 x}+2 e^x x^2+2 x^4} \, dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{e^{2 x}-2 e^x x^2+2 x^4} \, dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{e^{2 x}+2 e^x x^2+2 x^4} \, dx-\left (2+e^5\right ) \int \frac {x^7}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {e^x x^5}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx-\left (2+e^5\right ) \int \frac {x^7}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx-\int \frac {e^x x^6}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{2 x}-2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {e^x x^6}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx+\int \frac {x^8}{\left (e^{2 x}+2 e^x x^2+2 x^4\right )^2} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 1.52, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \left (-e^5 x^7+x^8\right )}{e^{4 x}+4 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 24, normalized size = 0.96
method | result | size |
risch | \(-\frac {2 \left ({\mathrm e}^{5}-x \right ) x^{7}}{4 x^{8}+{\mathrm e}^{4 x}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.84, size = 50, normalized size = 2.00 \begin {gather*} -\frac {2\,\left (2\,x^{14}\,{\mathrm {e}}^5-x^{15}\,{\mathrm {e}}^5-2\,x^{15}+x^{16}\right )}{\left (2\,x^7-x^8\right )\,\left ({\mathrm {e}}^{4\,x}+4\,x^8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.12, size = 22, normalized size = 0.88 \begin {gather*} \frac {2 x^{8} - 2 x^{7} e^{5}}{4 x^{8} + e^{4 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________