Integrand size = 173, antiderivative size = 28 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {(-4+x)^2+x}{e^{4+x}+x+x^3}\right )} \]
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \]
Integrate[(48*x + 141*x^3 - 42*x^4 + 3*x^5 + E^(4 + x)*(69*x - 27*x^2 + 3* x^3) + (-48*x + 21*x^2 - 51*x^3 + 21*x^4 - 3*x^5 + E^(4 + x)*(-48 + 21*x - 3*x^2))*Log[(16 - 7*x + x^2)/(E^(4 + x) + x + x^3)])/((16*x^3 - 7*x^4 + 1 7*x^5 - 7*x^6 + x^7 + E^(4 + x)*(16*x^2 - 7*x^3 + x^4))*Log[(16 - 7*x + x^ 2)/(E^(4 + x) + x + x^3)]^2),x]
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.
\(\displaystyle \int \frac {3 x^5-42 x^4+141 x^3+e^{x+4} \left (3 x^3-27 x^2+69 x\right )+\left (-3 x^5+21 x^4-51 x^3+21 x^2+e^{x+4} \left (-3 x^2+21 x-48\right )-48 x\right ) \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )+48 x}{\left (x^7-7 x^6+17 x^5-7 x^4+16 x^3+e^{x+4} \left (x^4-7 x^3+16 x^2\right )\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {3 \left (\frac {x \left (x^4-14 x^3+47 x^2+e^{x+4} \left (x^2-9 x+23\right )+16\right )}{\left (x^2-7 x+16\right ) \left (x^3+x+e^{x+4}\right )}-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )\right )}{x^2 \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {\frac {x \left (x^4-14 x^3+47 x^2+e^{x+4} \left (x^2-9 x+23\right )+16\right )}{\left (x^2-7 x+16\right ) \left (x^3+x+e^{x+4}\right )}-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}{x^2 \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 3 \int \left (\frac {x^3-\log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right ) x^2-9 x^2+7 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right ) x+23 x-16 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}{x^2 \left (x^2-7 x+16\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}-\frac {x^3-3 x^2+x-1}{x \left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\frac {17 i \int \frac {1}{\left (-2 x+i \sqrt {15}+7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx}{8 \sqrt {15}}+\frac {23}{16} \int \frac {1}{x \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\frac {7}{240} \left (15-7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x-i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\frac {7}{240} \left (15+7 i \sqrt {15}\right ) \int \frac {1}{\left (2 x+i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+\frac {17 i \int \frac {1}{\left (2 x+i \sqrt {15}-7\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx}{8 \sqrt {15}}-\int \frac {1}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+\int \frac {1}{x \left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx+3 \int \frac {x}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\int \frac {x^2}{\left (x^3+x+e^{x+4}\right ) \log ^2\left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx-\int \frac {1}{x^2 \log \left (\frac {x^2-7 x+16}{x^3+x+e^{x+4}}\right )}dx\right )\) |
Int[(48*x + 141*x^3 - 42*x^4 + 3*x^5 + E^(4 + x)*(69*x - 27*x^2 + 3*x^3) + (-48*x + 21*x^2 - 51*x^3 + 21*x^4 - 3*x^5 + E^(4 + x)*(-48 + 21*x - 3*x^2 ))*Log[(16 - 7*x + x^2)/(E^(4 + x) + x + x^3)])/((16*x^3 - 7*x^4 + 17*x^5 - 7*x^6 + x^7 + E^(4 + x)*(16*x^2 - 7*x^3 + x^4))*Log[(16 - 7*x + x^2)/(E^ (4 + x) + x + x^3)]^2),x]
3.29.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 7.57 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {3}{\ln \left (\frac {x^{2}-7 x +16}{{\mathrm e}^{4+x}+x^{3}+x}\right ) x}\) | \(29\) |
risch | \(\frac {6 i}{x \left (\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4+x}+x^{3}+x}\right ) \operatorname {csgn}\left (i \left (x^{2}-7 x +16\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4+x}+x^{3}+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{2}-7 x +16\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-7 x +16\right )}{{\mathrm e}^{4+x}+x^{3}+x}\right )}^{3}+2 i \ln \left (x^{2}-7 x +16\right )-2 i \ln \left ({\mathrm e}^{4+x}+x^{3}+x \right )\right )}\) | \(197\) |
int((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*ln((x^2-7 *x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+141*x^3 +48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ln((x^2 -7*x+16)/(exp(4+x)+x^3+x))^2,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {x^{2} - 7 \, x + 16}{x^{3} + x + e^{\left (x + 4\right )}}\right )} \]
integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log ((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm=\
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log {\left (\frac {x^{2} - 7 x + 16}{x^{3} + x + e^{x + 4}} \right )}} \]
integrate((((-3*x**2+21*x-48)*exp(4+x)-3*x**5+21*x**4-51*x**3+21*x**2-48*x )*ln((x**2-7*x+16)/(exp(4+x)+x**3+x))+(3*x**3-27*x**2+69*x)*exp(4+x)+3*x** 5-42*x**4+141*x**3+48*x)/((x**4-7*x**3+16*x**2)*exp(4+x)+x**7-7*x**6+17*x* *5-7*x**4+16*x**3)/ln((x**2-7*x+16)/(exp(4+x)+x**3+x))**2,x)
Time = 0.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=-\frac {3}{x \log \left (x^{3} + x + e^{\left (x + 4\right )}\right ) - x \log \left (x^{2} - 7 \, x + 16\right )} \]
integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log ((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm=\
Time = 0.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x \log \left (\frac {x^{2} - 7 \, x + 16}{x^{3} + x + e^{\left (x + 4\right )}}\right )} \]
integrate((((-3*x^2+21*x-48)*exp(4+x)-3*x^5+21*x^4-51*x^3+21*x^2-48*x)*log ((x^2-7*x+16)/(exp(4+x)+x^3+x))+(3*x^3-27*x^2+69*x)*exp(4+x)+3*x^5-42*x^4+ 141*x^3+48*x)/((x^4-7*x^3+16*x^2)*exp(4+x)+x^7-7*x^6+17*x^5-7*x^4+16*x^3)/ log((x^2-7*x+16)/(exp(4+x)+x^3+x))^2,x, algorithm=\
Time = 12.55 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {48 x+141 x^3-42 x^4+3 x^5+e^{4+x} \left (69 x-27 x^2+3 x^3\right )+\left (-48 x+21 x^2-51 x^3+21 x^4-3 x^5+e^{4+x} \left (-48+21 x-3 x^2\right )\right ) \log \left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )}{\left (16 x^3-7 x^4+17 x^5-7 x^6+x^7+e^{4+x} \left (16 x^2-7 x^3+x^4\right )\right ) \log ^2\left (\frac {16-7 x+x^2}{e^{4+x}+x+x^3}\right )} \, dx=\frac {3}{x\,\ln \left (\frac {x^2-7\,x+16}{x+{\mathrm {e}}^4\,{\mathrm {e}}^x+x^3}\right )} \]
int((48*x + exp(x + 4)*(69*x - 27*x^2 + 3*x^3) - log((x^2 - 7*x + 16)/(x + exp(x + 4) + x^3))*(48*x + exp(x + 4)*(3*x^2 - 21*x + 48) - 21*x^2 + 51*x ^3 - 21*x^4 + 3*x^5) + 141*x^3 - 42*x^4 + 3*x^5)/(log((x^2 - 7*x + 16)/(x + exp(x + 4) + x^3))^2*(exp(x + 4)*(16*x^2 - 7*x^3 + x^4) + 16*x^3 - 7*x^4 + 17*x^5 - 7*x^6 + x^7)),x)