Integrand size = 59, antiderivative size = 15 \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=\left (4 e^{-3 x}+e^x\right )^{20 x} \]
Time = 0.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=\left (4 e^{-3 x}+e^x\right )^{20 x} \]
Integrate[(4/E^(3*x) + E^x)^(-1 + 20*x)*((-240*x)/E^(3*x) + 20*E^x*x + (80 /E^(3*x) + 20*E^x)*Log[4/E^(3*x) + E^x]),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.
\(\displaystyle \int \left (4 e^{-3 x}+e^x\right )^{20 x-1} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \left (e^{-3 x} \left (e^{4 x}+4\right )\right )^{20 x-1} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-240 e^{-3 x} x \left (e^{-3 x} \left (e^{4 x}+4\right )\right )^{20 x-1}+20 e^x x \left (e^{-3 x} \left (e^{4 x}+4\right )\right )^{20 x-1}+20 e^{-3 x} \left (e^{4 x}+4\right ) \left (e^{-3 x} \left (e^{4 x}+4\right )\right )^{20 x-1} \log \left (e^{-3 x} \left (e^{4 x}+4\right )\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -240 \int e^{-3 x} \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x-1} xdx+20 \int e^x \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x-1} xdx-20 \int \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dxdx+40 \int \frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dx}{-2+2 e^x-e^{2 x}}dx+80 \int \frac {\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dx}{2-2 e^x+e^{2 x}}dx+80 \int \frac {\int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dx}{2+2 e^x+e^{2 x}}dx+40 \int \frac {e^x \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dx}{2+2 e^x+e^{2 x}}dx+20 \log \left (e^{-3 x} \left (e^{4 x}+4\right )\right ) \int \left (e^{-3 x} \left (4+e^{4 x}\right )\right )^{20 x}dx\) |
Int[(4/E^(3*x) + E^x)^(-1 + 20*x)*((-240*x)/E^(3*x) + 20*E^x*x + (80/E^(3* x) + 20*E^x)*Log[4/E^(3*x) + E^x]),x]
3.2.97.3.1 Defintions of rubi rules used
Time = 1.38 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \({\mathrm e}^{20 x \ln \left ({\mathrm e}^{x}+4 \,{\mathrm e}^{-3 x}\right )}\) | \(15\) |
risch | \(\left ({\mathrm e}^{x}\right )^{-60 x} \left ({\mathrm e}^{4 x}+4\right )^{20 x} {\mathrm e}^{10 i \pi x \left (\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )^{3}-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )-\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{3 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\operatorname {csgn}\left (i \left ({\mathrm e}^{4 x}+4\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{2}-\operatorname {csgn}\left (i \left ({\mathrm e}^{4 x}+4\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-3 x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-2 \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2}-\operatorname {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{3}+\operatorname {csgn}\left (i {\mathrm e}^{-3 x} \left ({\mathrm e}^{4 x}+4\right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-3 x}\right )\right )}\) | \(245\) |
int(((20*exp(x)+80*exp(-3*x))*ln(exp(x)+4*exp(-3*x))+20*exp(x)*x-240*x*exp (-3*x))*exp(20*x*ln(exp(x)+4*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x,method=_RE TURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=\left ({\left (e^{\left (4 \, x\right )} + 4\right )} e^{\left (-3 \, x\right )}\right )^{20 \, x} \]
integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-24 0*x*exp(-3*x))*exp(20*x*log(exp(x)+4*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, a lgorithm=\
Timed out. \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=\text {Timed out} \]
integrate(((20*exp(x)+80*exp(-3*x))*ln(exp(x)+4*exp(-3*x))+20*exp(x)*x-240 *x*exp(-3*x))*exp(20*x*ln(exp(x)+4*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (13) = 26\).
Time = 0.34 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.33 \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=e^{\left (-60 \, x^{2} + 20 \, x \log \left (e^{\left (2 \, x\right )} + 2 \, e^{x} + 2\right ) + 20 \, x \log \left (e^{\left (2 \, x\right )} - 2 \, e^{x} + 2\right )\right )} \]
integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-24 0*x*exp(-3*x))*exp(20*x*log(exp(x)+4*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, a lgorithm=\
\[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx=\int { -\frac {20 \, {\left (12 \, x e^{\left (-3 \, x\right )} - x e^{x} - {\left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )} \log \left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )\right )} {\left (4 \, e^{\left (-3 \, x\right )} + e^{x}\right )}^{20 \, x}}{4 \, e^{\left (-3 \, x\right )} + e^{x}} \,d x } \]
integrate(((20*exp(x)+80*exp(-3*x))*log(exp(x)+4*exp(-3*x))+20*exp(x)*x-24 0*x*exp(-3*x))*exp(20*x*log(exp(x)+4*exp(-3*x)))/(exp(x)+4*exp(-3*x)),x, a lgorithm=\
integrate(-20*(12*x*e^(-3*x) - x*e^x - (4*e^(-3*x) + e^x)*log(4*e^(-3*x) + e^x))*(4*e^(-3*x) + e^x)^(20*x)/(4*e^(-3*x) + e^x), x)
Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \left (4 e^{-3 x}+e^x\right )^{-1+20 x} \left (-240 e^{-3 x} x+20 e^x x+\left (80 e^{-3 x}+20 e^x\right ) \log \left (4 e^{-3 x}+e^x\right )\right ) \, dx={\left (4\,{\mathrm {e}}^{-3\,x}+{\mathrm {e}}^x\right )}^{20\,x} \]