Integrand size = 118, antiderivative size = 25 \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=-x+e^{4+x+8 x^2} \left (-3+\log \left (e^x+x\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=9 e^{4+x+8 x^2}-x-6 e^{4+x+8 x^2} \log \left (e^x+x\right )+e^{4+x+8 x^2} \log ^2\left (e^x+x\right ) \]
Integrate[(-E^x - x + E^(4 + x + 8*x^2)*(-6 + 9*x + 144*x^2 + E^x*(3 + 144 *x)) + E^(4 + x + 8*x^2)*(2 + E^x*(-4 - 96*x) - 6*x - 96*x^2)*Log[E^x + x] + E^(4 + x + 8*x^2)*(x + 16*x^2 + E^x*(1 + 16*x))*Log[E^x + x]^2)/(E^x + x),x]
9*E^(4 + x + 8*x^2) - x - 6*E^(4 + x + 8*x^2)*Log[E^x + x] + E^(4 + x + 8* x^2)*Log[E^x + x]^2
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.
\(\displaystyle \int \frac {e^{8 x^2+x+4} \left (144 x^2+9 x+e^x (144 x+3)-6\right )+e^{8 x^2+x+4} \left (16 x^2+x+e^x (16 x+1)\right ) \log ^2\left (x+e^x\right )+e^{8 x^2+x+4} \left (-96 x^2-6 x+e^x (-96 x-4)+2\right ) \log \left (x+e^x\right )-e^x-x}{x+e^x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{8 x^2+x+4} \left (\log \left (x+e^x\right )-3\right ) \left (-48 x^2+16 x^2 \log \left (x+e^x\right )-48 e^x x-3 x-e^x+16 e^x x \log \left (x+e^x\right )+x \log \left (x+e^x\right )+e^x \log \left (x+e^x\right )+2\right )}{x+e^x}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} e^{127/32} \sqrt {\frac {\pi }{2}} \int \frac {\text {erfi}\left (2 \sqrt {2} x+\frac {1}{4 \sqrt {2}}\right )}{-x-e^x}dx+\frac {1}{2} e^{127/32} \sqrt {\frac {\pi }{2}} \int \frac {x \text {erfi}\left (2 \sqrt {2} x+\frac {1}{4 \sqrt {2}}\right )}{x+e^x}dx-2 \int \int \frac {e^{8 x^2+x+4}}{x+e^x}dxdx-2 \int \frac {\int \frac {e^{8 x^2+x+4}}{x+e^x}dx}{x+e^x}dx+2 \int \frac {x \int \frac {e^{8 x^2+x+4}}{x+e^x}dx}{x+e^x}dx+2 \int \int \frac {e^{8 x^2+x+4} x}{x+e^x}dxdx+2 \int \frac {\int \frac {e^{8 x^2+x+4} x}{x+e^x}dx}{x+e^x}dx-2 \int \frac {x \int \frac {e^{8 x^2+x+4} x}{x+e^x}dx}{x+e^x}dx+\int e^{8 x^2+x+4} \log ^2\left (x+e^x\right )dx+16 \int e^{8 x^2+x+4} x \log ^2\left (x+e^x\right )dx+2 \log \left (x+e^x\right ) \int \frac {e^{8 x^2+x+4}}{x+e^x}dx-2 \log \left (x+e^x\right ) \int \frac {e^{8 x^2+x+4} x}{x+e^x}dx-\frac {1}{32} e^{127/32} \sqrt {\frac {\pi }{2}} (16 x+1) \text {erfi}\left (2 \sqrt {2} x+\frac {1}{4 \sqrt {2}}\right )+\frac {1}{2} e^{127/32} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {16 x+1}{4 \sqrt {2}}\right ) \log \left (x+e^x\right )+9 e^{8 x^2+x+4}-6 e^{8 x^2+x+4} \log \left (x+e^x\right )-x+\frac {1}{8} e^{\left (2 \sqrt {2} x+\frac {1}{4 \sqrt {2}}\right )^2+\frac {127}{32}}\) |
Int[(-E^x - x + E^(4 + x + 8*x^2)*(-6 + 9*x + 144*x^2 + E^x*(3 + 144*x)) + E^(4 + x + 8*x^2)*(2 + E^x*(-4 - 96*x) - 6*x - 96*x^2)*Log[E^x + x] + E^( 4 + x + 8*x^2)*(x + 16*x^2 + E^x*(1 + 16*x))*Log[E^x + x]^2)/(E^x + x),x]
3.1.35.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(23)=46\).
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96
method | result | size |
risch | \({\mathrm e}^{8 x^{2}+x +4} \ln \left ({\mathrm e}^{x}+x \right )^{2}-6 \,{\mathrm e}^{8 x^{2}+x +4} \ln \left ({\mathrm e}^{x}+x \right )-x +9 \,{\mathrm e}^{8 x^{2}+x +4}\) | \(49\) |
parallelrisch | \({\mathrm e}^{8 x^{2}+x +4} \ln \left ({\mathrm e}^{x}+x \right )^{2}-6 \,{\mathrm e}^{8 x^{2}+x +4} \ln \left ({\mathrm e}^{x}+x \right )-x +9 \,{\mathrm e}^{8 x^{2}+x +4}\) | \(49\) |
int((((16*x+1)*exp(x)+16*x^2+x)*exp(8*x^2+x+4)*ln(exp(x)+x)^2+((-96*x-4)*e xp(x)-96*x^2-6*x+2)*exp(8*x^2+x+4)*ln(exp(x)+x)+((144*x+3)*exp(x)+144*x^2+ 9*x-6)*exp(8*x^2+x+4)-exp(x)-x)/(exp(x)+x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right )^{2} - 6 \, e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right ) - x + 9 \, e^{\left (8 \, x^{2} + x + 4\right )} \]
integrate((((16*x+1)*exp(x)+16*x^2+x)*exp(8*x^2+x+4)*log(exp(x)+x)^2+((-96 *x-4)*exp(x)-96*x^2-6*x+2)*exp(8*x^2+x+4)*log(exp(x)+x)+((144*x+3)*exp(x)+ 144*x^2+9*x-6)*exp(8*x^2+x+4)-exp(x)-x)/(exp(x)+x),x, algorithm=\
e^(8*x^2 + x + 4)*log(x + e^x)^2 - 6*e^(8*x^2 + x + 4)*log(x + e^x) - x + 9*e^(8*x^2 + x + 4)
Timed out. \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=\text {Timed out} \]
integrate((((16*x+1)*exp(x)+16*x**2+x)*exp(8*x**2+x+4)*ln(exp(x)+x)**2+((- 96*x-4)*exp(x)-96*x**2-6*x+2)*exp(8*x**2+x+4)*ln(exp(x)+x)+((144*x+3)*exp( x)+144*x**2+9*x-6)*exp(8*x**2+x+4)-exp(x)-x)/(exp(x)+x),x)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (23) = 46\).
Time = 0.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.92 \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right )^{2} - 6 \, e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right ) - x + 9 \, e^{\left (8 \, x^{2} + x + 4\right )} \]
integrate((((16*x+1)*exp(x)+16*x^2+x)*exp(8*x^2+x+4)*log(exp(x)+x)^2+((-96 *x-4)*exp(x)-96*x^2-6*x+2)*exp(8*x^2+x+4)*log(exp(x)+x)+((144*x+3)*exp(x)+ 144*x^2+9*x-6)*exp(8*x^2+x+4)-exp(x)-x)/(exp(x)+x),x, algorithm=\
e^(8*x^2 + x + 4)*log(x + e^x)^2 - 6*e^(8*x^2 + x + 4)*log(x + e^x) - x + 9*e^(8*x^2 + x + 4)
\[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=\int { \frac {{\left (16 \, x^{2} + {\left (16 \, x + 1\right )} e^{x} + x\right )} e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right )^{2} - 2 \, {\left (48 \, x^{2} + 2 \, {\left (24 \, x + 1\right )} e^{x} + 3 \, x - 1\right )} e^{\left (8 \, x^{2} + x + 4\right )} \log \left (x + e^{x}\right ) + 3 \, {\left (48 \, x^{2} + {\left (48 \, x + 1\right )} e^{x} + 3 \, x - 2\right )} e^{\left (8 \, x^{2} + x + 4\right )} - x - e^{x}}{x + e^{x}} \,d x } \]
integrate((((16*x+1)*exp(x)+16*x^2+x)*exp(8*x^2+x+4)*log(exp(x)+x)^2+((-96 *x-4)*exp(x)-96*x^2-6*x+2)*exp(8*x^2+x+4)*log(exp(x)+x)+((144*x+3)*exp(x)+ 144*x^2+9*x-6)*exp(8*x^2+x+4)-exp(x)-x)/(exp(x)+x),x, algorithm=\
integrate(((16*x^2 + (16*x + 1)*e^x + x)*e^(8*x^2 + x + 4)*log(x + e^x)^2 - 2*(48*x^2 + 2*(24*x + 1)*e^x + 3*x - 1)*e^(8*x^2 + x + 4)*log(x + e^x) + 3*(48*x^2 + (48*x + 1)*e^x + 3*x - 2)*e^(8*x^2 + x + 4) - x - e^x)/(x + e ^x), x)
Timed out. \[ \int \frac {-e^x-x+e^{4+x+8 x^2} \left (-6+9 x+144 x^2+e^x (3+144 x)\right )+e^{4+x+8 x^2} \left (2+e^x (-4-96 x)-6 x-96 x^2\right ) \log \left (e^x+x\right )+e^{4+x+8 x^2} \left (x+16 x^2+e^x (1+16 x)\right ) \log ^2\left (e^x+x\right )}{e^x+x} \, dx=-\int \frac {-{\mathrm {e}}^{8\,x^2+x+4}\,\left (x+{\mathrm {e}}^x\,\left (16\,x+1\right )+16\,x^2\right )\,{\ln \left (x+{\mathrm {e}}^x\right )}^2+{\mathrm {e}}^{8\,x^2+x+4}\,\left (6\,x+{\mathrm {e}}^x\,\left (96\,x+4\right )+96\,x^2-2\right )\,\ln \left (x+{\mathrm {e}}^x\right )+x+{\mathrm {e}}^x-{\mathrm {e}}^{8\,x^2+x+4}\,\left (9\,x+{\mathrm {e}}^x\,\left (144\,x+3\right )+144\,x^2-6\right )}{x+{\mathrm {e}}^x} \,d x \]
int(-(x + exp(x) - exp(x + 8*x^2 + 4)*(9*x + exp(x)*(144*x + 3) + 144*x^2 - 6) + log(x + exp(x))*exp(x + 8*x^2 + 4)*(6*x + exp(x)*(96*x + 4) + 96*x^ 2 - 2) - log(x + exp(x))^2*exp(x + 8*x^2 + 4)*(x + exp(x)*(16*x + 1) + 16* x^2))/(x + exp(x)),x)