Integrand size = 212, antiderivative size = 30 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=\frac {x-\frac {x^2}{\left (4+e^x\right )^2}}{1+\left (\frac {2}{x}-2 x\right )^2} \]
Time = 6.45 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=\frac {\left (16+8 e^x+e^{2 x}-x\right ) x^3}{\left (4+e^x\right )^2 \left (4-7 x^2+4 x^4\right )} \]
Integrate[(768*x^2 - 64*x^3 - 448*x^4 + 56*x^5 - 256*x^6 + E^(2*x)*(144*x^ 2 - 84*x^4 - 48*x^6) + E^(3*x)*(12*x^2 - 7*x^4 - 4*x^6) + E^x*(576*x^2 - 1 6*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(1024 - 3584*x^2 + 5184*x^4 - 3584*x^6 + 1024*x^8 + E^(3*x)*(16 - 56*x^2 + 81*x^4 - 56*x^6 + 16*x^8) + E^(2*x)*(192 - 672*x^2 + 972*x^4 - 672*x^6 + 192*x^8) + E^x*(768 - 2688*x^ 2 + 3888*x^4 - 2688*x^6 + 768*x^8)),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.
| \(\displaystyle \int \frac {-256 x^6+56 x^5-448 x^4-64 x^3+768 x^2+e^{2 x} \left (-48 x^6-84 x^4+144 x^2\right )+e^{3 x} \left (-4 x^6-7 x^4+12 x^2\right )+e^x \left (8 x^8-206 x^6+14 x^5-328 x^4-16 x^3+576 x^2\right )}{1024 x^8-3584 x^6+5184 x^4-3584 x^2+e^{3 x} \left (16 x^8-56 x^6+81 x^4-56 x^2+16\right )+e^{2 x} \left (192 x^8-672 x^6+972 x^4-672 x^2+192\right )+e^x \left (768 x^8-2688 x^6+3888 x^4-2688 x^2+768\right )+1024} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^2 \left (-12 e^{2 x} \left (4 x^4+7 x^2-12\right )-e^{3 x} \left (4 x^4+7 x^2-12\right )-8 \left (32 x^4-7 x^3+56 x^2+8 x-96\right )+2 e^x \left (4 x^6-103 x^4+7 x^3-164 x^2-8 x+288\right )\right )}{\left (e^x+4\right )^3 \left (4 x^4-7 x^2+4\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 x^4}{\left (e^x+4\right )^3 \left (4 x^4-7 x^2+4\right )}-\frac {\left (4 x^4+7 x^2-12\right ) x^2}{\left (4 x^4-7 x^2+4\right )^2}+\frac {2 \left (4 x^5-7 x^3+7 x^2+4 x-8\right ) x^3}{\left (e^x+4\right )^2 \left (4 x^4-7 x^2+4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3}{4 x^4-7 x^2+4}+\frac {7 \left (7 i+\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (\sqrt {7-i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}-\frac {32 i \int \frac {1}{\left (4+e^x\right )^3 \left (\sqrt {7-i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}-\frac {7 \left (7 i+\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7-i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{4 \sqrt {15 \left (7-i \sqrt {15}\right )}}+\frac {8 i \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7-i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}-\frac {7 i \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7-i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {30}}-\frac {7 \left (7 i-\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (\sqrt {7+i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}+\frac {32 i \int \frac {1}{\left (4+e^x\right )^3 \left (\sqrt {7+i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}+\frac {7 \left (7 i-\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7+i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{4 \sqrt {15 \left (7+i \sqrt {15}\right )}}-\frac {8 i \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7+i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}+\frac {7 i \int \frac {1}{\left (4+e^x\right )^2 \left (\sqrt {7+i \sqrt {15}}-2 \sqrt {2} x\right )}dx}{\sqrt {30}}+\frac {7 \left (7 i+\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (2 \sqrt {2} x+\sqrt {7-i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}-\frac {32 i \int \frac {1}{\left (4+e^x\right )^3 \left (2 \sqrt {2} x+\sqrt {7-i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}-\frac {7 \left (7 i+\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7-i \sqrt {15}}\right )}dx}{4 \sqrt {15 \left (7-i \sqrt {15}\right )}}+\frac {8 i \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7-i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7-i \sqrt {15}\right )}}+\frac {7 i \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7-i \sqrt {15}}\right )}dx}{\sqrt {30}}-\frac {7 \left (7 i-\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^3 \left (2 \sqrt {2} x+\sqrt {7+i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}+\frac {32 i \int \frac {1}{\left (4+e^x\right )^3 \left (2 \sqrt {2} x+\sqrt {7+i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}+\frac {7 \left (7 i-\sqrt {15}\right ) \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7+i \sqrt {15}}\right )}dx}{4 \sqrt {15 \left (7+i \sqrt {15}\right )}}-\frac {8 i \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7+i \sqrt {15}}\right )}dx}{\sqrt {15 \left (7+i \sqrt {15}\right )}}-\frac {7 i \int \frac {1}{\left (4+e^x\right )^2 \left (2 \sqrt {2} x+\sqrt {7+i \sqrt {15}}\right )}dx}{\sqrt {30}}-14 \int \frac {x}{\left (4+e^x\right )^2 \left (4 x^4-7 x^2+4\right )^2}dx+\frac {17}{2} \int \frac {x^3}{\left (4+e^x\right )^2 \left (4 x^4-7 x^2+4\right )^2}dx-\frac {1}{4 \left (4+e^x\right )^2}\) |
Int[(768*x^2 - 64*x^3 - 448*x^4 + 56*x^5 - 256*x^6 + E^(2*x)*(144*x^2 - 84 *x^4 - 48*x^6) + E^(3*x)*(12*x^2 - 7*x^4 - 4*x^6) + E^x*(576*x^2 - 16*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(1024 - 3584*x^2 + 5184*x^4 - 3584* x^6 + 1024*x^8 + E^(3*x)*(16 - 56*x^2 + 81*x^4 - 56*x^6 + 16*x^8) + E^(2*x )*(192 - 672*x^2 + 972*x^4 - 672*x^6 + 192*x^8) + E^x*(768 - 2688*x^2 + 38 88*x^4 - 2688*x^6 + 768*x^8)),x]
3.2.86.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.61 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(\frac {x^{3} \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x}-x +16\right )}{\left (4 x^{4}-7 x^{2}+4\right ) \left ({\mathrm e}^{x}+4\right )^{2}}\) | \(38\) |
| parallelrisch | \(\frac {4 \,{\mathrm e}^{2 x} x^{3}-4 x^{4}+32 \,{\mathrm e}^{x} x^{3}+64 x^{3}}{4 \left ({\mathrm e}^{2 x}+8 \,{\mathrm e}^{x}+16\right ) \left (4 x^{4}-7 x^{2}+4\right )}\) | \(56\) |
int(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^2+(8*x ^8-206*x^6+14*x^5-328*x^4-16*x^3+576*x^2)*exp(x)-256*x^6+56*x^5-448*x^4-64 *x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-672*x^6+ 972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+768)*exp (x)+1024*x^8-3584*x^6+5184*x^4-3584*x^2+1024),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.47 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=-\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{64 \, x^{4} - 112 \, x^{2} + {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{x} + 64} \]
integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^ 2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x^3+576*x^2)*exp(x)-256*x^6+56*x^5-448* x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-67 2*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+76 8)*exp(x)+1024*x^8-3584*x^6+5184*x^4-3584*x^2+1024),x, algorithm=\
-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(64*x^4 - 112*x^2 + (4*x^4 - 7*x ^2 + 4)*e^(2*x) + 8*(4*x^4 - 7*x^2 + 4)*e^x + 64)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=- \frac {64 x^{4}}{4096 x^{4} - 7168 x^{2} + \left (256 x^{4} - 448 x^{2} + 256\right ) e^{2 x} + \left (2048 x^{4} - 3584 x^{2} + 2048\right ) e^{x} + 4096} + \frac {x^{3}}{4 x^{4} - 7 x^{2} + 4} \]
integrate(((-4*x**6-7*x**4+12*x**2)*exp(x)**3+(-48*x**6-84*x**4+144*x**2)* exp(x)**2+(8*x**8-206*x**6+14*x**5-328*x**4-16*x**3+576*x**2)*exp(x)-256*x **6+56*x**5-448*x**4-64*x**3+768*x**2)/((16*x**8-56*x**6+81*x**4-56*x**2+1 6)*exp(x)**3+(192*x**8-672*x**6+972*x**4-672*x**2+192)*exp(x)**2+(768*x**8 -2688*x**6+3888*x**4-2688*x**2+768)*exp(x)+1024*x**8-3584*x**6+5184*x**4-3 584*x**2+1024),x)
-64*x**4/(4096*x**4 - 7168*x**2 + (256*x**4 - 448*x**2 + 256)*exp(2*x) + ( 2048*x**4 - 3584*x**2 + 2048)*exp(x) + 4096) + x**3/(4*x**4 - 7*x**2 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (29) = 58\).
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.47 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=-\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{64 \, x^{4} - 112 \, x^{2} + {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{\left (2 \, x\right )} + 8 \, {\left (4 \, x^{4} - 7 \, x^{2} + 4\right )} e^{x} + 64} \]
integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^ 2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x^3+576*x^2)*exp(x)-256*x^6+56*x^5-448* x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-67 2*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+76 8)*exp(x)+1024*x^8-3584*x^6+5184*x^4-3584*x^2+1024),x, algorithm=\
-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(64*x^4 - 112*x^2 + (4*x^4 - 7*x ^2 + 4)*e^(2*x) + 8*(4*x^4 - 7*x^2 + 4)*e^x + 64)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.77 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=-\frac {x^{4} - x^{3} e^{\left (2 \, x\right )} - 8 \, x^{3} e^{x} - 16 \, x^{3}}{4 \, x^{4} e^{\left (2 \, x\right )} + 32 \, x^{4} e^{x} + 64 \, x^{4} - 7 \, x^{2} e^{\left (2 \, x\right )} - 56 \, x^{2} e^{x} - 112 \, x^{2} + 4 \, e^{\left (2 \, x\right )} + 32 \, e^{x} + 64} \]
integrate(((-4*x^6-7*x^4+12*x^2)*exp(x)^3+(-48*x^6-84*x^4+144*x^2)*exp(x)^ 2+(8*x^8-206*x^6+14*x^5-328*x^4-16*x^3+576*x^2)*exp(x)-256*x^6+56*x^5-448* x^4-64*x^3+768*x^2)/((16*x^8-56*x^6+81*x^4-56*x^2+16)*exp(x)^3+(192*x^8-67 2*x^6+972*x^4-672*x^2+192)*exp(x)^2+(768*x^8-2688*x^6+3888*x^4-2688*x^2+76 8)*exp(x)+1024*x^8-3584*x^6+5184*x^4-3584*x^2+1024),x, algorithm=\
-(x^4 - x^3*e^(2*x) - 8*x^3*e^x - 16*x^3)/(4*x^4*e^(2*x) + 32*x^4*e^x + 64 *x^4 - 7*x^2*e^(2*x) - 56*x^2*e^x - 112*x^2 + 4*e^(2*x) + 32*e^x + 64)
Time = 0.68 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {768 x^2-64 x^3-448 x^4+56 x^5-256 x^6+e^{2 x} \left (144 x^2-84 x^4-48 x^6\right )+e^{3 x} \left (12 x^2-7 x^4-4 x^6\right )+e^x \left (576 x^2-16 x^3-328 x^4+14 x^5-206 x^6+8 x^8\right )}{1024-3584 x^2+5184 x^4-3584 x^6+1024 x^8+e^{3 x} \left (16-56 x^2+81 x^4-56 x^6+16 x^8\right )+e^{2 x} \left (192-672 x^2+972 x^4-672 x^6+192 x^8\right )+e^x \left (768-2688 x^2+3888 x^4-2688 x^6+768 x^8\right )} \, dx=\frac {x^3\,\left ({\mathrm {e}}^{2\,x}-x+8\,{\mathrm {e}}^x+16\right )}{{\left ({\mathrm {e}}^x+4\right )}^2\,\left (4\,x^4-7\,x^2+4\right )} \]
int(-(exp(3*x)*(7*x^4 - 12*x^2 + 4*x^6) + exp(2*x)*(84*x^4 - 144*x^2 + 48* x^6) - 768*x^2 + 64*x^3 + 448*x^4 - 56*x^5 + 256*x^6 - exp(x)*(576*x^2 - 1 6*x^3 - 328*x^4 + 14*x^5 - 206*x^6 + 8*x^8))/(exp(x)*(3888*x^4 - 2688*x^2 - 2688*x^6 + 768*x^8 + 768) + exp(3*x)*(81*x^4 - 56*x^2 - 56*x^6 + 16*x^8 + 16) + exp(2*x)*(972*x^4 - 672*x^2 - 672*x^6 + 192*x^8 + 192) - 3584*x^2 + 5184*x^4 - 3584*x^6 + 1024*x^8 + 1024),x)