Integrand size = 157, antiderivative size = 29 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=4 \left (x^3-\left (-1+e^{2/x}-2 x\right )^4 (3+2 x)^2\right ) \] Output:
4*x^3-4*(3+2*x)^2*(exp(2/x)-1-2*x)^4
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(29)=58\).
Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.83 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=-4 \left (84 x+316 x^2+607 x^3+624 x^4+320 x^5+64 x^6+e^{8/x} (3+2 x)^2-4 e^{6/x} (1+2 x) (3+2 x)^2-4 e^{2/x} (1+2 x)^3 (3+2 x)^2+6 e^{4/x} \left (3+8 x+4 x^2\right )^2\right ) \] Input:
Integrate[(-336*x^2 - 2528*x^3 - 7284*x^4 - 9984*x^5 - 6400*x^6 - 1536*x^7 + E^(8/x)*(288 + 384*x + 80*x^2 - 32*x^3) + E^(6/x)*(-864 - 2880*x - 2208 *x^2 + 128*x^3 + 384*x^4) + E^(4/x)*(864 + 4608*x + 7296*x^2 + 1920*x^3 - 3072*x^4 - 1536*x^5) + E^(2/x)*(-288 - 2112*x - 4832*x^2 - 1792*x^3 + 6912 *x^4 + 8192*x^5 + 2560*x^6))/x^2,x]
Output:
-4*(84*x + 316*x^2 + 607*x^3 + 624*x^4 + 320*x^5 + 64*x^6 + E^(8/x)*(3 + 2 *x)^2 - 4*E^(6/x)*(1 + 2*x)*(3 + 2*x)^2 - 4*E^(2/x)*(1 + 2*x)^3*(3 + 2*x)^ 2 + 6*E^(4/x)*(3 + 8*x + 4*x^2)^2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.61 (sec) , antiderivative size = 217, normalized size of antiderivative = 7.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {2010, 2009}
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.
| \(\displaystyle \int \frac {-1536 x^7-6400 x^6-9984 x^5-7284 x^4-2528 x^3-336 x^2+e^{8/x} \left (-32 x^3+80 x^2+384 x+288\right )+e^{6/x} \left (384 x^4+128 x^3-2208 x^2-2880 x-864\right )+e^{4/x} \left (-1536 x^5-3072 x^4+1920 x^3+7296 x^2+4608 x+864\right )+e^{2/x} \left (2560 x^6+8192 x^5+6912 x^4-1792 x^3-4832 x^2-2112 x-288\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (-\frac {16 e^{8/x} (2 x+3) \left (x^2-4 x-6\right )}{x^2}+\frac {32 e^{2/x} (2 x+3) \left (10 x^3+7 x^2-8 x-3\right ) (2 x+1)^2}{x^2}-\frac {96 e^{4/x} (2 x+3) \left (4 x^3-8 x-3\right ) (2 x+1)}{x^2}+\frac {32 e^{6/x} (2 x+3) \left (6 x^3-7 x^2-24 x-9\right )}{x^2}-4 \left (384 x^5+1600 x^4+2496 x^3+1821 x^2+632 x+84\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 6144 \operatorname {ExpIntegralEi}\left (\frac {2}{x}\right )-16384 \operatorname {ExpIntegralEi}\left (\frac {4}{x}\right )-256 x^6-1280 x^5-2496 x^4+2304 e^{2/x} x^3-1024 e^{4/x} x^3+128 e^{6/x} x^3-2428 x^3+1408 e^{2/x} x^2-1088 e^{4/x} x^2+448 e^{6/x} x^2-1264 x^2-2016 e^{2/x} x+2944 e^{4/x} x+480 e^{6/x} x-336 x+144 e^{2/x}-216 e^{4/x}+144 e^{6/x}-4 e^{8/x} (2 x+3)^2-81920 \Gamma \left (-5,-\frac {2}{x}\right )-393216 \Gamma \left (-4,-\frac {4}{x}\right )+131072 \Gamma \left (-4,-\frac {2}{x}\right )\) |
Input:
Int[(-336*x^2 - 2528*x^3 - 7284*x^4 - 9984*x^5 - 6400*x^6 - 1536*x^7 + E^( 8/x)*(288 + 384*x + 80*x^2 - 32*x^3) + E^(6/x)*(-864 - 2880*x - 2208*x^2 + 128*x^3 + 384*x^4) + E^(4/x)*(864 + 4608*x + 7296*x^2 + 1920*x^3 - 3072*x ^4 - 1536*x^5) + E^(2/x)*(-288 - 2112*x - 4832*x^2 - 1792*x^3 + 6912*x^4 + 8192*x^5 + 2560*x^6))/x^2,x]
Output:
144*E^(2/x) - 216*E^(4/x) + 144*E^(6/x) - 336*x - 2016*E^(2/x)*x + 2944*E^ (4/x)*x + 480*E^(6/x)*x - 1264*x^2 + 1408*E^(2/x)*x^2 - 1088*E^(4/x)*x^2 + 448*E^(6/x)*x^2 - 2428*x^3 + 2304*E^(2/x)*x^3 - 1024*E^(4/x)*x^3 + 128*E^ (6/x)*x^3 - 2496*x^4 - 1280*x^5 - 256*x^6 - 4*E^(8/x)*(3 + 2*x)^2 + 6144*E xpIntegralEi[2/x] - 16384*ExpIntegralEi[4/x] - 81920*Gamma[-5, -2/x] - 393 216*Gamma[-4, -4/x] + 131072*Gamma[-4, -2/x]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(28)=56\).
Time = 7.67 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.41
| method | result | size |
| risch | \(-256 x^{6}-1280 x^{5}-2496 x^{4}-2428 x^{3}-1264 x^{2}-336 x +\left (-16 x^{2}-48 x -36\right ) {\mathrm e}^{\frac {8}{x}}+\left (128 x^{3}+448 x^{2}+480 x +144\right ) {\mathrm e}^{\frac {6}{x}}+\left (-384 x^{4}-1536 x^{3}-2112 x^{2}-1152 x -216\right ) {\mathrm e}^{\frac {4}{x}}+\left (512 x^{5}+2304 x^{4}+3840 x^{3}+2944 x^{2}+1056 x +144\right ) {\mathrm e}^{\frac {2}{x}}\) | \(128\) |
| derivativedivides | \(512 \,{\mathrm e}^{\frac {2}{x}} x^{5}-256 x^{6}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}-384 \,{\mathrm e}^{\frac {4}{x}} x^{4}-1280 x^{5}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}-2496 x^{4}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}-2112 \,{\mathrm e}^{\frac {4}{x}} x^{2}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}-2428 x^{3}+1056 x \,{\mathrm e}^{\frac {2}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}-48 \,{\mathrm e}^{\frac {8}{x}} x +480 \,{\mathrm e}^{\frac {6}{x}} x -1264 x^{2}+144 \,{\mathrm e}^{\frac {2}{x}}-216 \,{\mathrm e}^{\frac {4}{x}}-36 \,{\mathrm e}^{\frac {8}{x}}+144 \,{\mathrm e}^{\frac {6}{x}}-336 x\) | \(232\) |
| default | \(512 \,{\mathrm e}^{\frac {2}{x}} x^{5}-256 x^{6}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}-384 \,{\mathrm e}^{\frac {4}{x}} x^{4}-1280 x^{5}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}-2496 x^{4}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}-2112 \,{\mathrm e}^{\frac {4}{x}} x^{2}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}-2428 x^{3}+1056 x \,{\mathrm e}^{\frac {2}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}-48 \,{\mathrm e}^{\frac {8}{x}} x +480 \,{\mathrm e}^{\frac {6}{x}} x -1264 x^{2}+144 \,{\mathrm e}^{\frac {2}{x}}-216 \,{\mathrm e}^{\frac {4}{x}}-36 \,{\mathrm e}^{\frac {8}{x}}+144 \,{\mathrm e}^{\frac {6}{x}}-336 x\) | \(232\) |
| parallelrisch | \(512 \,{\mathrm e}^{\frac {2}{x}} x^{5}-256 x^{6}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}-384 \,{\mathrm e}^{\frac {4}{x}} x^{4}-1280 x^{5}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}-2496 x^{4}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}-2112 \,{\mathrm e}^{\frac {4}{x}} x^{2}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}-2428 x^{3}+1056 x \,{\mathrm e}^{\frac {2}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}-48 \,{\mathrm e}^{\frac {8}{x}} x +480 \,{\mathrm e}^{\frac {6}{x}} x -1264 x^{2}+144 \,{\mathrm e}^{\frac {2}{x}}-216 \,{\mathrm e}^{\frac {4}{x}}-36 \,{\mathrm e}^{\frac {8}{x}}+144 \,{\mathrm e}^{\frac {6}{x}}-336 x\) | \(232\) |
| parts | \(512 \,{\mathrm e}^{\frac {2}{x}} x^{5}-256 x^{6}+2304 \,{\mathrm e}^{\frac {2}{x}} x^{4}-384 \,{\mathrm e}^{\frac {4}{x}} x^{4}-1280 x^{5}+3840 x^{3} {\mathrm e}^{\frac {2}{x}}-1536 \,{\mathrm e}^{\frac {4}{x}} x^{3}+128 \,{\mathrm e}^{\frac {6}{x}} x^{3}-2496 x^{4}+2944 x^{2} {\mathrm e}^{\frac {2}{x}}-2112 \,{\mathrm e}^{\frac {4}{x}} x^{2}-16 \,{\mathrm e}^{\frac {8}{x}} x^{2}+448 \,{\mathrm e}^{\frac {6}{x}} x^{2}-2428 x^{3}+1056 x \,{\mathrm e}^{\frac {2}{x}}-1152 x \,{\mathrm e}^{\frac {4}{x}}-48 \,{\mathrm e}^{\frac {8}{x}} x +480 \,{\mathrm e}^{\frac {6}{x}} x -1264 x^{2}+144 \,{\mathrm e}^{\frac {2}{x}}-216 \,{\mathrm e}^{\frac {4}{x}}-36 \,{\mathrm e}^{\frac {8}{x}}+144 \,{\mathrm e}^{\frac {6}{x}}-336 x\) | \(232\) |
| orering | \(\text {Expression too large to display}\) | \(7772\) |
Input:
int(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880* x-864)*exp(2/x)^3+(-1536*x^5-3072*x^4+1920*x^3+7296*x^2+4608*x+864)*exp(2/ x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2/x)-15 36*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x,method=_RETURNVE RBOSE)
Output:
-256*x^6-1280*x^5-2496*x^4-2428*x^3-1264*x^2-336*x+(-16*x^2-48*x-36)*exp(8 /x)+(128*x^3+448*x^2+480*x+144)*exp(6/x)+(-384*x^4-1536*x^3-2112*x^2-1152* x-216)*exp(4/x)+(512*x^5+2304*x^4+3840*x^3+2944*x^2+1056*x+144)*exp(2/x)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.52 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=-256 \, x^{6} - 1280 \, x^{5} - 2496 \, x^{4} - 2428 \, x^{3} - 1264 \, x^{2} - 4 \, {\left (4 \, x^{2} + 12 \, x + 9\right )} e^{\frac {8}{x}} + 16 \, {\left (8 \, x^{3} + 28 \, x^{2} + 30 \, x + 9\right )} e^{\frac {6}{x}} - 24 \, {\left (16 \, x^{4} + 64 \, x^{3} + 88 \, x^{2} + 48 \, x + 9\right )} e^{\frac {4}{x}} + 16 \, {\left (32 \, x^{5} + 144 \, x^{4} + 240 \, x^{3} + 184 \, x^{2} + 66 \, x + 9\right )} e^{\frac {2}{x}} - 336 \, x \] Input:
integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2 -2880*x-864)*exp(2/x)^3+(-1536*x^5-3072*x^4+1920*x^3+7296*x^2+4608*x+864)* exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2 /x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm ="fricas")
Output:
-256*x^6 - 1280*x^5 - 2496*x^4 - 2428*x^3 - 1264*x^2 - 4*(4*x^2 + 12*x + 9 )*e^(8/x) + 16*(8*x^3 + 28*x^2 + 30*x + 9)*e^(6/x) - 24*(16*x^4 + 64*x^3 + 88*x^2 + 48*x + 9)*e^(4/x) + 16*(32*x^5 + 144*x^4 + 240*x^3 + 184*x^2 + 6 6*x + 9)*e^(2/x) - 336*x
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.21 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=- 256 x^{6} - 1280 x^{5} - 2496 x^{4} - 2428 x^{3} - 1264 x^{2} - 336 x + \left (- 16 x^{2} - 48 x - 36\right ) e^{\frac {8}{x}} + \left (128 x^{3} + 448 x^{2} + 480 x + 144\right ) e^{\frac {6}{x}} + \left (- 384 x^{4} - 1536 x^{3} - 2112 x^{2} - 1152 x - 216\right ) e^{\frac {4}{x}} + \left (512 x^{5} + 2304 x^{4} + 3840 x^{3} + 2944 x^{2} + 1056 x + 144\right ) e^{\frac {2}{x}} \] Input:
integrate(((-32*x**3+80*x**2+384*x+288)*exp(2/x)**4+(384*x**4+128*x**3-220 8*x**2-2880*x-864)*exp(2/x)**3+(-1536*x**5-3072*x**4+1920*x**3+7296*x**2+4 608*x+864)*exp(2/x)**2+(2560*x**6+8192*x**5+6912*x**4-1792*x**3-4832*x**2- 2112*x-288)*exp(2/x)-1536*x**7-6400*x**6-9984*x**5-7284*x**4-2528*x**3-336 *x**2)/x**2,x)
Output:
-256*x**6 - 1280*x**5 - 2496*x**4 - 2428*x**3 - 1264*x**2 - 336*x + (-16*x **2 - 48*x - 36)*exp(8/x) + (128*x**3 + 448*x**2 + 480*x + 144)*exp(6/x) + (-384*x**4 - 1536*x**3 - 2112*x**2 - 1152*x - 216)*exp(4/x) + (512*x**5 + 2304*x**4 + 3840*x**3 + 2944*x**2 + 1056*x + 144)*exp(2/x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 219, normalized size of antiderivative = 7.55 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=-256 \, x^{6} - 1280 \, x^{5} - 2496 \, x^{4} - 2428 \, x^{3} - 1264 \, x^{2} - 336 \, x - 384 \, {\rm Ei}\left (\frac {8}{x}\right ) + 2880 \, {\rm Ei}\left (\frac {6}{x}\right ) - 4608 \, {\rm Ei}\left (\frac {4}{x}\right ) + 2112 \, {\rm Ei}\left (\frac {2}{x}\right ) - 36 \, e^{\frac {8}{x}} + 144 \, e^{\frac {6}{x}} - 216 \, e^{\frac {4}{x}} + 144 \, e^{\frac {2}{x}} + 9664 \, \Gamma \left (-1, -\frac {2}{x}\right ) - 29184 \, \Gamma \left (-1, -\frac {4}{x}\right ) + 13248 \, \Gamma \left (-1, -\frac {6}{x}\right ) - 640 \, \Gamma \left (-1, -\frac {8}{x}\right ) - 7168 \, \Gamma \left (-2, -\frac {2}{x}\right ) + 30720 \, \Gamma \left (-2, -\frac {4}{x}\right ) + 4608 \, \Gamma \left (-2, -\frac {6}{x}\right ) - 2048 \, \Gamma \left (-2, -\frac {8}{x}\right ) - 55296 \, \Gamma \left (-3, -\frac {2}{x}\right ) + 196608 \, \Gamma \left (-3, -\frac {4}{x}\right ) - 82944 \, \Gamma \left (-3, -\frac {6}{x}\right ) + 131072 \, \Gamma \left (-4, -\frac {2}{x}\right ) - 393216 \, \Gamma \left (-4, -\frac {4}{x}\right ) - 81920 \, \Gamma \left (-5, -\frac {2}{x}\right ) \] Input:
integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2 -2880*x-864)*exp(2/x)^3+(-1536*x^5-3072*x^4+1920*x^3+7296*x^2+4608*x+864)* exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2 /x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm ="maxima")
Output:
-256*x^6 - 1280*x^5 - 2496*x^4 - 2428*x^3 - 1264*x^2 - 336*x - 384*Ei(8/x) + 2880*Ei(6/x) - 4608*Ei(4/x) + 2112*Ei(2/x) - 36*e^(8/x) + 144*e^(6/x) - 216*e^(4/x) + 144*e^(2/x) + 9664*gamma(-1, -2/x) - 29184*gamma(-1, -4/x) + 13248*gamma(-1, -6/x) - 640*gamma(-1, -8/x) - 7168*gamma(-2, -2/x) + 307 20*gamma(-2, -4/x) + 4608*gamma(-2, -6/x) - 2048*gamma(-2, -8/x) - 55296*g amma(-3, -2/x) + 196608*gamma(-3, -4/x) - 82944*gamma(-3, -6/x) + 131072*g amma(-4, -2/x) - 393216*gamma(-4, -4/x) - 81920*gamma(-5, -2/x)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 230, normalized size of antiderivative = 7.93 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=4 \, x^{6} {\left (\frac {128 \, e^{\frac {2}{x}}}{x} - \frac {320}{x} - \frac {96 \, e^{\frac {4}{x}}}{x^{2}} + \frac {576 \, e^{\frac {2}{x}}}{x^{2}} - \frac {624}{x^{2}} + \frac {32 \, e^{\frac {6}{x}}}{x^{3}} - \frac {384 \, e^{\frac {4}{x}}}{x^{3}} + \frac {960 \, e^{\frac {2}{x}}}{x^{3}} - \frac {607}{x^{3}} - \frac {4 \, e^{\frac {8}{x}}}{x^{4}} + \frac {112 \, e^{\frac {6}{x}}}{x^{4}} - \frac {528 \, e^{\frac {4}{x}}}{x^{4}} + \frac {736 \, e^{\frac {2}{x}}}{x^{4}} - \frac {316}{x^{4}} - \frac {12 \, e^{\frac {8}{x}}}{x^{5}} + \frac {120 \, e^{\frac {6}{x}}}{x^{5}} - \frac {288 \, e^{\frac {4}{x}}}{x^{5}} + \frac {264 \, e^{\frac {2}{x}}}{x^{5}} - \frac {84}{x^{5}} - \frac {9 \, e^{\frac {8}{x}}}{x^{6}} + \frac {36 \, e^{\frac {6}{x}}}{x^{6}} - \frac {54 \, e^{\frac {4}{x}}}{x^{6}} + \frac {36 \, e^{\frac {2}{x}}}{x^{6}} - 64\right )} \] Input:
integrate(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2 -2880*x-864)*exp(2/x)^3+(-1536*x^5-3072*x^4+1920*x^3+7296*x^2+4608*x+864)* exp(2/x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2 /x)-1536*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x, algorithm ="giac")
Output:
4*x^6*(128*e^(2/x)/x - 320/x - 96*e^(4/x)/x^2 + 576*e^(2/x)/x^2 - 624/x^2 + 32*e^(6/x)/x^3 - 384*e^(4/x)/x^3 + 960*e^(2/x)/x^3 - 607/x^3 - 4*e^(8/x) /x^4 + 112*e^(6/x)/x^4 - 528*e^(4/x)/x^4 + 736*e^(2/x)/x^4 - 316/x^4 - 12* e^(8/x)/x^5 + 120*e^(6/x)/x^5 - 288*e^(4/x)/x^5 + 264*e^(2/x)/x^5 - 84/x^5 - 9*e^(8/x)/x^6 + 36*e^(6/x)/x^6 - 54*e^(4/x)/x^6 + 36*e^(2/x)/x^6 - 64)
Time = 3.58 (sec) , antiderivative size = 207, normalized size of antiderivative = 7.14 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=144\,{\mathrm {e}}^{2/x}-336\,x-216\,{\mathrm {e}}^{4/x}+144\,{\mathrm {e}}^{6/x}-36\,{\mathrm {e}}^{8/x}+1056\,x\,{\mathrm {e}}^{2/x}-1152\,x\,{\mathrm {e}}^{4/x}+480\,x\,{\mathrm {e}}^{6/x}-48\,x\,{\mathrm {e}}^{8/x}+2944\,x^2\,{\mathrm {e}}^{2/x}+3840\,x^3\,{\mathrm {e}}^{2/x}-2112\,x^2\,{\mathrm {e}}^{4/x}+2304\,x^4\,{\mathrm {e}}^{2/x}-1536\,x^3\,{\mathrm {e}}^{4/x}+512\,x^5\,{\mathrm {e}}^{2/x}+448\,x^2\,{\mathrm {e}}^{6/x}-384\,x^4\,{\mathrm {e}}^{4/x}+128\,x^3\,{\mathrm {e}}^{6/x}-16\,x^2\,{\mathrm {e}}^{8/x}-1264\,x^2-2428\,x^3-2496\,x^4-1280\,x^5-256\,x^6 \] Input:
int(-(exp(2/x)*(2112*x + 4832*x^2 + 1792*x^3 - 6912*x^4 - 8192*x^5 - 2560* x^6 + 288) - exp(4/x)*(4608*x + 7296*x^2 + 1920*x^3 - 3072*x^4 - 1536*x^5 + 864) - exp(8/x)*(384*x + 80*x^2 - 32*x^3 + 288) + exp(6/x)*(2880*x + 220 8*x^2 - 128*x^3 - 384*x^4 + 864) + 336*x^2 + 2528*x^3 + 7284*x^4 + 9984*x^ 5 + 6400*x^6 + 1536*x^7)/x^2,x)
Output:
144*exp(2/x) - 336*x - 216*exp(4/x) + 144*exp(6/x) - 36*exp(8/x) + 1056*x* exp(2/x) - 1152*x*exp(4/x) + 480*x*exp(6/x) - 48*x*exp(8/x) + 2944*x^2*exp (2/x) + 3840*x^3*exp(2/x) - 2112*x^2*exp(4/x) + 2304*x^4*exp(2/x) - 1536*x ^3*exp(4/x) + 512*x^5*exp(2/x) + 448*x^2*exp(6/x) - 384*x^4*exp(4/x) + 128 *x^3*exp(6/x) - 16*x^2*exp(8/x) - 1264*x^2 - 2428*x^3 - 2496*x^4 - 1280*x^ 5 - 256*x^6
Time = 0.17 (sec) , antiderivative size = 225, normalized size of antiderivative = 7.76 \[ \int \frac {-336 x^2-2528 x^3-7284 x^4-9984 x^5-6400 x^6-1536 x^7+e^{8/x} \left (288+384 x+80 x^2-32 x^3\right )+e^{6/x} \left (-864-2880 x-2208 x^2+128 x^3+384 x^4\right )+e^{4/x} \left (864+4608 x+7296 x^2+1920 x^3-3072 x^4-1536 x^5\right )+e^{2/x} \left (-288-2112 x-4832 x^2-1792 x^3+6912 x^4+8192 x^5+2560 x^6\right )}{x^2} \, dx=-16 e^{\frac {8}{x}} x^{2}-48 e^{\frac {8}{x}} x -36 e^{\frac {8}{x}}+128 e^{\frac {6}{x}} x^{3}+448 e^{\frac {6}{x}} x^{2}+480 e^{\frac {6}{x}} x +144 e^{\frac {6}{x}}-384 e^{\frac {4}{x}} x^{4}-1536 e^{\frac {4}{x}} x^{3}-2112 e^{\frac {4}{x}} x^{2}-1152 e^{\frac {4}{x}} x -216 e^{\frac {4}{x}}+512 e^{\frac {2}{x}} x^{5}+2304 e^{\frac {2}{x}} x^{4}+3840 e^{\frac {2}{x}} x^{3}+2944 e^{\frac {2}{x}} x^{2}+1056 e^{\frac {2}{x}} x +144 e^{\frac {2}{x}}-256 x^{6}-1280 x^{5}-2496 x^{4}-2428 x^{3}-1264 x^{2}-336 x \] Input:
int(((-32*x^3+80*x^2+384*x+288)*exp(2/x)^4+(384*x^4+128*x^3-2208*x^2-2880* x-864)*exp(2/x)^3+(-1536*x^5-3072*x^4+1920*x^3+7296*x^2+4608*x+864)*exp(2/ x)^2+(2560*x^6+8192*x^5+6912*x^4-1792*x^3-4832*x^2-2112*x-288)*exp(2/x)-15 36*x^7-6400*x^6-9984*x^5-7284*x^4-2528*x^3-336*x^2)/x^2,x)
Output:
4*( - 4*e**(8/x)*x**2 - 12*e**(8/x)*x - 9*e**(8/x) + 32*e**(6/x)*x**3 + 11 2*e**(6/x)*x**2 + 120*e**(6/x)*x + 36*e**(6/x) - 96*e**(4/x)*x**4 - 384*e* *(4/x)*x**3 - 528*e**(4/x)*x**2 - 288*e**(4/x)*x - 54*e**(4/x) + 128*e**(2 /x)*x**5 + 576*e**(2/x)*x**4 + 960*e**(2/x)*x**3 + 736*e**(2/x)*x**2 + 264 *e**(2/x)*x + 36*e**(2/x) - 64*x**6 - 320*x**5 - 624*x**4 - 607*x**3 - 316 *x**2 - 84*x)