Integrand size = 116, antiderivative size = 25 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=x^2 \left (9+e^x-2 \left (3+\frac {1}{4 x^2}+x\right )^2\right )^2 \] Output:
x^2*(9+exp(x)-2*(3+1/4/x^2+x)^2)^2
Leaf count is larger than twice the leaf count of optimal. \(101\) vs. \(2(25)=50\).
Time = 0.55 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=\frac {1}{64 x^6}+\frac {3}{4 x^4}+\frac {1}{4 x^3}+\frac {45}{4 x^2}+\frac {9}{x}+90 x+117 x^2+e^{2 x} x^2+220 x^3+180 x^4+48 x^5+4 x^6-\frac {1}{4} e^x \left (24+\frac {1}{x^2}+8 x+72 x^2+96 x^3+16 x^4\right ) \] Input:
Integrate[(-3 - 96*x^2 - 24*x^3 - 720*x^4 - 288*x^5 + 2880*x^7 + 7488*x^8 + 21120*x^9 + 23040*x^10 + 7680*x^11 + 768*x^12 + E^(2*x)*(64*x^8 + 64*x^9 ) + E^x*(16*x^4 - 8*x^5 - 256*x^7 - 1216*x^8 - 2880*x^9 - 1280*x^10 - 128* x^11))/(32*x^7),x]
Output:
1/(64*x^6) + 3/(4*x^4) + 1/(4*x^3) + 45/(4*x^2) + 9/x + 90*x + 117*x^2 + E ^(2*x)*x^2 + 220*x^3 + 180*x^4 + 48*x^5 + 4*x^6 - (E^x*(24 + x^(-2) + 8*x + 72*x^2 + 96*x^3 + 16*x^4))/4
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(25)=50\).
Time = 0.55 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {768 x^{12}+7680 x^{11}+23040 x^{10}+21120 x^9+7488 x^8+2880 x^7-288 x^5-720 x^4-24 x^3-96 x^2+e^{2 x} \left (64 x^9+64 x^8\right )+e^x \left (-128 x^{11}-1280 x^{10}-2880 x^9-1216 x^8-256 x^7-8 x^5+16 x^4\right )-3}{32 x^7} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{32} \int -\frac {-768 x^{12}-7680 x^{11}-23040 x^{10}-21120 x^9-7488 x^8-2880 x^7+288 x^5+720 x^4+24 x^3+96 x^2-64 e^{2 x} \left (x^9+x^8\right )-8 e^x \left (-16 x^{11}-160 x^{10}-360 x^9-152 x^8-32 x^7-x^5+2 x^4\right )+3}{x^7}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{32} \int \frac {-768 x^{12}-7680 x^{11}-23040 x^{10}-21120 x^9-7488 x^8-2880 x^7+288 x^5+720 x^4+24 x^3+96 x^2-64 e^{2 x} \left (x^9+x^8\right )-8 e^x \left (-16 x^{11}-160 x^{10}-360 x^9-152 x^8-32 x^7-x^5+2 x^4\right )+3}{x^7}dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle -\frac {1}{32} \int \left (-64 e^{2 x} x (x+1)+\frac {8 e^x \left (16 x^7+160 x^6+360 x^5+152 x^4+32 x^3+x-2\right )}{x^3}-\frac {3 \left (256 x^{12}+2560 x^{11}+7680 x^{10}+7040 x^9+2496 x^8+960 x^7-96 x^5-240 x^4-8 x^3-32 x^2-1\right )}{x^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{32} \left (128 x^6+\frac {1}{2 x^6}+1536 x^5-128 e^x x^4+5760 x^4+\frac {24}{x^4}-768 e^x x^3+7040 x^3+\frac {8}{x^3}-576 e^x x^2+32 e^{2 x} x^2+3744 x^2-\frac {8 e^x}{x^2}+\frac {360}{x^2}-64 e^x x+2880 x-192 e^x+\frac {288}{x}\right )\) |
Input:
Int[(-3 - 96*x^2 - 24*x^3 - 720*x^4 - 288*x^5 + 2880*x^7 + 7488*x^8 + 2112 0*x^9 + 23040*x^10 + 7680*x^11 + 768*x^12 + E^(2*x)*(64*x^8 + 64*x^9) + E^ x*(16*x^4 - 8*x^5 - 256*x^7 - 1216*x^8 - 2880*x^9 - 1280*x^10 - 128*x^11)) /(32*x^7),x]
Output:
(-192*E^x + 1/(2*x^6) + 24/x^4 + 8/x^3 + 360/x^2 - (8*E^x)/x^2 + 288/x + 2 880*x - 64*E^x*x + 3744*x^2 - 576*E^x*x^2 + 32*E^(2*x)*x^2 + 7040*x^3 - 76 8*E^x*x^3 + 5760*x^4 - 128*E^x*x^4 + 1536*x^5 + 128*x^6)/32
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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. \(98\) vs. \(2(22)=44\).
Time = 4.01 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96
method | result | size |
risch | \(4 x^{6}+48 x^{5}+180 x^{4}+220 x^{3}+117 x^{2}+90 x +\frac {288 x^{5}+360 x^{4}+8 x^{3}+24 x^{2}+\frac {1}{2}}{32 x^{6}}+{\mathrm e}^{2 x} x^{2}-\frac {\left (16 x^{6}+96 x^{5}+72 x^{4}+8 x^{3}+24 x^{2}+1\right ) {\mathrm e}^{x}}{4 x^{2}}\) | \(99\) |
default | \(117 x^{2}+90 x +\frac {1}{64 x^{6}}+\frac {3}{4 x^{4}}+\frac {1}{4 x^{3}}+\frac {45}{4 x^{2}}+\frac {9}{x}+220 x^{3}+180 x^{4}+48 x^{5}+4 x^{6}-\frac {{\mathrm e}^{x}}{4 x^{2}}-2 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}-18 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}+{\mathrm e}^{2 x} x^{2}\) | \(100\) |
parts | \(117 x^{2}+90 x +\frac {1}{64 x^{6}}+\frac {3}{4 x^{4}}+\frac {1}{4 x^{3}}+\frac {45}{4 x^{2}}+\frac {9}{x}+220 x^{3}+180 x^{4}+48 x^{5}+4 x^{6}-\frac {{\mathrm e}^{x}}{4 x^{2}}-2 \,{\mathrm e}^{x} x -6 \,{\mathrm e}^{x}-18 \,{\mathrm e}^{x} x^{2}-24 \,{\mathrm e}^{x} x^{3}-4 \,{\mathrm e}^{x} x^{4}+{\mathrm e}^{2 x} x^{2}\) | \(100\) |
parallelrisch | \(\frac {256 x^{12}+3072 x^{11}-256 \,{\mathrm e}^{x} x^{10}+11520 x^{10}-1536 x^{9} {\mathrm e}^{x}+64 \,{\mathrm e}^{2 x} x^{8}+14080 x^{9}-1152 x^{8} {\mathrm e}^{x}+7488 x^{8}-128 x^{7} {\mathrm e}^{x}+5760 x^{7}-384 x^{6} {\mathrm e}^{x}+1+576 x^{5}-16 \,{\mathrm e}^{x} x^{4}+720 x^{4}+16 x^{3}+48 x^{2}}{64 x^{6}}\) | \(109\) |
orering | \(\text {Expression too large to display}\) | \(1574\) |
Input:
int(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8- 256*x^7-8*x^5+16*x^4)*exp(x)+768*x^12+7680*x^11+23040*x^10+21120*x^9+7488* x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x,method=_RETURNVERBOSE)
Output:
4*x^6+48*x^5+180*x^4+220*x^3+117*x^2+90*x+1/32*(288*x^5+360*x^4+8*x^3+24*x ^2+1/2)/x^6+exp(x)^2*x^2-1/4*(16*x^6+96*x^5+72*x^4+8*x^3+24*x^2+1)/x^2*exp (x)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (23) = 46\).
Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.96 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=\frac {256 \, x^{12} + 3072 \, x^{11} + 11520 \, x^{10} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} + 7488 \, x^{8} + 5760 \, x^{7} + 576 \, x^{5} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} - 16 \, {\left (16 \, x^{10} + 96 \, x^{9} + 72 \, x^{8} + 8 \, x^{7} + 24 \, x^{6} + x^{4}\right )} e^{x} + 1}{64 \, x^{6}} \] Input:
integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-121 6*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+768*x^12+7680*x^11+23040*x^10+21120*x^9 +7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="fric as")
Output:
1/64*(256*x^12 + 3072*x^11 + 11520*x^10 + 14080*x^9 + 64*x^8*e^(2*x) + 748 8*x^8 + 5760*x^7 + 576*x^5 + 720*x^4 + 16*x^3 + 48*x^2 - 16*(16*x^10 + 96* x^9 + 72*x^8 + 8*x^7 + 24*x^6 + x^4)*e^x + 1)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.08 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=4 x^{6} + 48 x^{5} + 180 x^{4} + 220 x^{3} + 117 x^{2} + 90 x + \frac {4 x^{4} e^{2 x} + \left (- 16 x^{6} - 96 x^{5} - 72 x^{4} - 8 x^{3} - 24 x^{2} - 1\right ) e^{x}}{4 x^{2}} + \frac {576 x^{5} + 720 x^{4} + 16 x^{3} + 48 x^{2} + 1}{64 x^{6}} \] Input:
integrate(1/32*((64*x**9+64*x**8)*exp(x)**2+(-128*x**11-1280*x**10-2880*x* *9-1216*x**8-256*x**7-8*x**5+16*x**4)*exp(x)+768*x**12+7680*x**11+23040*x* *10+21120*x**9+7488*x**8+2880*x**7-288*x**5-720*x**4-24*x**3-96*x**2-3)/x* *7,x)
Output:
4*x**6 + 48*x**5 + 180*x**4 + 220*x**3 + 117*x**2 + 90*x + (4*x**4*exp(2*x ) + (-16*x**6 - 96*x**5 - 72*x**4 - 8*x**3 - 24*x**2 - 1)*exp(x))/(4*x**2) + (576*x**5 + 720*x**4 + 16*x**3 + 48*x**2 + 1)/(64*x**6)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.28 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=4 \, x^{6} + 48 \, x^{5} + 180 \, x^{4} + 220 \, x^{3} + 117 \, x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} - 40 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} - 90 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 38 \, {\left (x - 1\right )} e^{x} + 90 \, x + \frac {9}{x} + \frac {45}{4 \, x^{2}} + \frac {1}{4 \, x^{3}} + \frac {3}{4 \, x^{4}} + \frac {1}{64 \, x^{6}} - 8 \, e^{x} - \frac {1}{4} \, \Gamma \left (-1, -x\right ) - \frac {1}{2} \, \Gamma \left (-2, -x\right ) \] Input:
integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-121 6*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+768*x^12+7680*x^11+23040*x^10+21120*x^9 +7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="maxi ma")
Output:
4*x^6 + 48*x^5 + 180*x^4 + 220*x^3 + 117*x^2 + 1/2*(2*x^2 - 2*x + 1)*e^(2* x) + 1/2*(2*x - 1)*e^(2*x) - 4*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x - 40 *(x^3 - 3*x^2 + 6*x - 6)*e^x - 90*(x^2 - 2*x + 2)*e^x - 38*(x - 1)*e^x + 9 0*x + 9/x + 45/4/x^2 + 1/4/x^3 + 3/4/x^4 + 1/64/x^6 - 8*e^x - 1/4*gamma(-1 , -x) - 1/2*gamma(-2, -x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (23) = 46\).
Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.32 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=\frac {256 \, x^{12} + 3072 \, x^{11} - 256 \, x^{10} e^{x} + 11520 \, x^{10} - 1536 \, x^{9} e^{x} + 14080 \, x^{9} + 64 \, x^{8} e^{\left (2 \, x\right )} - 1152 \, x^{8} e^{x} + 7488 \, x^{8} - 128 \, x^{7} e^{x} + 5760 \, x^{7} - 384 \, x^{6} e^{x} + 576 \, x^{5} - 16 \, x^{4} e^{x} + 720 \, x^{4} + 16 \, x^{3} + 48 \, x^{2} + 1}{64 \, x^{6}} \] Input:
integrate(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-121 6*x^8-256*x^7-8*x^5+16*x^4)*exp(x)+768*x^12+7680*x^11+23040*x^10+21120*x^9 +7488*x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x, algorithm="giac ")
Output:
1/64*(256*x^12 + 3072*x^11 - 256*x^10*e^x + 11520*x^10 - 1536*x^9*e^x + 14 080*x^9 + 64*x^8*e^(2*x) - 1152*x^8*e^x + 7488*x^8 - 128*x^7*e^x + 5760*x^ 7 - 384*x^6*e^x + 576*x^5 - 16*x^4*e^x + 720*x^4 + 16*x^3 + 48*x^2 + 1)/x^ 6
Time = 3.86 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.68 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=\frac {\frac {3\,x^2}{4}-x^4\,\left (\frac {{\mathrm {e}}^x}{4}-\frac {45}{4}\right )+\frac {x^3}{4}+9\,x^5+\frac {1}{64}}{x^6}-6\,{\mathrm {e}}^x+x^2\,\left ({\mathrm {e}}^{2\,x}-18\,{\mathrm {e}}^x+117\right )-x\,\left (2\,{\mathrm {e}}^x-90\right )-x^4\,\left (4\,{\mathrm {e}}^x-180\right )-x^3\,\left (24\,{\mathrm {e}}^x-220\right )+48\,x^5+4\,x^6 \] Input:
int(((exp(2*x)*(64*x^8 + 64*x^9))/32 - (exp(x)*(8*x^5 - 16*x^4 + 256*x^7 + 1216*x^8 + 2880*x^9 + 1280*x^10 + 128*x^11))/32 - 3*x^2 - (3*x^3)/4 - (45 *x^4)/2 - 9*x^5 + 90*x^7 + 234*x^8 + 660*x^9 + 720*x^10 + 240*x^11 + 24*x^ 12 - 3/32)/x^7,x)
Output:
((3*x^2)/4 - x^4*(exp(x)/4 - 45/4) + x^3/4 + 9*x^5 + 1/64)/x^6 - 6*exp(x) + x^2*(exp(2*x) - 18*exp(x) + 117) - x*(2*exp(x) - 90) - x^4*(4*exp(x) - 1 80) - x^3*(24*exp(x) - 220) + 48*x^5 + 4*x^6
Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 4.60 \[ \int \frac {-3-96 x^2-24 x^3-720 x^4-288 x^5+2880 x^7+7488 x^8+21120 x^9+23040 x^{10}+7680 x^{11}+768 x^{12}+e^{2 x} \left (64 x^8+64 x^9\right )+e^x \left (16 x^4-8 x^5-256 x^7-1216 x^8-2880 x^9-1280 x^{10}-128 x^{11}\right )}{32 x^7} \, dx=\frac {64 e^{2 x} x^{8}-256 e^{x} x^{10}-1536 e^{x} x^{9}-1152 e^{x} x^{8}-128 e^{x} x^{7}-384 e^{x} x^{6}-16 e^{x} x^{4}+256 x^{12}+3072 x^{11}+11520 x^{10}+14080 x^{9}+7488 x^{8}+5760 x^{7}+576 x^{5}+720 x^{4}+16 x^{3}+48 x^{2}+1}{64 x^{6}} \] Input:
int(1/32*((64*x^9+64*x^8)*exp(x)^2+(-128*x^11-1280*x^10-2880*x^9-1216*x^8- 256*x^7-8*x^5+16*x^4)*exp(x)+768*x^12+7680*x^11+23040*x^10+21120*x^9+7488* x^8+2880*x^7-288*x^5-720*x^4-24*x^3-96*x^2-3)/x^7,x)
Output:
(64*e**(2*x)*x**8 - 256*e**x*x**10 - 1536*e**x*x**9 - 1152*e**x*x**8 - 128 *e**x*x**7 - 384*e**x*x**6 - 16*e**x*x**4 + 256*x**12 + 3072*x**11 + 11520 *x**10 + 14080*x**9 + 7488*x**8 + 5760*x**7 + 576*x**5 + 720*x**4 + 16*x** 3 + 48*x**2 + 1)/(64*x**6)