Integrand size = 194, antiderivative size = 27 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {\left (2+4 e^{2 x}+\frac {8 x}{9}\right )^4 x^2}{\left (-4+x^2\right )^4} \] Output:
x^2*(8/9*x+4*exp(x)^2+2)^4/(x^2-4)^4
Time = 12.54 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {16 x^2 \left (9+18 e^{2 x}+4 x\right )^4}{6561 \left (-4+x^2\right )^4} \] Input:
Integrate[(-839808*x - 2239488*x^2 - 2620512*x^3 - 1670400*x^4 - 595968*x^ 5 - 110592*x^6 - 8192*x^7 + E^(8*x)*(-13436928*x - 53747712*x^2 - 10077696 *x^3 + 13436928*x^4) + E^(6*x)*(-26873856*x - 98537472*x^2 - 55987200*x^3 + 12690432*x^4 + 8957952*x^5) + E^(4*x)*(-20155392*x - 67184640*x^2 - 5891 0976*x^3 - 9082368*x^4 + 6967296*x^5 + 1990656*x^6) + E^(2*x)*(-6718464*x - 20155392*x^2 - 21959424*x^3 - 9374976*x^4 - 340992*x^5 + 774144*x^6 + 14 7456*x^7))/(-6718464 + 8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^ 8 + 6561*x^10),x]
Output:
(16*x^2*(9 + 18*E^(2*x) + 4*x)^4)/(6561*(-4 + x^2)^4)
Leaf count is larger than twice the leaf count of optimal. \(514\) vs. \(2(27)=54\).
Time = 3.74 (sec) , antiderivative size = 514, normalized size of antiderivative = 19.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {2070, 7239, 27, 7293, 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.
| \(\displaystyle \int \frac {-8192 x^7-110592 x^6-595968 x^5-1670400 x^4-2620512 x^3-2239488 x^2+e^{8 x} \left (13436928 x^4-10077696 x^3-53747712 x^2-13436928 x\right )+e^{6 x} \left (8957952 x^5+12690432 x^4-55987200 x^3-98537472 x^2-26873856 x\right )+e^{4 x} \left (1990656 x^6+6967296 x^5-9082368 x^4-58910976 x^3-67184640 x^2-20155392 x\right )+e^{2 x} \left (147456 x^7+774144 x^6-340992 x^5-9374976 x^4-21959424 x^3-20155392 x^2-6718464 x\right )-839808 x}{6561 x^{10}-131220 x^8+1049760 x^6-4199040 x^4+8398080 x^2-6718464} \, dx\) |
| \(\Big \downarrow \) 2070 |
| \(\displaystyle \int \frac {-8192 x^7-110592 x^6-595968 x^5-1670400 x^4-2620512 x^3-2239488 x^2+e^{8 x} \left (13436928 x^4-10077696 x^3-53747712 x^2-13436928 x\right )+e^{6 x} \left (8957952 x^5+12690432 x^4-55987200 x^3-98537472 x^2-26873856 x\right )+e^{4 x} \left (1990656 x^6+6967296 x^5-9082368 x^4-58910976 x^3-67184640 x^2-20155392 x\right )+e^{2 x} \left (147456 x^7+774144 x^6-340992 x^5-9374976 x^4-21959424 x^3-20155392 x^2-6718464 x\right )-839808 x}{\left (3\ 3^{3/5} x^2-12\ 3^{3/5}\right )^5}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {32 x \left (4 x+18 e^{2 x}+9\right )^3 \left (4 x^3+27 x^2-18 e^{2 x} \left (4 x^3-3 x^2-16 x-4\right )+48 x+36\right )}{6561 \left (4-x^2\right )^5}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {32 \int \frac {x \left (4 x+18 e^{2 x}+9\right )^3 \left (4 x^3+27 x^2+48 x+18 e^{2 x} \left (-4 x^3+3 x^2+16 x+4\right )+36\right )}{\left (4-x^2\right )^5}dx}{6561}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {32 \int \left (-\frac {256 x^7}{\left (x^2-4\right )^5}-\frac {3456 x^6}{\left (x^2-4\right )^5}-\frac {18624 x^5}{\left (x^2-4\right )^5}-\frac {52200 x^4}{\left (x^2-4\right )^5}-\frac {81891 x^3}{\left (x^2-4\right )^5}-\frac {69984 x^2}{\left (x^2-4\right )^5}+\frac {104976 e^{8 x} \left (4 x^3-3 x^2-16 x-4\right ) x}{\left (x^2-4\right )^5}+\frac {72 e^{2 x} (4 x+9)^2 \left (4 x^4+3 x^3-43 x^2-76 x-36\right ) x}{\left (x^2-4\right )^5}+\frac {23328 e^{6 x} \left (12 x^4+17 x^3-75 x^2-132 x-36\right ) x}{\left (x^2-4\right )^5}+\frac {1944 e^{4 x} \left (32 x^5+112 x^4-146 x^3-947 x^2-1080 x-324\right ) x}{\left (x^2-4\right )^5}-\frac {26244 x}{\left (x^2-4\right )^5}\right )dx}{6561}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {32 \left (\frac {360 x}{\left (4-x^2\right )^2}-\frac {3627 x}{\left (4-x^2\right )^3}+\frac {8748 x}{\left (4-x^2\right )^4}+\frac {4656}{\left (4-x^2\right )^2}-\frac {76961}{2 \left (4-x^2\right )^3}+\frac {81474}{\left (4-x^2\right )^4}+\frac {8 x^8}{\left (4-x^2\right )^4}+\frac {432 x^5}{\left (4-x^2\right )^4}-\frac {360 x^3}{\left (4-x^2\right )^3}+\frac {6525 x^3}{\left (4-x^2\right )^4}+\frac {52488 e^{8 x} \left (4 x-x^3\right ) x}{\left (4-x^2\right )^5}+\frac {8991 e^{2 x}}{256 (2-x)}-\frac {4131 e^{4 x}}{256 (2-x)}-\frac {6561 e^{6 x}}{64 (2-x)}+\frac {8991 e^{2 x}}{256 (x+2)}-\frac {4131 e^{4 x}}{256 (x+2)}-\frac {6561 e^{6 x}}{64 (x+2)}+\frac {14535 e^{2 x}}{128 (2-x)^2}-\frac {39123 e^{4 x}}{128 (2-x)^2}-\frac {12393 e^{6 x}}{32 (2-x)^2}+\frac {3447 e^{2 x}}{128 (x+2)^2}+\frac {30861 e^{4 x}}{128 (x+2)^2}-\frac {729 e^{6 x}}{32 (x+2)^2}-\frac {7803 e^{2 x}}{4 (2-x)^3}-\frac {4131 e^{4 x}}{2 (2-x)^3}-\frac {729 e^{6 x}}{(2-x)^3}+\frac {27 e^{2 x}}{4 (x+2)^3}+\frac {243 e^{4 x}}{2 (x+2)^3}+\frac {729 e^{6 x}}{(x+2)^3}+\frac {44217 e^{2 x}}{16 (2-x)^4}+\frac {70227 e^{4 x}}{16 (2-x)^4}+\frac {12393 e^{6 x}}{4 (2-x)^4}+\frac {9 e^{2 x}}{16 (x+2)^4}+\frac {243 e^{4 x}}{16 (x+2)^4}+\frac {729 e^{6 x}}{4 (x+2)^4}\right )}{6561}\) |
Input:
Int[(-839808*x - 2239488*x^2 - 2620512*x^3 - 1670400*x^4 - 595968*x^5 - 11 0592*x^6 - 8192*x^7 + E^(8*x)*(-13436928*x - 53747712*x^2 - 10077696*x^3 + 13436928*x^4) + E^(6*x)*(-26873856*x - 98537472*x^2 - 55987200*x^3 + 1269 0432*x^4 + 8957952*x^5) + E^(4*x)*(-20155392*x - 67184640*x^2 - 58910976*x ^3 - 9082368*x^4 + 6967296*x^5 + 1990656*x^6) + E^(2*x)*(-6718464*x - 2015 5392*x^2 - 21959424*x^3 - 9374976*x^4 - 340992*x^5 + 774144*x^6 + 147456*x ^7))/(-6718464 + 8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^8 + 65 61*x^10),x]
Output:
(32*((44217*E^(2*x))/(16*(2 - x)^4) + (70227*E^(4*x))/(16*(2 - x)^4) + (12 393*E^(6*x))/(4*(2 - x)^4) - (7803*E^(2*x))/(4*(2 - x)^3) - (4131*E^(4*x)) /(2*(2 - x)^3) - (729*E^(6*x))/(2 - x)^3 + (14535*E^(2*x))/(128*(2 - x)^2) - (39123*E^(4*x))/(128*(2 - x)^2) - (12393*E^(6*x))/(32*(2 - x)^2) + (899 1*E^(2*x))/(256*(2 - x)) - (4131*E^(4*x))/(256*(2 - x)) - (6561*E^(6*x))/( 64*(2 - x)) + (9*E^(2*x))/(16*(2 + x)^4) + (243*E^(4*x))/(16*(2 + x)^4) + (729*E^(6*x))/(4*(2 + x)^4) + (27*E^(2*x))/(4*(2 + x)^3) + (243*E^(4*x))/( 2*(2 + x)^3) + (729*E^(6*x))/(2 + x)^3 + (3447*E^(2*x))/(128*(2 + x)^2) + (30861*E^(4*x))/(128*(2 + x)^2) - (729*E^(6*x))/(32*(2 + x)^2) + (8991*E^( 2*x))/(256*(2 + x)) - (4131*E^(4*x))/(256*(2 + x)) - (6561*E^(6*x))/(64*(2 + x)) + 81474/(4 - x^2)^4 + (8748*x)/(4 - x^2)^4 + (6525*x^3)/(4 - x^2)^4 + (432*x^5)/(4 - x^2)^4 + (8*x^8)/(4 - x^2)^4 - 76961/(2*(4 - x^2)^3) - ( 3627*x)/(4 - x^2)^3 - (360*x^3)/(4 - x^2)^3 + 4656/(4 - x^2)^2 + (360*x)/( 4 - x^2)^2 + (52488*E^(8*x)*x*(4*x - x^3))/(4 - x^2)^5))/6561
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x^2, 0], Expon[Px , x^2]], b = Rt[Coeff[Px, x^2, Expon[Px, x^2]], Expon[Px, x^2]]}, Int[u*(a + b*x^2)^(Expon[Px, x^2]*p), x] /; EqQ[Px, (a + b*x^2)^Expon[Px, x^2]]] /; IntegerQ[p] && PolyQ[Px, x^2] && GtQ[Expon[Px, x^2], 1] && NeQ[Coeff[Px, x^ 2, 0], 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(141\) vs. \(2(24)=48\).
Time = 7.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 5.26
| method | result | size |
| parallelrisch | \(\frac {161243136 x^{2} {\mathrm e}^{8 x}-31850496+143327232 x^{3} {\mathrm e}^{6 x}-124416 x^{8}+47775744 x^{4} {\mathrm e}^{4 x}+322486272 x^{2} {\mathrm e}^{6 x}+7077888 \,{\mathrm e}^{2 x} x^{5}+214990848 \,{\mathrm e}^{4 x} x^{3}+2383872 x^{6}+47775744 \,{\mathrm e}^{2 x} x^{4}+241864704 \,{\mathrm e}^{4 x} x^{2}+3538944 x^{5}+107495424 \,{\mathrm e}^{2 x} x^{3}+80621568 \,{\mathrm e}^{2 x} x^{2}+17915904 x^{3}+41928192 x^{2}}{629856 x^{8}-10077696 x^{6}+60466176 x^{4}-161243136 x^{2}+161243136}\) | \(142\) |
| risch | \(\frac {\frac {4096}{6561} x^{6}+\frac {4096}{729} x^{5}+\frac {512}{27} x^{4}+\frac {256}{9} x^{3}+16 x^{2}}{x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256}+\frac {256 x^{2} {\mathrm e}^{8 x}}{\left (x^{2}-4\right )^{4}}+\frac {512 x^{2} \left (4 x +9\right ) {\mathrm e}^{6 x}}{9 \left (x^{2}-4\right )^{4}}+\frac {128 x^{2} \left (16 x^{2}+72 x +81\right ) {\mathrm e}^{4 x}}{27 \left (x^{2}-4\right )^{4}}+\frac {128 x^{2} \left (64 x^{3}+432 x^{2}+972 x +729\right ) {\mathrm e}^{2 x}}{729 \left (x^{2}-4\right )^{4}}\) | \(145\) |
| parts | \(\frac {4063}{419904 \left (2+x \right )}-\frac {4063}{419904 \left (-2+x \right )}+\frac {767}{209952 \left (2+x \right )^{2}}+\frac {83521}{26244 \left (-2+x \right )^{4}}+\frac {19652}{6561 \left (-2+x \right )^{3}}+\frac {138431}{209952 \left (-2+x \right )^{2}}+\frac {1}{26244 \left (2+x \right )^{4}}+\frac {4}{6561 \left (2+x \right )^{3}}+\frac {{\mathrm e}^{6 x}}{2 x -4}+\frac {{\mathrm e}^{8 x}}{4 x -8}+\frac {4 \,{\mathrm e}^{8 x}}{\left (2+x \right )^{4}}-\frac {{\mathrm e}^{8 x}}{2 \left (-2+x \right )^{2}}-\frac {{\mathrm e}^{8 x}}{4 \left (2+x \right )}-\frac {{\mathrm e}^{8 x}}{2 \left (2+x \right )^{2}}+\frac {4 \,{\mathrm e}^{8 x}}{\left (-2+x \right )^{4}}+\frac {8 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{4}}-\frac {17 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{3}}-\frac {{\mathrm e}^{6 x}}{2 \left (2+x \right )}-\frac {{\mathrm e}^{6 x}}{9 \left (2+x \right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{3}}+\frac {136 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{4}}+\frac {272 \,{\mathrm e}^{4 x}}{27 \left (-2+x \right )^{3}}+\frac {2 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{4}}+\frac {16 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{3}}+\frac {17 \,{\mathrm e}^{4 x}}{216 \left (-2+x \right )}-\frac {161 \,{\mathrm e}^{4 x}}{108 \left (-2+x \right )^{2}}-\frac {17 \,{\mathrm e}^{4 x}}{216 \left (2+x \right )}+\frac {127 \,{\mathrm e}^{4 x}}{108 \left (2+x \right )^{2}}+\frac {578 \,{\mathrm e}^{4 x}}{27 \left (-2+x \right )^{4}}-\frac {37 \,{\mathrm e}^{2 x}}{216 \left (-2+x \right )}+\frac {1615 \,{\mathrm e}^{2 x}}{2916 \left (-2+x \right )^{2}}+\frac {2312 \,{\mathrm e}^{2 x}}{243 \left (-2+x \right )^{3}}+\frac {2 \,{\mathrm e}^{2 x}}{729 \left (2+x \right )^{4}}+\frac {8 \,{\mathrm e}^{2 x}}{243 \left (2+x \right )^{3}}+\frac {37 \,{\mathrm e}^{2 x}}{216 \left (2+x \right )}+\frac {383 \,{\mathrm e}^{2 x}}{2916 \left (2+x \right )^{2}}+\frac {9826 \,{\mathrm e}^{2 x}}{729 \left (-2+x \right )^{4}}\) | \(388\) |
| default | \(-\frac {239}{729 \left (2+x \right )^{2}}+\frac {57856}{2187 \left (x^{2}-4\right )^{2}}+\frac {432304}{2187 \left (x^{2}-4\right )^{3}}+\frac {1160}{729 \left (-2+x \right )^{4}}+\frac {364}{243 \left (-2+x \right )^{3}}+\frac {239}{729 \left (-2+x \right )^{2}}-\frac {1160}{729 \left (2+x \right )^{4}}+\frac {364}{243 \left (2+x \right )^{3}}+\frac {2672704}{6561 \left (x^{2}-4\right )^{4}}+\frac {4096}{6561 \left (x^{2}-4\right )}+\frac {{\mathrm e}^{6 x}}{2 x -4}+\frac {{\mathrm e}^{8 x}}{4 x -8}+\frac {4 \,{\mathrm e}^{8 x}}{\left (2+x \right )^{4}}-\frac {{\mathrm e}^{8 x}}{2 \left (-2+x \right )^{2}}-\frac {{\mathrm e}^{8 x}}{4 \left (2+x \right )}-\frac {{\mathrm e}^{8 x}}{2 \left (2+x \right )^{2}}+\frac {4 \,{\mathrm e}^{8 x}}{\left (-2+x \right )^{4}}+\frac {8 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{4}}-\frac {17 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{3}}-\frac {{\mathrm e}^{6 x}}{2 \left (2+x \right )}-\frac {{\mathrm e}^{6 x}}{9 \left (2+x \right )^{2}}+\frac {32 \,{\mathrm e}^{6 x}}{9 \left (2+x \right )^{3}}+\frac {136 \,{\mathrm e}^{6 x}}{9 \left (-2+x \right )^{4}}+\frac {272 \,{\mathrm e}^{4 x}}{27 \left (-2+x \right )^{3}}+\frac {2 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{4}}+\frac {16 \,{\mathrm e}^{4 x}}{27 \left (2+x \right )^{3}}+\frac {17 \,{\mathrm e}^{4 x}}{216 \left (-2+x \right )}-\frac {161 \,{\mathrm e}^{4 x}}{108 \left (-2+x \right )^{2}}-\frac {17 \,{\mathrm e}^{4 x}}{216 \left (2+x \right )}+\frac {127 \,{\mathrm e}^{4 x}}{108 \left (2+x \right )^{2}}+\frac {578 \,{\mathrm e}^{4 x}}{27 \left (-2+x \right )^{4}}-\frac {37 \,{\mathrm e}^{2 x}}{216 \left (-2+x \right )}+\frac {1615 \,{\mathrm e}^{2 x}}{2916 \left (-2+x \right )^{2}}+\frac {2312 \,{\mathrm e}^{2 x}}{243 \left (-2+x \right )^{3}}+\frac {2 \,{\mathrm e}^{2 x}}{729 \left (2+x \right )^{4}}+\frac {8 \,{\mathrm e}^{2 x}}{243 \left (2+x \right )^{3}}+\frac {37 \,{\mathrm e}^{2 x}}{216 \left (2+x \right )}+\frac {383 \,{\mathrm e}^{2 x}}{2916 \left (2+x \right )^{2}}+\frac {9826 \,{\mathrm e}^{2 x}}{729 \left (-2+x \right )^{4}}\) | \(410\) |
| orering | \(\text {Expression too large to display}\) | \(8685\) |
Input:
int(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952 *x^5+12690432*x^4-55987200*x^3-98537472*x^2-26873856*x)*exp(x)^6+(1990656* x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*exp(x)^4 +(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6 718464*x)*exp(x)^2-8192*x^7-110592*x^6-595968*x^5-1670400*x^4-2620512*x^3- 2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+839808 0*x^2-6718464),x,method=_RETURNVERBOSE)
Output:
1/629856*(161243136*x^2*exp(x)^8-31850496+143327232*x^3*exp(x)^6-124416*x^ 8+47775744*x^4*exp(x)^4+322486272*x^2*exp(x)^6+7077888*x^5*exp(x)^2+214990 848*x^3*exp(x)^4+2383872*x^6+47775744*exp(x)^2*x^4+241864704*x^2*exp(x)^4+ 3538944*x^5+107495424*exp(x)^2*x^3+80621568*exp(x)^2*x^2+17915904*x^3+4192 8192*x^2)/(x^8-16*x^6+96*x^4-256*x^2+256)
Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 125, normalized size of antiderivative = 4.63 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {16 \, {\left (256 \, x^{6} + 2304 \, x^{5} + 7776 \, x^{4} + 11664 \, x^{3} + 104976 \, x^{2} e^{\left (8 \, x\right )} + 6561 \, x^{2} + 23328 \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e^{\left (6 \, x\right )} + 1944 \, {\left (16 \, x^{4} + 72 \, x^{3} + 81 \, x^{2}\right )} e^{\left (4 \, x\right )} + 72 \, {\left (64 \, x^{5} + 432 \, x^{4} + 972 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} \] Input:
integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8 957952*x^5+12690432*x^4-55987200*x^3-98537472*x^2-26873856*x)*exp(x)^6+(19 90656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*ex p(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392 *x^2-6718464*x)*exp(x)^2-8192*x^7-110592*x^6-595968*x^5-1670400*x^4-262051 2*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+ 8398080*x^2-6718464),x, algorithm="fricas")
Output:
16/6561*(256*x^6 + 2304*x^5 + 7776*x^4 + 11664*x^3 + 104976*x^2*e^(8*x) + 6561*x^2 + 23328*(4*x^3 + 9*x^2)*e^(6*x) + 1944*(16*x^4 + 72*x^3 + 81*x^2) *e^(4*x) + 72*(64*x^5 + 432*x^4 + 972*x^3 + 729*x^2)*e^(2*x))/(x^8 - 16*x^ 6 + 96*x^4 - 256*x^2 + 256)
Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (24) = 48\).
Time = 0.27 (sec) , antiderivative size = 631, normalized size of antiderivative = 23.37 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx =\text {Too large to display} \] Input:
integrate(((13436928*x**4-10077696*x**3-53747712*x**2-13436928*x)*exp(x)** 8+(8957952*x**5+12690432*x**4-55987200*x**3-98537472*x**2-26873856*x)*exp( x)**6+(1990656*x**6+6967296*x**5-9082368*x**4-58910976*x**3-67184640*x**2- 20155392*x)*exp(x)**4+(147456*x**7+774144*x**6-340992*x**5-9374976*x**4-21 959424*x**3-20155392*x**2-6718464*x)*exp(x)**2-8192*x**7-110592*x**6-59596 8*x**5-1670400*x**4-2620512*x**3-2239488*x**2-839808*x)/(6561*x**10-131220 *x**8+1049760*x**6-4199040*x**4+8398080*x**2-6718464),x)
Output:
((45349632*x**26 - 2176782336*x**24 + 47889211392*x**22 - 638522818560*x** 20 + 5746705367040*x**18 - 36778914349056*x**16 + 171634933628928*x**14 - 588462629584896*x**12 + 1471156573962240*x**10 - 2615389464821760*x**8 + 3 138467357786112*x**6 - 2282521714753536*x**4 + 760840571584512*x**2)*exp(8 *x) + (40310784*x**27 + 90699264*x**26 - 1934917632*x**25 - 4353564672*x** 24 + 42568187904*x**23 + 95778422784*x**22 - 567575838720*x**21 - 12770456 37120*x**20 + 5108182548480*x**19 + 11493410734080*x**18 - 32692368310272* x**17 - 73557828698112*x**16 + 152564385447936*x**15 + 343269867257856*x** 14 - 523077892964352*x**13 - 1176925259169792*x**12 + 1307694732410880*x** 11 + 2942313147924480*x**10 - 2324790635397120*x**9 - 5230778929643520*x** 8 + 2789748762476544*x**7 + 6276934715572224*x**6 - 2028908190892032*x**5 - 4565043429507072*x**4 + 676302730297344*x**3 + 1521681143169024*x**2)*ex p(6*x) + (13436928*x**28 + 60466176*x**27 - 576948096*x**26 - 2902376448*x **25 + 10924222464*x**24 + 63852281856*x**23 - 117358129152*x**22 - 851363 758080*x**21 + 744943288320*x**20 + 7662273822720*x**19 - 2277398052864*x* *18 - 49038552465408*x**17 - 4313576374272*x**16 + 228846578171904*x**15 + 83093102788608*x**14 - 784616839446528*x**13 - 446795700240384*x**12 + 19 61542098616320*x**11 + 1431804649144320*x**10 - 3487185953095680*x**9 - 29 93167943073792*x**8 + 4184623143714816*x**7 + 4031398306381824*x**6 - 3043 362286338048*x**5 - 3198348328697856*x**4 + 1014454095446016*x**3 + 114...
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (25) = 50\).
Time = 0.09 (sec) , antiderivative size = 357, normalized size of antiderivative = 13.22 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {8 \, {\left (15 \, x^{7} + 292 \, x^{5} - 880 \, x^{3} + 960 \, x\right )}}{729 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {15 \, x^{7} - 220 \, x^{5} + 1168 \, x^{3} + 960 \, x}{72 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} - \frac {725 \, {\left (3 \, x^{7} - 44 \, x^{5} - 176 \, x^{3} + 192 \, x\right )}}{5832 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {4096 \, {\left (x^{6} - 6 \, x^{4} + 16 \, x^{2} - 16\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {49664 \, {\left (3 \, x^{4} - 8 \, x^{2} + 8\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {128 \, {\left (1458 \, x^{2} e^{\left (8 \, x\right )} + 324 \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e^{\left (6 \, x\right )} + 27 \, {\left (16 \, x^{4} + 72 \, x^{3} + 81 \, x^{2}\right )} e^{\left (4 \, x\right )} + {\left (64 \, x^{5} + 432 \, x^{4} + 972 \, x^{3} + 729 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )}}{729 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {5392 \, {\left (x^{2} - 1\right )}}{81 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} + \frac {16}{x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256} \] Input:
integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8 957952*x^5+12690432*x^4-55987200*x^3-98537472*x^2-26873856*x)*exp(x)^6+(19 90656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*ex p(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392 *x^2-6718464*x)*exp(x)^2-8192*x^7-110592*x^6-595968*x^5-1670400*x^4-262051 2*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+ 8398080*x^2-6718464),x, algorithm="maxima")
Output:
8/729*(15*x^7 + 292*x^5 - 880*x^3 + 960*x)/(x^8 - 16*x^6 + 96*x^4 - 256*x^ 2 + 256) + 1/72*(15*x^7 - 220*x^5 + 1168*x^3 + 960*x)/(x^8 - 16*x^6 + 96*x ^4 - 256*x^2 + 256) - 725/5832*(3*x^7 - 44*x^5 - 176*x^3 + 192*x)/(x^8 - 1 6*x^6 + 96*x^4 - 256*x^2 + 256) + 4096/6561*(x^6 - 6*x^4 + 16*x^2 - 16)/(x ^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 49664/6561*(3*x^4 - 8*x^2 + 8)/(x^ 8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 128/729*(1458*x^2*e^(8*x) + 324*(4* x^3 + 9*x^2)*e^(6*x) + 27*(16*x^4 + 72*x^3 + 81*x^2)*e^(4*x) + (64*x^5 + 4 32*x^4 + 972*x^3 + 729*x^2)*e^(2*x))/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 25 6) + 5392/81*(x^2 - 1)/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256) + 16/(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (25) = 50\).
Time = 0.13 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.19 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {16 \, {\left (256 \, x^{6} + 4608 \, x^{5} e^{\left (2 \, x\right )} + 2304 \, x^{5} + 31104 \, x^{4} e^{\left (4 \, x\right )} + 31104 \, x^{4} e^{\left (2 \, x\right )} + 7776 \, x^{4} + 93312 \, x^{3} e^{\left (6 \, x\right )} + 139968 \, x^{3} e^{\left (4 \, x\right )} + 69984 \, x^{3} e^{\left (2 \, x\right )} + 11664 \, x^{3} + 104976 \, x^{2} e^{\left (8 \, x\right )} + 209952 \, x^{2} e^{\left (6 \, x\right )} + 157464 \, x^{2} e^{\left (4 \, x\right )} + 52488 \, x^{2} e^{\left (2 \, x\right )} + 6561 \, x^{2}\right )}}{6561 \, {\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}} \] Input:
integrate(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8 957952*x^5+12690432*x^4-55987200*x^3-98537472*x^2-26873856*x)*exp(x)^6+(19 90656*x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*ex p(x)^4+(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392 *x^2-6718464*x)*exp(x)^2-8192*x^7-110592*x^6-595968*x^5-1670400*x^4-262051 2*x^3-2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+ 8398080*x^2-6718464),x, algorithm="giac")
Output:
16/6561*(256*x^6 + 4608*x^5*e^(2*x) + 2304*x^5 + 31104*x^4*e^(4*x) + 31104 *x^4*e^(2*x) + 7776*x^4 + 93312*x^3*e^(6*x) + 139968*x^3*e^(4*x) + 69984*x ^3*e^(2*x) + 11664*x^3 + 104976*x^2*e^(8*x) + 209952*x^2*e^(6*x) + 157464* x^2*e^(4*x) + 52488*x^2*e^(2*x) + 6561*x^2)/(x^8 - 16*x^6 + 96*x^4 - 256*x ^2 + 256)
Time = 0.46 (sec) , antiderivative size = 210, normalized size of antiderivative = 7.78 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {\frac {4096\,x^6}{6561}+\frac {4096\,x^5}{729}+\frac {512\,x^4}{27}+\frac {256\,x^3}{9}+16\,x^2}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{2\,x}\,\left (\frac {8192\,x^5}{729}+\frac {2048\,x^4}{27}+\frac {512\,x^3}{3}+128\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{6\,x}\,\left (\frac {2048\,x^3}{9}+512\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {256\,x^2\,{\mathrm {e}}^{8\,x}}{x^8-16\,x^6+96\,x^4-256\,x^2+256}+\frac {{\mathrm {e}}^{4\,x}\,\left (\frac {2048\,x^4}{27}+\frac {1024\,x^3}{3}+384\,x^2\right )}{x^8-16\,x^6+96\,x^4-256\,x^2+256} \] Input:
int(-(839808*x + exp(2*x)*(6718464*x + 20155392*x^2 + 21959424*x^3 + 93749 76*x^4 + 340992*x^5 - 774144*x^6 - 147456*x^7) + exp(8*x)*(13436928*x + 53 747712*x^2 + 10077696*x^3 - 13436928*x^4) + exp(6*x)*(26873856*x + 9853747 2*x^2 + 55987200*x^3 - 12690432*x^4 - 8957952*x^5) + 2239488*x^2 + 2620512 *x^3 + 1670400*x^4 + 595968*x^5 + 110592*x^6 + 8192*x^7 + exp(4*x)*(201553 92*x + 67184640*x^2 + 58910976*x^3 + 9082368*x^4 - 6967296*x^5 - 1990656*x ^6))/(8398080*x^2 - 4199040*x^4 + 1049760*x^6 - 131220*x^8 + 6561*x^10 - 6 718464),x)
Output:
(16*x^2 + (256*x^3)/9 + (512*x^4)/27 + (4096*x^5)/729 + (4096*x^6)/6561)/( 96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256) + (exp(2*x)*(128*x^2 + (512*x^3)/3 + (2048*x^4)/27 + (8192*x^5)/729))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256) + (exp(6*x)*(512*x^2 + (2048*x^3)/9))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256) + (256*x^2*exp(8*x))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256) + (exp(4 *x)*(384*x^2 + (1024*x^3)/3 + (2048*x^4)/27))/(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256)
Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.67 \[ \int \frac {-839808 x-2239488 x^2-2620512 x^3-1670400 x^4-595968 x^5-110592 x^6-8192 x^7+e^{8 x} \left (-13436928 x-53747712 x^2-10077696 x^3+13436928 x^4\right )+e^{6 x} \left (-26873856 x-98537472 x^2-55987200 x^3+12690432 x^4+8957952 x^5\right )+e^{4 x} \left (-20155392 x-67184640 x^2-58910976 x^3-9082368 x^4+6967296 x^5+1990656 x^6\right )+e^{2 x} \left (-6718464 x-20155392 x^2-21959424 x^3-9374976 x^4-340992 x^5+774144 x^6+147456 x^7\right )}{-6718464+8398080 x^2-4199040 x^4+1049760 x^6-131220 x^8+6561 x^{10}} \, dx=\frac {1679616 e^{8 x} x^{2}+1492992 e^{6 x} x^{3}+3359232 e^{6 x} x^{2}+497664 e^{4 x} x^{4}+2239488 e^{4 x} x^{3}+2519424 e^{4 x} x^{2}+73728 e^{2 x} x^{5}+497664 e^{2 x} x^{4}+1119744 e^{2 x} x^{3}+839808 e^{2 x} x^{2}+256 x^{8}+36864 x^{5}+148992 x^{4}+186624 x^{3}+39440 x^{2}+65536}{6561 x^{8}-104976 x^{6}+629856 x^{4}-1679616 x^{2}+1679616} \] Input:
int(((13436928*x^4-10077696*x^3-53747712*x^2-13436928*x)*exp(x)^8+(8957952 *x^5+12690432*x^4-55987200*x^3-98537472*x^2-26873856*x)*exp(x)^6+(1990656* x^6+6967296*x^5-9082368*x^4-58910976*x^3-67184640*x^2-20155392*x)*exp(x)^4 +(147456*x^7+774144*x^6-340992*x^5-9374976*x^4-21959424*x^3-20155392*x^2-6 718464*x)*exp(x)^2-8192*x^7-110592*x^6-595968*x^5-1670400*x^4-2620512*x^3- 2239488*x^2-839808*x)/(6561*x^10-131220*x^8+1049760*x^6-4199040*x^4+839808 0*x^2-6718464),x)
Output:
(16*(104976*e**(8*x)*x**2 + 93312*e**(6*x)*x**3 + 209952*e**(6*x)*x**2 + 3 1104*e**(4*x)*x**4 + 139968*e**(4*x)*x**3 + 157464*e**(4*x)*x**2 + 4608*e* *(2*x)*x**5 + 31104*e**(2*x)*x**4 + 69984*e**(2*x)*x**3 + 52488*e**(2*x)*x **2 + 16*x**8 + 2304*x**5 + 9312*x**4 + 11664*x**3 + 2465*x**2 + 4096))/(6 561*(x**8 - 16*x**6 + 96*x**4 - 256*x**2 + 256))