Integrand size = 184, antiderivative size = 25 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=\frac {16 (e-x)^2 \left (-e^{2 x^2}+x\right )^4}{x^4} \]
Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(25)=50\).
Time = 10.51 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.88 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=16 \left (8 e^{1+2 x^2}+\frac {e^{8 x^2} (e-x)^2}{x^4}-\frac {4 e^{2+6 x^2}}{x^3}+\frac {8 e^{1+6 x^2}}{x^2}+\frac {6 e^{4 x^2} (e-x)^2}{x^2}-\frac {4 e^{6 x^2}}{x}-\frac {4 e^{2+2 x^2}}{x}-2 e x-4 e^{2 x^2} x+x^2\right ) \]
Integrate[(-32*E*x^5 + 32*x^6 + E^(8*x^2)*(-32*x^2 + 256*x^4 + E^2*(-64 + 256*x^2) + E*(96*x - 512*x^3)) + E^(6*x^2)*(64*x^3 - 768*x^5 + E^2*(192*x - 768*x^3) + E*(-256*x^2 + 1536*x^4)) + E^(4*x^2)*(768*x^6 + E^2*(-192*x^2 + 768*x^4) + E*(192*x^3 - 1536*x^5)) + E^(2*x^2)*(-64*x^5 + 512*E*x^6 - 2 56*x^7 + E^2*(64*x^3 - 256*x^5)))/x^5,x]
16*(8*E^(1 + 2*x^2) + (E^(8*x^2)*(E - x)^2)/x^4 - (4*E^(2 + 6*x^2))/x^3 + (8*E^(1 + 6*x^2))/x^2 + (6*E^(4*x^2)*(E - x)^2)/x^2 - (4*E^(6*x^2))/x - (4 *E^(2 + 2*x^2))/x - 2*E*x - 4*E^(2*x^2)*x + x^2)
Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(25)=50\).
Time = 0.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.88, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, 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.
| \(\displaystyle \int \frac {32 x^6-32 e x^5+e^{8 x^2} \left (256 x^4+e \left (96 x-512 x^3\right )-32 x^2+e^2 \left (256 x^2-64\right )\right )+e^{6 x^2} \left (-768 x^5+64 x^3+e^2 \left (192 x-768 x^3\right )+e \left (1536 x^4-256 x^2\right )\right )+e^{2 x^2} \left (-256 x^7+512 e x^6-64 x^5+e^2 \left (64 x^3-256 x^5\right )\right )+e^{4 x^2} \left (768 x^6+e \left (192 x^3-1536 x^5\right )+e^2 \left (768 x^4-192 x^2\right )\right )}{x^5} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {192 e^{4 x^2} \left (-4 x^3+4 e x^2-e\right ) (e-x)}{x^3}-\frac {64 e^{2 x^2} \left (-4 x^3+4 e x^2-x-e\right ) (e-x)}{x^2}+\frac {32 e^{8 x^2} \left (-8 x^3+8 e x^2+x-2 e\right ) (e-x)}{x^5}-\frac {64 e^{6 x^2} \left (-12 x^3+12 e x^2+x-3 e\right ) (e-x)}{x^4}-32 (e-x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {64 e^{2 x^2} \left (e x^2-x^3\right ) (e-x)}{x^3}+\frac {16 e^{8 x^2} \left (e x^2-x^3\right ) (e-x)}{x^6}-\frac {64 e^{6 x^2} \left (e x^2-x^3\right ) (e-x)}{x^5}+\frac {96 e^{4 x^2} \left (e x^2-x^3\right ) (e-x)}{x^4}+16 (e-x)^2\) |
Int[(-32*E*x^5 + 32*x^6 + E^(8*x^2)*(-32*x^2 + 256*x^4 + E^2*(-64 + 256*x^ 2) + E*(96*x - 512*x^3)) + E^(6*x^2)*(64*x^3 - 768*x^5 + E^2*(192*x - 768* x^3) + E*(-256*x^2 + 1536*x^4)) + E^(4*x^2)*(768*x^6 + E^2*(-192*x^2 + 768 *x^4) + E*(192*x^3 - 1536*x^5)) + E^(2*x^2)*(-64*x^5 + 512*E*x^6 - 256*x^7 + E^2*(64*x^3 - 256*x^5)))/x^5,x]
16*(E - x)^2 + (16*E^(8*x^2)*(E - x)*(E*x^2 - x^3))/x^6 - (64*E^(6*x^2)*(E - x)*(E*x^2 - x^3))/x^5 + (96*E^(4*x^2)*(E - x)*(E*x^2 - x^3))/x^4 - (64* E^(2*x^2)*(E - x)*(E*x^2 - x^3))/x^3
3.19.85.3.1 Defintions of rubi rules used
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. \(99\) vs. \(2(25)=50\).
Time = 0.41 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.00
| method | result | size |
| risch | \(-32 x \,{\mathrm e}+16 x^{2}+\frac {16 \left ({\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{8 x^{2}}}{x^{4}}-\frac {64 \left ({\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{6 x^{2}}}{x^{3}}+\frac {96 \left ({\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{4 x^{2}}}{x^{2}}-\frac {64 \left ({\mathrm e}^{2}-2 x \,{\mathrm e}+x^{2}\right ) {\mathrm e}^{2 x^{2}}}{x}\) | \(100\) |
| parallelrisch | \(-\frac {-16 \,{\mathrm e}^{8 x^{2}} {\mathrm e}^{2}+32 \,{\mathrm e}^{8 x^{2}} {\mathrm e} x -16 \,{\mathrm e}^{8 x^{2}} x^{2}+64 \,{\mathrm e}^{6 x^{2}} {\mathrm e}^{2} x -128 \,{\mathrm e}^{6 x^{2}} {\mathrm e} x^{2}+64 \,{\mathrm e}^{6 x^{2}} x^{3}-96 \,{\mathrm e}^{4 x^{2}} {\mathrm e}^{2} x^{2}+192 \,{\mathrm e}^{4 x^{2}} {\mathrm e} x^{3}-96 \,{\mathrm e}^{4 x^{2}} x^{4}+64 \,{\mathrm e}^{2 x^{2}} {\mathrm e}^{2} x^{3}-128 \,{\mathrm e}^{2 x^{2}} {\mathrm e} x^{4}+64 \,{\mathrm e}^{2 x^{2}} x^{5}+32 x^{5} {\mathrm e}-16 x^{6}}{x^{4}}\) | \(168\) |
| default | \(16 x^{2}+\frac {16 \,{\mathrm e}^{8 x^{2}}}{x^{2}}-\frac {64 \,{\mathrm e}^{6 x^{2}}}{x}+96 \,{\mathrm e}^{4 x^{2}}-64 x \,{\mathrm e}^{2 x^{2}}-64 \,{\mathrm e}^{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {2}\, x \right )-384 \,{\mathrm e} \sqrt {\pi }\, \operatorname {erfi}\left (2 x \right )-64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{8 x^{2}}}{4 x^{4}}-\frac {2 \,{\mathrm e}^{8 x^{2}}}{x^{2}}-16 \,\operatorname {Ei}_{1}\left (-8 x^{2}\right )\right )+192 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{6 x^{2}}}{3 x^{3}}-\frac {4 \,{\mathrm e}^{6 x^{2}}}{x}+4 \sqrt {6}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {6}\, x \right )\right )+96 \,{\mathrm e} \left (-\frac {{\mathrm e}^{8 x^{2}}}{3 x^{3}}-\frac {16 \,{\mathrm e}^{8 x^{2}}}{3 x}+\frac {32 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (2 \sqrt {2}\, x \right )}{3}\right )-192 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{4 x^{2}}}{2 x^{2}}-2 \,\operatorname {Ei}_{1}\left (-4 x^{2}\right )\right )-256 \,{\mathrm e} \left (-\frac {{\mathrm e}^{6 x^{2}}}{2 x^{2}}-3 \,\operatorname {Ei}_{1}\left (-6 x^{2}\right )\right )+256 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{8 x^{2}}}{2 x^{2}}-4 \,\operatorname {Ei}_{1}\left (-8 x^{2}\right )\right )+64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{2 x^{2}}}{x}+\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {2}\, x \right )\right )+192 \,{\mathrm e} \left (-\frac {{\mathrm e}^{4 x^{2}}}{x}+2 \sqrt {\pi }\, \operatorname {erfi}\left (2 x \right )\right )-768 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{6 x^{2}}}{x}+\sqrt {6}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {6}\, x \right )\right )-512 \,{\mathrm e} \left (-\frac {{\mathrm e}^{8 x^{2}}}{x}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (2 \sqrt {2}\, x \right )\right )-384 \,{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-4 x^{2}\right )-768 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-6 x^{2}\right )+128 \,{\mathrm e}^{2 x^{2}} {\mathrm e}-32 x \,{\mathrm e}\) | \(434\) |
| parts | \(16 x^{2}+\frac {16 \,{\mathrm e}^{8 x^{2}}}{x^{2}}-\frac {64 \,{\mathrm e}^{6 x^{2}}}{x}+96 \,{\mathrm e}^{4 x^{2}}-64 x \,{\mathrm e}^{2 x^{2}}-64 \,{\mathrm e}^{2} \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {2}\, x \right )-384 \,{\mathrm e} \sqrt {\pi }\, \operatorname {erfi}\left (2 x \right )-64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{8 x^{2}}}{4 x^{4}}-\frac {2 \,{\mathrm e}^{8 x^{2}}}{x^{2}}-16 \,\operatorname {Ei}_{1}\left (-8 x^{2}\right )\right )+192 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{6 x^{2}}}{3 x^{3}}-\frac {4 \,{\mathrm e}^{6 x^{2}}}{x}+4 \sqrt {6}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {6}\, x \right )\right )+96 \,{\mathrm e} \left (-\frac {{\mathrm e}^{8 x^{2}}}{3 x^{3}}-\frac {16 \,{\mathrm e}^{8 x^{2}}}{3 x}+\frac {32 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (2 \sqrt {2}\, x \right )}{3}\right )-192 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{4 x^{2}}}{2 x^{2}}-2 \,\operatorname {Ei}_{1}\left (-4 x^{2}\right )\right )-256 \,{\mathrm e} \left (-\frac {{\mathrm e}^{6 x^{2}}}{2 x^{2}}-3 \,\operatorname {Ei}_{1}\left (-6 x^{2}\right )\right )+256 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{8 x^{2}}}{2 x^{2}}-4 \,\operatorname {Ei}_{1}\left (-8 x^{2}\right )\right )+64 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{2 x^{2}}}{x}+\sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {2}\, x \right )\right )+192 \,{\mathrm e} \left (-\frac {{\mathrm e}^{4 x^{2}}}{x}+2 \sqrt {\pi }\, \operatorname {erfi}\left (2 x \right )\right )-768 \,{\mathrm e}^{2} \left (-\frac {{\mathrm e}^{6 x^{2}}}{x}+\sqrt {6}\, \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {6}\, x \right )\right )-512 \,{\mathrm e} \left (-\frac {{\mathrm e}^{8 x^{2}}}{x}+2 \sqrt {2}\, \sqrt {\pi }\, \operatorname {erfi}\left (2 \sqrt {2}\, x \right )\right )-384 \,{\mathrm e}^{2} \operatorname {Ei}_{1}\left (-4 x^{2}\right )-768 \,{\mathrm e} \,\operatorname {Ei}_{1}\left (-6 x^{2}\right )+128 \,{\mathrm e}^{2 x^{2}} {\mathrm e}-32 x \,{\mathrm e}\) | \(434\) |
int((((256*x^2-64)*exp(1)^2+(-512*x^3+96*x)*exp(1)+256*x^4-32*x^2)*exp(x^2 )^8+((-768*x^3+192*x)*exp(1)^2+(1536*x^4-256*x^2)*exp(1)-768*x^5+64*x^3)*e xp(x^2)^6+((768*x^4-192*x^2)*exp(1)^2+(-1536*x^5+192*x^3)*exp(1)+768*x^6)* exp(x^2)^4+((-256*x^5+64*x^3)*exp(1)^2+512*x^6*exp(1)-256*x^7-64*x^5)*exp( x^2)^2-32*x^5*exp(1)+32*x^6)/x^5,x,method=_RETURNVERBOSE)
-32*x*exp(1)+16*x^2+16*(exp(2)-2*x*exp(1)+x^2)/x^4*exp(8*x^2)-64*(exp(2)-2 *x*exp(1)+x^2)/x^3*exp(6*x^2)+96*(exp(2)-2*x*exp(1)+x^2)/x^2*exp(4*x^2)-64 *(exp(2)-2*x*exp(1)+x^2)/x*exp(2*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 4.28 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=\frac {16 \, {\left (x^{6} - 2 \, x^{5} e + {\left (x^{2} - 2 \, x e + e^{2}\right )} e^{\left (8 \, x^{2}\right )} - 4 \, {\left (x^{3} - 2 \, x^{2} e + x e^{2}\right )} e^{\left (6 \, x^{2}\right )} + 6 \, {\left (x^{4} - 2 \, x^{3} e + x^{2} e^{2}\right )} e^{\left (4 \, x^{2}\right )} - 4 \, {\left (x^{5} - 2 \, x^{4} e + x^{3} e^{2}\right )} e^{\left (2 \, x^{2}\right )}\right )}}{x^{4}} \]
integrate((((256*x^2-64)*exp(1)^2+(-512*x^3+96*x)*exp(1)+256*x^4-32*x^2)*e xp(x^2)^8+((-768*x^3+192*x)*exp(1)^2+(1536*x^4-256*x^2)*exp(1)-768*x^5+64* x^3)*exp(x^2)^6+((768*x^4-192*x^2)*exp(1)^2+(-1536*x^5+192*x^3)*exp(1)+768 *x^6)*exp(x^2)^4+((-256*x^5+64*x^3)*exp(1)^2+512*x^6*exp(1)-256*x^7-64*x^5 )*exp(x^2)^2-32*x^5*exp(1)+32*x^6)/x^5,x, algorithm=\
16*(x^6 - 2*x^5*e + (x^2 - 2*x*e + e^2)*e^(8*x^2) - 4*(x^3 - 2*x^2*e + x*e ^2)*e^(6*x^2) + 6*(x^4 - 2*x^3*e + x^2*e^2)*e^(4*x^2) - 4*(x^5 - 2*x^4*e + x^3*e^2)*e^(2*x^2))/x^4
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (20) = 40\).
Time = 0.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.16 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=16 x^{2} - 32 e x + \frac {\left (16 x^{8} - 32 e x^{7} + 16 x^{6} e^{2}\right ) e^{8 x^{2}} + \left (- 64 x^{9} + 128 e x^{8} - 64 x^{7} e^{2}\right ) e^{6 x^{2}} + \left (96 x^{10} - 192 e x^{9} + 96 x^{8} e^{2}\right ) e^{4 x^{2}} + \left (- 64 x^{11} + 128 e x^{10} - 64 x^{9} e^{2}\right ) e^{2 x^{2}}}{x^{10}} \]
integrate((((256*x**2-64)*exp(1)**2+(-512*x**3+96*x)*exp(1)+256*x**4-32*x* *2)*exp(x**2)**8+((-768*x**3+192*x)*exp(1)**2+(1536*x**4-256*x**2)*exp(1)- 768*x**5+64*x**3)*exp(x**2)**6+((768*x**4-192*x**2)*exp(1)**2+(-1536*x**5+ 192*x**3)*exp(1)+768*x**6)*exp(x**2)**4+((-256*x**5+64*x**3)*exp(1)**2+512 *x**6*exp(1)-256*x**7-64*x**5)*exp(x**2)**2-32*x**5*exp(1)+32*x**6)/x**5,x )
16*x**2 - 32*E*x + ((16*x**8 - 32*E*x**7 + 16*x**6*exp(2))*exp(8*x**2) + ( -64*x**9 + 128*E*x**8 - 64*x**7*exp(2))*exp(6*x**2) + (96*x**10 - 192*E*x* *9 + 96*x**8*exp(2))*exp(4*x**2) + (-64*x**11 + 128*E*x**10 - 64*x**9*exp( 2))*exp(2*x**2))/x**10
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.36 (sec) , antiderivative size = 325, normalized size of antiderivative = 13.00 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=64 i \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {2} x\right ) e^{2} + 384 i \, \sqrt {\pi } \operatorname {erf}\left (2 i \, x\right ) e + 16 \, x^{2} + 64 i \, \sqrt {6} \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {6} x\right ) + 384 \, {\rm Ei}\left (4 \, x^{2}\right ) e^{2} - 32 \, x e + 768 \, {\rm Ei}\left (6 \, x^{2}\right ) e - 64 \, x e^{\left (2 \, x^{2}\right )} - \frac {32 \, \sqrt {2} \sqrt {-x^{2}} e^{2} \Gamma \left (-\frac {1}{2}, -2 \, x^{2}\right )}{x} + \frac {384 \, \sqrt {6} \sqrt {-x^{2}} e^{2} \Gamma \left (-\frac {1}{2}, -6 \, x^{2}\right )}{x} + \frac {512 \, \sqrt {2} \sqrt {-x^{2}} e \Gamma \left (-\frac {1}{2}, -8 \, x^{2}\right )}{x} - 384 \, e^{2} \Gamma \left (-1, -4 \, x^{2}\right ) - 768 \, e \Gamma \left (-1, -6 \, x^{2}\right ) + 1024 \, e^{2} \Gamma \left (-1, -8 \, x^{2}\right ) + 2048 \, e^{2} \Gamma \left (-2, -8 \, x^{2}\right ) - \frac {192 \, \sqrt {-x^{2}} e \Gamma \left (-\frac {1}{2}, -4 \, x^{2}\right )}{x} - \frac {32 \, \sqrt {6} \sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -6 \, x^{2}\right )}{x} - \frac {576 \, \sqrt {6} \left (-x^{2}\right )^{\frac {3}{2}} e^{2} \Gamma \left (-\frac {3}{2}, -6 \, x^{2}\right )}{x^{3}} - \frac {768 \, \sqrt {2} \left (-x^{2}\right )^{\frac {3}{2}} e \Gamma \left (-\frac {3}{2}, -8 \, x^{2}\right )}{x^{3}} + 128 \, {\rm Ei}\left (8 \, x^{2}\right ) + 96 \, e^{\left (4 \, x^{2}\right )} + 128 \, e^{\left (2 \, x^{2} + 1\right )} - 128 \, \Gamma \left (-1, -8 \, x^{2}\right ) \]
integrate((((256*x^2-64)*exp(1)^2+(-512*x^3+96*x)*exp(1)+256*x^4-32*x^2)*e xp(x^2)^8+((-768*x^3+192*x)*exp(1)^2+(1536*x^4-256*x^2)*exp(1)-768*x^5+64* x^3)*exp(x^2)^6+((768*x^4-192*x^2)*exp(1)^2+(-1536*x^5+192*x^3)*exp(1)+768 *x^6)*exp(x^2)^4+((-256*x^5+64*x^3)*exp(1)^2+512*x^6*exp(1)-256*x^7-64*x^5 )*exp(x^2)^2-32*x^5*exp(1)+32*x^6)/x^5,x, algorithm=\
64*I*sqrt(2)*sqrt(pi)*erf(I*sqrt(2)*x)*e^2 + 384*I*sqrt(pi)*erf(2*I*x)*e + 16*x^2 + 64*I*sqrt(6)*sqrt(pi)*erf(I*sqrt(6)*x) + 384*Ei(4*x^2)*e^2 - 32* x*e + 768*Ei(6*x^2)*e - 64*x*e^(2*x^2) - 32*sqrt(2)*sqrt(-x^2)*e^2*gamma(- 1/2, -2*x^2)/x + 384*sqrt(6)*sqrt(-x^2)*e^2*gamma(-1/2, -6*x^2)/x + 512*sq rt(2)*sqrt(-x^2)*e*gamma(-1/2, -8*x^2)/x - 384*e^2*gamma(-1, -4*x^2) - 768 *e*gamma(-1, -6*x^2) + 1024*e^2*gamma(-1, -8*x^2) + 2048*e^2*gamma(-2, -8* x^2) - 192*sqrt(-x^2)*e*gamma(-1/2, -4*x^2)/x - 32*sqrt(6)*sqrt(-x^2)*gamm a(-1/2, -6*x^2)/x - 576*sqrt(6)*(-x^2)^(3/2)*e^2*gamma(-3/2, -6*x^2)/x^3 - 768*sqrt(2)*(-x^2)^(3/2)*e*gamma(-3/2, -8*x^2)/x^3 + 128*Ei(8*x^2) + 96*e ^(4*x^2) + 128*e^(2*x^2 + 1) - 128*gamma(-1, -8*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 154, normalized size of antiderivative = 6.16 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=\frac {16 \, {\left (x^{6} - 2 \, x^{5} e - 4 \, x^{5} e^{\left (2 \, x^{2}\right )} + 6 \, x^{4} e^{\left (4 \, x^{2}\right )} + 8 \, x^{4} e^{\left (2 \, x^{2} + 1\right )} - 4 \, x^{3} e^{\left (6 \, x^{2}\right )} - 12 \, x^{3} e^{\left (4 \, x^{2} + 1\right )} - 4 \, x^{3} e^{\left (2 \, x^{2} + 2\right )} + x^{2} e^{\left (8 \, x^{2}\right )} + 8 \, x^{2} e^{\left (6 \, x^{2} + 1\right )} + 6 \, x^{2} e^{\left (4 \, x^{2} + 2\right )} - 2 \, x e^{\left (8 \, x^{2} + 1\right )} - 4 \, x e^{\left (6 \, x^{2} + 2\right )} + e^{\left (8 \, x^{2} + 2\right )}\right )}}{x^{4}} \]
integrate((((256*x^2-64)*exp(1)^2+(-512*x^3+96*x)*exp(1)+256*x^4-32*x^2)*e xp(x^2)^8+((-768*x^3+192*x)*exp(1)^2+(1536*x^4-256*x^2)*exp(1)-768*x^5+64* x^3)*exp(x^2)^6+((768*x^4-192*x^2)*exp(1)^2+(-1536*x^5+192*x^3)*exp(1)+768 *x^6)*exp(x^2)^4+((-256*x^5+64*x^3)*exp(1)^2+512*x^6*exp(1)-256*x^7-64*x^5 )*exp(x^2)^2-32*x^5*exp(1)+32*x^6)/x^5,x, algorithm=\
16*(x^6 - 2*x^5*e - 4*x^5*e^(2*x^2) + 6*x^4*e^(4*x^2) + 8*x^4*e^(2*x^2 + 1 ) - 4*x^3*e^(6*x^2) - 12*x^3*e^(4*x^2 + 1) - 4*x^3*e^(2*x^2 + 2) + x^2*e^( 8*x^2) + 8*x^2*e^(6*x^2 + 1) + 6*x^2*e^(4*x^2 + 2) - 2*x*e^(8*x^2 + 1) - 4 *x*e^(6*x^2 + 2) + e^(8*x^2 + 2))/x^4
Time = 12.08 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.04 \[ \int \frac {-32 e x^5+32 x^6+e^{8 x^2} \left (-32 x^2+256 x^4+e^2 \left (-64+256 x^2\right )+e \left (96 x-512 x^3\right )\right )+e^{6 x^2} \left (64 x^3-768 x^5+e^2 \left (192 x-768 x^3\right )+e \left (-256 x^2+1536 x^4\right )\right )+e^{4 x^2} \left (768 x^6+e^2 \left (-192 x^2+768 x^4\right )+e \left (192 x^3-1536 x^5\right )\right )+e^{2 x^2} \left (-64 x^5+512 e x^6-256 x^7+e^2 \left (64 x^3-256 x^5\right )\right )}{x^5} \, dx=96\,{\mathrm {e}}^{4\,x^2}-32\,x\,\mathrm {e}+128\,\mathrm {e}\,{\mathrm {e}}^{2\,x^2}-64\,x\,{\mathrm {e}}^{2\,x^2}-\frac {64\,{\mathrm {e}}^{6\,x^2}}{x}+\frac {16\,{\mathrm {e}}^{8\,x^2}}{x^2}+16\,x^2-\frac {64\,{\mathrm {e}}^2\,{\mathrm {e}}^{2\,x^2}}{x}-\frac {192\,\mathrm {e}\,{\mathrm {e}}^{4\,x^2}}{x}+\frac {96\,{\mathrm {e}}^2\,{\mathrm {e}}^{4\,x^2}}{x^2}+\frac {128\,\mathrm {e}\,{\mathrm {e}}^{6\,x^2}}{x^2}-\frac {64\,{\mathrm {e}}^2\,{\mathrm {e}}^{6\,x^2}}{x^3}-\frac {32\,\mathrm {e}\,{\mathrm {e}}^{8\,x^2}}{x^3}+\frac {16\,{\mathrm {e}}^2\,{\mathrm {e}}^{8\,x^2}}{x^4} \]
int((exp(8*x^2)*(exp(1)*(96*x - 512*x^3) + exp(2)*(256*x^2 - 64) - 32*x^2 + 256*x^4) + exp(6*x^2)*(exp(2)*(192*x - 768*x^3) - exp(1)*(256*x^2 - 1536 *x^4) + 64*x^3 - 768*x^5) - 32*x^5*exp(1) + exp(4*x^2)*(exp(1)*(192*x^3 - 1536*x^5) - exp(2)*(192*x^2 - 768*x^4) + 768*x^6) + 32*x^6 + exp(2*x^2)*(e xp(2)*(64*x^3 - 256*x^5) + 512*x^6*exp(1) - 64*x^5 - 256*x^7))/x^5,x)
96*exp(4*x^2) - 32*x*exp(1) + 128*exp(1)*exp(2*x^2) - 64*x*exp(2*x^2) - (6 4*exp(6*x^2))/x + (16*exp(8*x^2))/x^2 + 16*x^2 - (64*exp(2)*exp(2*x^2))/x - (192*exp(1)*exp(4*x^2))/x + (96*exp(2)*exp(4*x^2))/x^2 + (128*exp(1)*exp (6*x^2))/x^2 - (64*exp(2)*exp(6*x^2))/x^3 - (32*exp(1)*exp(8*x^2))/x^3 + ( 16*exp(2)*exp(8*x^2))/x^4