\(\int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} (458752-1376256 x-4587520 x^4)+e^{18 x+15 x^4} (229376-1032192 x-3440640 x^4)+e^{6 x+5 x^4} (524288-786432 x-2621440 x^4)+e^{24 x+20 x^4} (71680-430080 x-1433600 x^4)+e^{30 x+25 x^4} (14336-107520 x-358400 x^4)+e^{36 x+30 x^4} (1792-16128 x-53760 x^4)+e^{42 x+35 x^4} (128-1344 x-4480 x^4)+e^{48 x+40 x^4} (4-48 x-160 x^4)}{x^5} \, dx\) [2322]

Optimal result

Integrand size = 190, antiderivative size = 28 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=5-\frac {\left (4+e^{x+5 \left (x+x^4\right )}\right )^8}{x^4}+x-x^2 \] Output:

x-(4+exp(5*x^4+6*x))^8/x^4-x^2+5
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(124\) vs. \(2(28)=56\).

Time = 10.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.43 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=-\frac {65536+114688 e^{2 x \left (6+5 x^3\right )}+57344 e^{3 x \left (6+5 x^3\right )}+17920 e^{4 x \left (6+5 x^3\right )}+3584 e^{5 x \left (6+5 x^3\right )}+448 e^{6 x \left (6+5 x^3\right )}+32 e^{7 x \left (6+5 x^3\right )}+e^{8 x \left (6+5 x^3\right )}+131072 e^{6 x+5 x^4}-x^5+x^6}{x^4} \] Input:

Integrate[(262144 + x^5 - 2*x^6 + E^(12*x + 10*x^4)*(458752 - 1376256*x - 
4587520*x^4) + E^(18*x + 15*x^4)*(229376 - 1032192*x - 3440640*x^4) + E^(6 
*x + 5*x^4)*(524288 - 786432*x - 2621440*x^4) + E^(24*x + 20*x^4)*(71680 - 
 430080*x - 1433600*x^4) + E^(30*x + 25*x^4)*(14336 - 107520*x - 358400*x^ 
4) + E^(36*x + 30*x^4)*(1792 - 16128*x - 53760*x^4) + E^(42*x + 35*x^4)*(1 
28 - 1344*x - 4480*x^4) + E^(48*x + 40*x^4)*(4 - 48*x - 160*x^4))/x^5,x]
 

Output:

-((65536 + 114688*E^(2*x*(6 + 5*x^3)) + 57344*E^(3*x*(6 + 5*x^3)) + 17920* 
E^(4*x*(6 + 5*x^3)) + 3584*E^(5*x*(6 + 5*x^3)) + 448*E^(6*x*(6 + 5*x^3)) + 
 32*E^(7*x*(6 + 5*x^3)) + E^(8*x*(6 + 5*x^3)) + 131072*E^(6*x + 5*x^4) - x 
^5 + x^6)/x^4)
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(291\) vs. \(2(28)=56\).

Time = 0.95 (sec) , antiderivative size = 291, normalized size of antiderivative = 10.39, 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.

\(\displaystyle \int \frac {-2 x^6+x^5+e^{10 x^4+12 x} \left (-4587520 x^4-1376256 x+458752\right )+e^{15 x^4+18 x} \left (-3440640 x^4-1032192 x+229376\right )+e^{5 x^4+6 x} \left (-2621440 x^4-786432 x+524288\right )+e^{20 x^4+24 x} \left (-1433600 x^4-430080 x+71680\right )+e^{25 x^4+30 x} \left (-358400 x^4-107520 x+14336\right )+e^{30 x^4+36 x} \left (-53760 x^4-16128 x+1792\right )+e^{35 x^4+42 x} \left (-4480 x^4-1344 x+128\right )+e^{40 x^4+48 x} \left (-160 x^4-48 x+4\right )+262144}{x^5} \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (\frac {-2 x^6+x^5+262144}{x^5}-\frac {262144 e^{5 x^4+6 x} \left (10 x^4+3 x-2\right )}{x^5}-\frac {458752 e^{2 x \left (5 x^3+6\right )} \left (10 x^4+3 x-1\right )}{x^5}-\frac {71680 e^{4 x \left (5 x^3+6\right )} \left (20 x^4+6 x-1\right )}{x^5}-\frac {114688 e^{3 x \left (5 x^3+6\right )} \left (30 x^4+9 x-2\right )}{x^5}-\frac {1792 e^{6 x \left (5 x^3+6\right )} \left (30 x^4+9 x-1\right )}{x^5}-\frac {4 e^{8 x \left (5 x^3+6\right )} \left (40 x^4+12 x-1\right )}{x^5}-\frac {7168 e^{5 x \left (5 x^3+6\right )} \left (50 x^4+15 x-2\right )}{x^5}-\frac {64 e^{7 x \left (5 x^3+6\right )} \left (70 x^4+21 x-2\right )}{x^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {65536}{x^4}-x^2-\frac {114688 e^{2 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {57344 e^{3 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {17920 e^{4 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {3584 e^{5 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {448 e^{6 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {32 e^{7 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {e^{8 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {131072 e^{5 x^4+6 x} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}+x\)

Input:

Int[(262144 + x^5 - 2*x^6 + E^(12*x + 10*x^4)*(458752 - 1376256*x - 458752 
0*x^4) + E^(18*x + 15*x^4)*(229376 - 1032192*x - 3440640*x^4) + E^(6*x + 5 
*x^4)*(524288 - 786432*x - 2621440*x^4) + E^(24*x + 20*x^4)*(71680 - 43008 
0*x - 1433600*x^4) + E^(30*x + 25*x^4)*(14336 - 107520*x - 358400*x^4) + E 
^(36*x + 30*x^4)*(1792 - 16128*x - 53760*x^4) + E^(42*x + 35*x^4)*(128 - 1 
344*x - 4480*x^4) + E^(48*x + 40*x^4)*(4 - 48*x - 160*x^4))/x^5,x]
 

Output:

-65536/x^4 + x - x^2 - (114688*E^(2*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 
 + 10*x^3)) - (57344*E^(3*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3) 
) - (17920*E^(4*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (3584* 
E^(5*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (448*E^(6*x*(6 + 
5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (32*E^(7*x*(6 + 5*x^3))*(3*x 
+ 10*x^4))/(x^5*(3 + 10*x^3)) - (E^(8*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5* 
(3 + 10*x^3)) - (131072*E^(6*x + 5*x^4)*(3*x + 10*x^4))/(x^5*(3 + 10*x^3))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

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]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(139\) vs. \(2(27)=54\).

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00

\[-x^{2}+x -\frac {65536}{x^{4}}-\frac {{\mathrm e}^{8 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {32 \,{\mathrm e}^{7 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {448 \,{\mathrm e}^{6 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {3584 \,{\mathrm e}^{5 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {17920 \,{\mathrm e}^{4 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {57344 \,{\mathrm e}^{3 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {114688 \,{\mathrm e}^{2 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {131072 \,{\mathrm e}^{x \left (5 x^{3}+6\right )}}{x^{4}}\]

Input:

int(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6 
*x)^7+(-53760*x^4-16128*x+1792)*exp(5*x^4+6*x)^6+(-358400*x^4-107520*x+143 
36)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+(-3440 
640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752) 
*exp(5*x^4+6*x)^2+(-2621440*x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+ 
262144)/x^5,x)
 

Output:

-x^2+x-65536/x^4-1/x^4*exp(8*x*(5*x^3+6))-32/x^4*exp(7*x*(5*x^3+6))-448/x^ 
4*exp(6*x*(5*x^3+6))-3584/x^4*exp(5*x*(5*x^3+6))-17920/x^4*exp(4*x*(5*x^3+ 
6))-57344/x^4*exp(3*x*(5*x^3+6))-114688/x^4*exp(2*x*(5*x^3+6))-131072*exp( 
x*(5*x^3+6))/x^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (27) = 54\).

Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.89 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=-\frac {x^{6} - x^{5} + e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )} + 65536}{x^{4}} \] Input:

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5 
*x^4+6*x)^7+(-53760*x^4-16128*x+1792)*exp(5*x^4+6*x)^6+(-358400*x^4-107520 
*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+ 
(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+4 
58752)*exp(5*x^4+6*x)^2+(-2621440*x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^ 
6+x^5+262144)/x^5,x, algorithm="fricas")
 

Output:

-(x^6 - x^5 + e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 3 
6*x) + 3584*e^(25*x^4 + 30*x) + 17920*e^(20*x^4 + 24*x) + 57344*e^(15*x^4 
+ 18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x) + 65536)/x^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (22) = 44\).

Time = 0.18 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.79 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=- x^{2} + x - \frac {65536}{x^{4}} + \frac {- 131072 x^{28} e^{5 x^{4} + 6 x} - 114688 x^{28} e^{10 x^{4} + 12 x} - 57344 x^{28} e^{15 x^{4} + 18 x} - 17920 x^{28} e^{20 x^{4} + 24 x} - 3584 x^{28} e^{25 x^{4} + 30 x} - 448 x^{28} e^{30 x^{4} + 36 x} - 32 x^{28} e^{35 x^{4} + 42 x} - x^{28} e^{40 x^{4} + 48 x}}{x^{32}} \] Input:

integrate(((-160*x**4-48*x+4)*exp(5*x**4+6*x)**8+(-4480*x**4-1344*x+128)*e 
xp(5*x**4+6*x)**7+(-53760*x**4-16128*x+1792)*exp(5*x**4+6*x)**6+(-358400*x 
**4-107520*x+14336)*exp(5*x**4+6*x)**5+(-1433600*x**4-430080*x+71680)*exp( 
5*x**4+6*x)**4+(-3440640*x**4-1032192*x+229376)*exp(5*x**4+6*x)**3+(-45875 
20*x**4-1376256*x+458752)*exp(5*x**4+6*x)**2+(-2621440*x**4-786432*x+52428 
8)*exp(5*x**4+6*x)-2*x**6+x**5+262144)/x**5,x)
 

Output:

-x**2 + x - 65536/x**4 + (-131072*x**28*exp(5*x**4 + 6*x) - 114688*x**28*e 
xp(10*x**4 + 12*x) - 57344*x**28*exp(15*x**4 + 18*x) - 17920*x**28*exp(20* 
x**4 + 24*x) - 3584*x**28*exp(25*x**4 + 30*x) - 448*x**28*exp(30*x**4 + 36 
*x) - 32*x**28*exp(35*x**4 + 42*x) - x**28*exp(40*x**4 + 48*x))/x**32
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.00 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=-x^{2} + x - \frac {e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )}}{x^{4}} - \frac {65536}{x^{4}} \] Input:

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5 
*x^4+6*x)^7+(-53760*x^4-16128*x+1792)*exp(5*x^4+6*x)^6+(-358400*x^4-107520 
*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+ 
(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+4 
58752)*exp(5*x^4+6*x)^2+(-2621440*x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^ 
6+x^5+262144)/x^5,x, algorithm="maxima")
 

Output:

-x^2 + x - (e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 36* 
x) + 3584*e^(25*x^4 + 30*x) + 17920*e^(20*x^4 + 24*x) + 57344*e^(15*x^4 + 
18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x))/x^4 - 65536/x^4
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (27) = 54\).

Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.89 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=-\frac {x^{6} - x^{5} + e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )} + 65536}{x^{4}} \] Input:

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5 
*x^4+6*x)^7+(-53760*x^4-16128*x+1792)*exp(5*x^4+6*x)^6+(-358400*x^4-107520 
*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+ 
(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+4 
58752)*exp(5*x^4+6*x)^2+(-2621440*x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^ 
6+x^5+262144)/x^5,x, algorithm="giac")
 

Output:

-(x^6 - x^5 + e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 3 
6*x) + 3584*e^(25*x^4 + 30*x) + 17920*e^(20*x^4 + 24*x) + 57344*e^(15*x^4 
+ 18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x) + 65536)/x^4
 

Mupad [B] (verification not implemented)

Time = 3.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=-x\,\left (x-1\right )-\frac {{\left ({\mathrm {e}}^{5\,x^4+6\,x}+4\right )}^8}{x^4} \] Input:

int(-(exp(48*x + 40*x^4)*(48*x + 160*x^4 - 4) + exp(42*x + 35*x^4)*(1344*x 
 + 4480*x^4 - 128) + exp(36*x + 30*x^4)*(16128*x + 53760*x^4 - 1792) + exp 
(30*x + 25*x^4)*(107520*x + 358400*x^4 - 14336) + exp(24*x + 20*x^4)*(4300 
80*x + 1433600*x^4 - 71680) + exp(6*x + 5*x^4)*(786432*x + 2621440*x^4 - 5 
24288) + exp(18*x + 15*x^4)*(1032192*x + 3440640*x^4 - 229376) + exp(12*x 
+ 10*x^4)*(1376256*x + 4587520*x^4 - 458752) - x^5 + 2*x^6 - 262144)/x^5,x 
)
 

Output:

- x*(x - 1) - (exp(6*x + 5*x^4) + 4)^8/x^4
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.21 \[ \int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} \left (458752-1376256 x-4587520 x^4\right )+e^{18 x+15 x^4} \left (229376-1032192 x-3440640 x^4\right )+e^{6 x+5 x^4} \left (524288-786432 x-2621440 x^4\right )+e^{24 x+20 x^4} \left (71680-430080 x-1433600 x^4\right )+e^{30 x+25 x^4} \left (14336-107520 x-358400 x^4\right )+e^{36 x+30 x^4} \left (1792-16128 x-53760 x^4\right )+e^{42 x+35 x^4} \left (128-1344 x-4480 x^4\right )+e^{48 x+40 x^4} \left (4-48 x-160 x^4\right )}{x^5} \, dx=\frac {-e^{40 x^{4}+48 x}-32 e^{35 x^{4}+42 x}-448 e^{30 x^{4}+36 x}-3584 e^{25 x^{4}+30 x}-17920 e^{20 x^{4}+24 x}-57344 e^{15 x^{4}+18 x}-114688 e^{10 x^{4}+12 x}-131072 e^{5 x^{4}+6 x}-x^{6}+x^{5}-65536}{x^{4}} \] Input:

int(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6 
*x)^7+(-53760*x^4-16128*x+1792)*exp(5*x^4+6*x)^6+(-358400*x^4-107520*x+143 
36)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+(-3440 
640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752) 
*exp(5*x^4+6*x)^2+(-2621440*x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+ 
262144)/x^5,x)
 

Output:

( - e**(40*x**4 + 48*x) - 32*e**(35*x**4 + 42*x) - 448*e**(30*x**4 + 36*x) 
 - 3584*e**(25*x**4 + 30*x) - 17920*e**(20*x**4 + 24*x) - 57344*e**(15*x** 
4 + 18*x) - 114688*e**(10*x**4 + 12*x) - 131072*e**(5*x**4 + 6*x) - x**6 + 
 x**5 - 65536)/x**4