3.24.62 \(\int \frac {-10 x-44 x^2-55 x^3-31 x^4-13 x^5-5 x^6+e^{2 x} (-192 x-320 x^2-176 x^3-80 x^4-32 x^5)+e^x (96 x+256 x^2+216 x^3+96 x^4+48 x^5+8 x^6)+(-8 x-24 x^2-18 x^3-6 x^4-4 x^5+e^{2 x} (-128 x-128 x^2-32 x^3-32 x^4)+e^x (64 x+128 x^2+48 x^3+32 x^4+8 x^5)) \log (x)}{4+x^2} \, dx\)

Optimal. Leaf size=28 \[ -x \left (1-4 e^x+x\right )^2 (x+x (x+\log (x)))+\log \left (4+x^2\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 2.27, antiderivative size = 124, normalized size of antiderivative = 4.43, number of steps used = 64, number of rules used = 15, integrand size = 180, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6688, 6725, 260, 321, 203, 266, 43, 302, 2304, 6742, 2196, 2176, 2194, 2554, 12} \begin {gather*} -x^5+8 e^x x^4-3 x^4-x^4 \log (x)+16 e^x x^3-16 e^{2 x} x^3-3 x^3+8 e^x x^3 \log (x)-2 x^3 \log (x)+8 e^x x^2-16 e^{2 x} x^2-x^2+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-x^2 \log (x)+\log \left (x^2+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10*x - 44*x^2 - 55*x^3 - 31*x^4 - 13*x^5 - 5*x^6 + E^(2*x)*(-192*x - 320*x^2 - 176*x^3 - 80*x^4 - 32*x^5
) + E^x*(96*x + 256*x^2 + 216*x^3 + 96*x^4 + 48*x^5 + 8*x^6) + (-8*x - 24*x^2 - 18*x^3 - 6*x^4 - 4*x^5 + E^(2*
x)*(-128*x - 128*x^2 - 32*x^3 - 32*x^4) + E^x*(64*x + 128*x^2 + 48*x^3 + 32*x^4 + 8*x^5))*Log[x])/(4 + x^2),x]

[Out]

-x^2 + 8*E^x*x^2 - 16*E^(2*x)*x^2 - 3*x^3 + 16*E^x*x^3 - 16*E^(2*x)*x^3 - 3*x^4 + 8*E^x*x^4 - x^5 - x^2*Log[x]
+ 8*E^x*x^2*Log[x] - 16*E^(2*x)*x^2*Log[x] - 2*x^3*Log[x] + 8*E^x*x^3*Log[x] - x^4*Log[x] + Log[4 + x^2]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-10-44 x-55 x^2-31 x^3-13 x^4-5 x^5-16 e^{2 x} \left (12+20 x+11 x^2+5 x^3+2 x^4\right )+8 e^x \left (12+32 x+27 x^2+12 x^3+6 x^4+x^5\right )-2 \left (4+x^2\right ) \left (1+3 x+2 x^2+16 e^{2 x} (1+x)-4 e^x \left (2+4 x+x^2\right )\right ) \log (x)\right )}{4+x^2} \, dx\\ &=\int \left (-\frac {10 x}{4+x^2}-\frac {44 x^2}{4+x^2}-\frac {55 x^3}{4+x^2}-\frac {31 x^4}{4+x^2}-\frac {13 x^5}{4+x^2}-\frac {5 x^6}{4+x^2}-2 x \log (x)-6 x^2 \log (x)-4 x^3 \log (x)-16 e^{2 x} x (1+x) (3+2 x+2 \log (x))+8 e^x x \left (3+8 x+6 x^2+x^3+2 \log (x)+4 x \log (x)+x^2 \log (x)\right )\right ) \, dx\\ &=-(2 \int x \log (x) \, dx)-4 \int x^3 \log (x) \, dx-5 \int \frac {x^6}{4+x^2} \, dx-6 \int x^2 \log (x) \, dx+8 \int e^x x \left (3+8 x+6 x^2+x^3+2 \log (x)+4 x \log (x)+x^2 \log (x)\right ) \, dx-10 \int \frac {x}{4+x^2} \, dx-13 \int \frac {x^5}{4+x^2} \, dx-16 \int e^{2 x} x (1+x) (3+2 x+2 \log (x)) \, dx-31 \int \frac {x^4}{4+x^2} \, dx-44 \int \frac {x^2}{4+x^2} \, dx-55 \int \frac {x^3}{4+x^2} \, dx\\ &=-44 x+\frac {x^2}{2}+\frac {2 x^3}{3}+\frac {x^4}{4}-x^2 \log (x)-2 x^3 \log (x)-x^4 \log (x)-5 \log \left (4+x^2\right )-5 \int \left (16-4 x^2+x^4-\frac {64}{4+x^2}\right ) \, dx-\frac {13}{2} \operatorname {Subst}\left (\int \frac {x^2}{4+x} \, dx,x,x^2\right )+8 \int \left (e^x x \left (3+8 x+6 x^2+x^3\right )+e^x x \left (2+4 x+x^2\right ) \log (x)\right ) \, dx-16 \int \left (e^{2 x} x \left (3+5 x+2 x^2\right )+2 e^{2 x} x (1+x) \log (x)\right ) \, dx-\frac {55}{2} \operatorname {Subst}\left (\int \frac {x}{4+x} \, dx,x,x^2\right )-31 \int \left (-4+x^2+\frac {16}{4+x^2}\right ) \, dx+176 \int \frac {1}{4+x^2} \, dx\\ &=\frac {x^2}{2}-3 x^3+\frac {x^4}{4}-x^5+88 \tan ^{-1}\left (\frac {x}{2}\right )-x^2 \log (x)-2 x^3 \log (x)-x^4 \log (x)-5 \log \left (4+x^2\right )-\frac {13}{2} \operatorname {Subst}\left (\int \left (-4+x+\frac {16}{4+x}\right ) \, dx,x,x^2\right )+8 \int e^x x \left (3+8 x+6 x^2+x^3\right ) \, dx+8 \int e^x x \left (2+4 x+x^2\right ) \log (x) \, dx-16 \int e^{2 x} x \left (3+5 x+2 x^2\right ) \, dx-\frac {55}{2} \operatorname {Subst}\left (\int \left (1-\frac {4}{4+x}\right ) \, dx,x,x^2\right )-32 \int e^{2 x} x (1+x) \log (x) \, dx+320 \int \frac {1}{4+x^2} \, dx-496 \int \frac {1}{4+x^2} \, dx\\ &=-x^2-3 x^3-3 x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )-8 \int e^x x (1+x) \, dx+8 \int \left (3 e^x x+8 e^x x^2+6 e^x x^3+e^x x^4\right ) \, dx-16 \int \left (3 e^{2 x} x+5 e^{2 x} x^2+2 e^{2 x} x^3\right ) \, dx+32 \int \frac {1}{2} e^{2 x} x \, dx\\ &=-x^2-3 x^3-3 x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )+8 \int e^x x^4 \, dx-8 \int \left (e^x x+e^x x^2\right ) \, dx+16 \int e^{2 x} x \, dx+24 \int e^x x \, dx-32 \int e^{2 x} x^3 \, dx-48 \int e^{2 x} x \, dx+48 \int e^x x^3 \, dx+64 \int e^x x^2 \, dx-80 \int e^{2 x} x^2 \, dx\\ &=24 e^x x-16 e^{2 x} x-x^2+64 e^x x^2-40 e^{2 x} x^2-3 x^3+48 e^x x^3-16 e^{2 x} x^3-3 x^4+8 e^x x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )-8 \int e^{2 x} \, dx-8 \int e^x x \, dx-8 \int e^x x^2 \, dx-24 \int e^x \, dx+24 \int e^{2 x} \, dx-32 \int e^x x^3 \, dx+48 \int e^{2 x} x^2 \, dx+80 \int e^{2 x} x \, dx-128 \int e^x x \, dx-144 \int e^x x^2 \, dx\\ &=-24 e^x+8 e^{2 x}-112 e^x x+24 e^{2 x} x-x^2-88 e^x x^2-16 e^{2 x} x^2-3 x^3+16 e^x x^3-16 e^{2 x} x^3-3 x^4+8 e^x x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )+8 \int e^x \, dx+16 \int e^x x \, dx-40 \int e^{2 x} \, dx-48 \int e^{2 x} x \, dx+96 \int e^x x^2 \, dx+128 \int e^x \, dx+288 \int e^x x \, dx\\ &=112 e^x-12 e^{2 x}+192 e^x x-x^2+8 e^x x^2-16 e^{2 x} x^2-3 x^3+16 e^x x^3-16 e^{2 x} x^3-3 x^4+8 e^x x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )-16 \int e^x \, dx+24 \int e^{2 x} \, dx-192 \int e^x x \, dx-288 \int e^x \, dx\\ &=-192 e^x-x^2+8 e^x x^2-16 e^{2 x} x^2-3 x^3+16 e^x x^3-16 e^{2 x} x^3-3 x^4+8 e^x x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )+192 \int e^x \, dx\\ &=-x^2+8 e^x x^2-16 e^{2 x} x^2-3 x^3+16 e^x x^3-16 e^{2 x} x^3-3 x^4+8 e^x x^4-x^5-x^2 \log (x)+8 e^x x^2 \log (x)-16 e^{2 x} x^2 \log (x)-2 x^3 \log (x)+8 e^x x^3 \log (x)-x^4 \log (x)+\log \left (4+x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.13, size = 42, normalized size = 1.50 \begin {gather*} -x^2 (1+x) \left (1-4 e^x+x\right )^2-x^2 \left (1-4 e^x+x\right )^2 \log (x)+\log \left (4+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*x - 44*x^2 - 55*x^3 - 31*x^4 - 13*x^5 - 5*x^6 + E^(2*x)*(-192*x - 320*x^2 - 176*x^3 - 80*x^4 -
32*x^5) + E^x*(96*x + 256*x^2 + 216*x^3 + 96*x^4 + 48*x^5 + 8*x^6) + (-8*x - 24*x^2 - 18*x^3 - 6*x^4 - 4*x^5 +
E^(2*x)*(-128*x - 128*x^2 - 32*x^3 - 32*x^4) + E^x*(64*x + 128*x^2 + 48*x^3 + 32*x^4 + 8*x^5))*Log[x])/(4 + x
^2),x]

[Out]

-(x^2*(1 + x)*(1 - 4*E^x + x)^2) - x^2*(1 - 4*E^x + x)^2*Log[x] + Log[4 + x^2]

________________________________________________________________________________________

fricas [B]  time = 0.58, size = 92, normalized size = 3.29 \begin {gather*} -x^{5} - 3 \, x^{4} - 3 \, x^{3} - x^{2} - 16 \, {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{x} - {\left (x^{4} + 2 \, x^{3} + 16 \, x^{2} e^{\left (2 \, x\right )} + x^{2} - 8 \, {\left (x^{3} + x^{2}\right )} e^{x}\right )} \log \relax (x) + \log \left (x^{2} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-32*x^4-32*x^3-128*x^2-128*x)*exp(x)^2+(8*x^5+32*x^4+48*x^3+128*x^2+64*x)*exp(x)-4*x^5-6*x^4-18*x
^3-24*x^2-8*x)*log(x)+(-32*x^5-80*x^4-176*x^3-320*x^2-192*x)*exp(x)^2+(8*x^6+48*x^5+96*x^4+216*x^3+256*x^2+96*
x)*exp(x)-5*x^6-13*x^5-31*x^4-55*x^3-44*x^2-10*x)/(x^2+4),x, algorithm="fricas")

[Out]

-x^5 - 3*x^4 - 3*x^3 - x^2 - 16*(x^3 + x^2)*e^(2*x) + 8*(x^4 + 2*x^3 + x^2)*e^x - (x^4 + 2*x^3 + 16*x^2*e^(2*x
) + x^2 - 8*(x^3 + x^2)*e^x)*log(x) + log(x^2 + 4)

________________________________________________________________________________________

giac [B]  time = 0.23, size = 116, normalized size = 4.14 \begin {gather*} -x^{5} + 8 \, x^{4} e^{x} - x^{4} \log \relax (x) + 8 \, x^{3} e^{x} \log \relax (x) - 3 \, x^{4} - 16 \, x^{3} e^{\left (2 \, x\right )} + 16 \, x^{3} e^{x} - 2 \, x^{3} \log \relax (x) - 16 \, x^{2} e^{\left (2 \, x\right )} \log \relax (x) + 8 \, x^{2} e^{x} \log \relax (x) - 3 \, x^{3} - 16 \, x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - x^{2} \log \relax (x) - x^{2} + \log \left (x^{2} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-32*x^4-32*x^3-128*x^2-128*x)*exp(x)^2+(8*x^5+32*x^4+48*x^3+128*x^2+64*x)*exp(x)-4*x^5-6*x^4-18*x
^3-24*x^2-8*x)*log(x)+(-32*x^5-80*x^4-176*x^3-320*x^2-192*x)*exp(x)^2+(8*x^6+48*x^5+96*x^4+216*x^3+256*x^2+96*
x)*exp(x)-5*x^6-13*x^5-31*x^4-55*x^3-44*x^2-10*x)/(x^2+4),x, algorithm="giac")

[Out]

-x^5 + 8*x^4*e^x - x^4*log(x) + 8*x^3*e^x*log(x) - 3*x^4 - 16*x^3*e^(2*x) + 16*x^3*e^x - 2*x^3*log(x) - 16*x^2
*e^(2*x)*log(x) + 8*x^2*e^x*log(x) - 3*x^3 - 16*x^2*e^(2*x) + 8*x^2*e^x - x^2*log(x) - x^2 + log(x^2 + 4)

________________________________________________________________________________________

maple [B]  time = 0.47, size = 109, normalized size = 3.89

method	result	size
risch	\(\left (-x^{4}+8 \,{\mathrm e}^{x} x^{3}-16 \,{\mathrm e}^{2 x} x^{2}-2 x^{3}+8 \,{\mathrm e}^{x} x^{2}-x^{2}\right ) \ln \relax (x )-x^{5}+8 \,{\mathrm e}^{x} x^{4}-16 \,{\mathrm e}^{2 x} x^{3}-3 x^{4}+16 \,{\mathrm e}^{x} x^{3}-16 \,{\mathrm e}^{2 x} x^{2}-3 x^{3}+8 \,{\mathrm e}^{x} x^{2}-x^{2}+\ln \left (x^{2}+4\right )\)	\(109\)
default	\(-16 \,{\mathrm e}^{2 x} x^{3}-16 \,{\mathrm e}^{2 x} x^{2}-16 \ln \relax (x ) {\mathrm e}^{2 x} x^{2}+8 \,{\mathrm e}^{x} x^{2}+16 \,{\mathrm e}^{x} x^{3}+8 \,{\mathrm e}^{x} x^{4}+8 x^{2} {\mathrm e}^{x} \ln \relax (x )+8 x^{3} {\mathrm e}^{x} \ln \relax (x )-x^{5}-3 x^{4}-3 x^{3}-x^{2}+\ln \left (x^{2}+4\right )-x^{4} \ln \relax (x )-2 x^{3} \ln \relax (x )-x^{2} \ln \relax (x )\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-32*x^4-32*x^3-128*x^2-128*x)*exp(x)^2+(8*x^5+32*x^4+48*x^3+128*x^2+64*x)*exp(x)-4*x^5-6*x^4-18*x^3-24*
x^2-8*x)*ln(x)+(-32*x^5-80*x^4-176*x^3-320*x^2-192*x)*exp(x)^2+(8*x^6+48*x^5+96*x^4+216*x^3+256*x^2+96*x)*exp(
x)-5*x^6-13*x^5-31*x^4-55*x^3-44*x^2-10*x)/(x^2+4),x,method=_RETURNVERBOSE)

[Out]

(-x^4+8*exp(x)*x^3-16*exp(2*x)*x^2-2*x^3+8*exp(x)*x^2-x^2)*ln(x)-x^5+8*exp(x)*x^4-16*exp(2*x)*x^3-3*x^4+16*exp
(x)*x^3-16*exp(2*x)*x^2-3*x^3+8*exp(x)*x^2-x^2+ln(x^2+4)

________________________________________________________________________________________

maxima [B]  time = 0.76, size = 88, normalized size = 3.14 \begin {gather*} -x^{5} - 3 \, x^{4} - 3 \, x^{3} - x^{2} - 16 \, {\left (x^{3} + x^{2} \log \relax (x) + x^{2}\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x^{4} + 2 \, x^{3} + x^{2} + {\left (x^{3} + x^{2}\right )} \log \relax (x)\right )} e^{x} - {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (x) + \log \left (x^{2} + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-32*x^4-32*x^3-128*x^2-128*x)*exp(x)^2+(8*x^5+32*x^4+48*x^3+128*x^2+64*x)*exp(x)-4*x^5-6*x^4-18*x
^3-24*x^2-8*x)*log(x)+(-32*x^5-80*x^4-176*x^3-320*x^2-192*x)*exp(x)^2+(8*x^6+48*x^5+96*x^4+216*x^3+256*x^2+96*
x)*exp(x)-5*x^6-13*x^5-31*x^4-55*x^3-44*x^2-10*x)/(x^2+4),x, algorithm="maxima")

[Out]

-x^5 - 3*x^4 - 3*x^3 - x^2 - 16*(x^3 + x^2*log(x) + x^2)*e^(2*x) + 8*(x^4 + 2*x^3 + x^2 + (x^3 + x^2)*log(x))*
e^x - (x^4 + 2*x^3 + x^2)*log(x) + log(x^2 + 4)

________________________________________________________________________________________

mupad [B]  time = 1.72, size = 103, normalized size = 3.68 \begin {gather*} \ln \left (x^2+4\right )-\ln \relax (x)\,\left (16\,x^2\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (8\,x^3+8\,x^2\right )+x^2+2\,x^3+x^4\right )+{\mathrm {e}}^x\,\left (8\,x^4+16\,x^3+8\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (16\,x^3+16\,x^2\right )-x^2-3\,x^3-3\,x^4-x^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(10*x + log(x)*(8*x - exp(x)*(64*x + 128*x^2 + 48*x^3 + 32*x^4 + 8*x^5) + exp(2*x)*(128*x + 128*x^2 + 32*
x^3 + 32*x^4) + 24*x^2 + 18*x^3 + 6*x^4 + 4*x^5) - exp(x)*(96*x + 256*x^2 + 216*x^3 + 96*x^4 + 48*x^5 + 8*x^6)
+ exp(2*x)*(192*x + 320*x^2 + 176*x^3 + 80*x^4 + 32*x^5) + 44*x^2 + 55*x^3 + 31*x^4 + 13*x^5 + 5*x^6)/(x^2 +
4),x)

[Out]

log(x^2 + 4) - log(x)*(16*x^2*exp(2*x) - exp(x)*(8*x^2 + 8*x^3) + x^2 + 2*x^3 + x^4) + exp(x)*(8*x^2 + 16*x^3
+ 8*x^4) - exp(2*x)*(16*x^2 + 16*x^3) - x^2 - 3*x^3 - 3*x^4 - x^5

________________________________________________________________________________________

sympy [B]  time = 0.63, size = 100, normalized size = 3.57 \begin {gather*} - x^{5} - 3 x^{4} - 3 x^{3} - x^{2} + \left (- 16 x^{3} - 16 x^{2} \log {\relax (x )} - 16 x^{2}\right ) e^{2 x} + \left (- x^{4} - 2 x^{3} - x^{2}\right ) \log {\relax (x )} + \left (8 x^{4} + 8 x^{3} \log {\relax (x )} + 16 x^{3} + 8 x^{2} \log {\relax (x )} + 8 x^{2}\right ) e^{x} + \log {\left (x^{2} + 4 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-32*x**4-32*x**3-128*x**2-128*x)*exp(x)**2+(8*x**5+32*x**4+48*x**3+128*x**2+64*x)*exp(x)-4*x**5-6
*x**4-18*x**3-24*x**2-8*x)*ln(x)+(-32*x**5-80*x**4-176*x**3-320*x**2-192*x)*exp(x)**2+(8*x**6+48*x**5+96*x**4+
216*x**3+256*x**2+96*x)*exp(x)-5*x**6-13*x**5-31*x**4-55*x**3-44*x**2-10*x)/(x**2+4),x)

[Out]

-x**5 - 3*x**4 - 3*x**3 - x**2 + (-16*x**3 - 16*x**2*log(x) - 16*x**2)*exp(2*x) + (-x**4 - 2*x**3 - x**2)*log(
x) + (8*x**4 + 8*x**3*log(x) + 16*x**3 + 8*x**2*log(x) + 8*x**2)*exp(x) + log(x**2 + 4)

________________________________________________________________________________________