\(\int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} (e^8 (-8 x-8 x^3)+e^{8+x} (2 x+x^2+2 x^3)) \log ^3(x)}{e^8 \log ^3(x)} \, dx\) [7436]

Optimal result

Integrand size = 63, antiderivative size = 28 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=2+e^{x^2} \left (-4+e^x\right ) x^2-\frac {x^3}{e^8 \log ^2(x)} \]

[Out]

2+x^2*exp(x^2)*(exp(x)-4)-x^3/exp(4)^2/ln(x)^2

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 5.36, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.127, Rules used = {12, 6874, 2326, 2343, 2346, 2209, 2413, 6617} \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x))}{2 e^8}+\frac {9 (2-3 \log (x)) \operatorname {ExpIntegralEi}(3 \log (x))}{2 e^8}+\frac {27 \log (x) \operatorname {ExpIntegralEi}(3 \log (x))}{e^8}-\frac {9 (3 \log (x)+1) \operatorname {ExpIntegralEi}(3 \log (x))}{2 e^8}-\frac {9 x^3}{e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}+\frac {3 x^3 (3 \log (x)+1)}{2 e^8 \log (x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}-e^{x^2} \left (4 x^2-e^x x^2\right ) \]

[In]

Int[(2*x^2 - 3*x^2*Log[x] + E^x^2*(E^8*(-8*x - 8*x^3) + E^(8 + x)*(2*x + x^2 + 2*x^3))*Log[x]^3)/(E^8*Log[x]^3
),x]

[Out]

(-9*x^3)/E^8 - E^x^2*(4*x^2 - E^x*x^2) - (9*ExpIntegralEi[3*Log[x]])/(2*E^8) + (9*ExpIntegralEi[3*Log[x]]*(2 -
3*Log[x]))/(2*E^8) - (x^3*(2 - 3*Log[x]))/(2*E^8*Log[x]^2) - (3*x^3*(2 - 3*Log[x]))/(2*E^8*Log[x]) + (27*ExpI
ntegralEi[3*Log[x]]*Log[x])/E^8 - (9*ExpIntegralEi[3*Log[x]]*(1 + 3*Log[x]))/(2*E^8) + (3*x^3*(1 + 3*Log[x]))/
(2*E^8*Log[x])

Rule 12

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

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2343

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

Rule 2346

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

Rule 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 6617

Int[ExpIntegralEi[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(ExpIntegralEi[a + b*x]/b), x] - Simp[E^(a
+ b*x)/b, x] /; FreeQ[{a, b}, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{\log ^3(x)} \, dx}{e^8} \\ & = \frac {\int \left (e^{8+x^2} x \left (-8+2 e^x+e^x x-8 x^2+2 e^x x^2\right )-\frac {x^2 (-2+3 \log (x))}{\log ^3(x)}\right ) \, dx}{e^8} \\ & = \frac {\int e^{8+x^2} x \left (-8+2 e^x+e^x x-8 x^2+2 e^x x^2\right ) \, dx}{e^8}-\frac {\int \frac {x^2 (-2+3 \log (x))}{\log ^3(x)} \, dx}{e^8} \\ & = -e^{x^2} \left (4 x^2-e^x x^2\right )+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}+\frac {3 \int \left (\frac {9 \operatorname {ExpIntegralEi}(3 \log (x))}{2 x}-\frac {x^2 (1+3 \log (x))}{2 \log ^2(x)}\right ) \, dx}{e^8} \\ & = -e^{x^2} \left (4 x^2-e^x x^2\right )+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}-\frac {3 \int \frac {x^2 (1+3 \log (x))}{\log ^2(x)} \, dx}{2 e^8}+\frac {27 \int \frac {\operatorname {ExpIntegralEi}(3 \log (x))}{x} \, dx}{2 e^8} \\ & = -e^{x^2} \left (4 x^2-e^x x^2\right )+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (1+3 \log (x))}{2 e^8}+\frac {3 x^3 (1+3 \log (x))}{2 e^8 \log (x)}+\frac {9 \int \left (\frac {3 \operatorname {ExpIntegralEi}(3 \log (x))}{x}-\frac {x^2}{\log (x)}\right ) \, dx}{2 e^8}+\frac {27 \text {Subst}(\int \operatorname {ExpIntegralEi}(3 x) \, dx,x,\log (x))}{2 e^8} \\ & = -\frac {9 x^3}{2 e^8}-e^{x^2} \left (4 x^2-e^x x^2\right )+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}+\frac {27 \operatorname {ExpIntegralEi}(3 \log (x)) \log (x)}{2 e^8}-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (1+3 \log (x))}{2 e^8}+\frac {3 x^3 (1+3 \log (x))}{2 e^8 \log (x)}-\frac {9 \int \frac {x^2}{\log (x)} \, dx}{2 e^8}+\frac {27 \int \frac {\operatorname {ExpIntegralEi}(3 \log (x))}{x} \, dx}{2 e^8} \\ & = -\frac {9 x^3}{2 e^8}-e^{x^2} \left (4 x^2-e^x x^2\right )+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}+\frac {27 \operatorname {ExpIntegralEi}(3 \log (x)) \log (x)}{2 e^8}-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (1+3 \log (x))}{2 e^8}+\frac {3 x^3 (1+3 \log (x))}{2 e^8 \log (x)}-\frac {9 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )}{2 e^8}+\frac {27 \text {Subst}(\int \operatorname {ExpIntegralEi}(3 x) \, dx,x,\log (x))}{2 e^8} \\ & = -\frac {9 x^3}{e^8}-e^{x^2} \left (4 x^2-e^x x^2\right )-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x))}{2 e^8}+\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (2-3 \log (x))}{2 e^8}-\frac {x^3 (2-3 \log (x))}{2 e^8 \log ^2(x)}-\frac {3 x^3 (2-3 \log (x))}{2 e^8 \log (x)}+\frac {27 \operatorname {ExpIntegralEi}(3 \log (x)) \log (x)}{e^8}-\frac {9 \operatorname {ExpIntegralEi}(3 \log (x)) (1+3 \log (x))}{2 e^8}+\frac {3 x^3 (1+3 \log (x))}{2 e^8 \log (x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {x^2 \left (e^{8+x^2} \left (-4+e^x\right )-\frac {x}{\log ^2(x)}\right )}{e^8} \]

[In]

Integrate[(2*x^2 - 3*x^2*Log[x] + E^x^2*(E^8*(-8*x - 8*x^3) + E^(8 + x)*(2*x + x^2 + 2*x^3))*Log[x]^3)/(E^8*Lo
g[x]^3),x]

[Out]

(x^2*(E^(8 + x^2)*(-4 + E^x) - x/Log[x]^2))/E^8

Maple [A] (verified)

Time = 53.46 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89

method	result	size
risch	\(x^{2} {\mathrm e}^{x^{2}} \left ({\mathrm e}^{x}-4\right )-\frac {x^{3} {\mathrm e}^{-8}}{\ln \left (x \right )^{2}}\)	\(25\)
default	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{2} {\mathrm e}^{x^{2}+x}-\frac {x^{3}}{\ln \left (x \right )^{2}}-4 \,{\mathrm e}^{8} x^{2} {\mathrm e}^{x^{2}}\right )\)	\(43\)
parallelrisch	\(\frac {{\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{2} \ln \left (x \right )^{2} {\mathrm e}^{x} {\mathrm e}^{x^{2}}-4 \,{\mathrm e}^{8} x^{2} \ln \left (x \right )^{2} {\mathrm e}^{x^{2}}-x^{3}\right )}{\ln \left (x \right )^{2}}\)	\(51\)
parts	\(-4 x^{2} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{x^{2}+x}+2 \,{\mathrm e}^{-8} \left (-\frac {x^{3}}{2 \ln \left (x \right )^{2}}-\frac {3 x^{3}}{2 \ln \left (x \right )}-\frac {9 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x \right )\right )}{2}\right )-3 \,{\mathrm e}^{-8} \left (-\frac {x^{3}}{\ln \left (x \right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x \right )\right )\right )\)	\(78\)

[In]

int((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*ln(x)^3-3*x^2*ln(x)+2*x^2)/exp(4)^2/ln(x
)^3,x,method=_RETURNVERBOSE)

[Out]

x^2*exp(x^2)*(exp(x)-4)-x^3*exp(-8)/ln(x)^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=-\frac {{\left ({\left (4 \, x^{2} e^{8} - x^{2} e^{\left (x + 8\right )}\right )} e^{\left (x^{2}\right )} \log \left (x\right )^{2} + x^{3}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \]

[In]

integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4
)^2/log(x)^3,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=- \frac {x^{3}}{e^{8} \log {\left (x \right )}^{2}} + \left (x^{2} e^{x} - 4 x^{2}\right ) e^{x^{2}} \]

[In]

integrate((((2*x**3+x**2+2*x)*exp(4)**2*exp(x)+(-8*x**3-8*x)*exp(4)**2)*exp(x**2)*ln(x)**3-3*x**2*ln(x)+2*x**2
)/exp(4)**2/ln(x)**3,x)

[Out]

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

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.26 (sec) , antiderivative size = 281, normalized size of antiderivative = 10.04 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {1}{8} \, {\left ({\left (\frac {12 \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{\left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} + 6 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )} - 8 \, \Gamma \left (2, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )\right )} e^{\frac {31}{4}} - {\left (\frac {4 \, {\left (2 \, x + 1\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}{\left (-{\left (2 \, x + 1\right )}^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} + 4 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\frac {31}{4}} - 4 \, {\left (\frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} - 2 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\frac {31}{4}} - 32 \, {\left (x^{2} e^{8} - e^{8}\right )} e^{\left (x^{2}\right )} - 32 \, e^{\left (x^{2} + 8\right )} - 72 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) - 144 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right )\right )} e^{\left (-8\right )} \]

[In]

integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4
)^2/log(x)^3,x, algorithm="maxima")

[Out]

1/8*((12*(2*x + 1)^3*gamma(3/2, -1/4*(2*x + 1)^2)/(-(2*x + 1)^2)^(3/2) - sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*
x + 1)^2)) - 1)/sqrt(-(2*x + 1)^2) + 6*e^(1/4*(2*x + 1)^2) - 8*gamma(2, -1/4*(2*x + 1)^2))*e^(31/4) - (4*(2*x
+ 1)^3*gamma(3/2, -1/4*(2*x + 1)^2)/(-(2*x + 1)^2)^(3/2) - sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1
)/sqrt(-(2*x + 1)^2) + 4*e^(1/4*(2*x + 1)^2))*e^(31/4) - 4*(sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) -
1)/sqrt(-(2*x + 1)^2) - 2*e^(1/4*(2*x + 1)^2))*e^(31/4) - 32*(x^2*e^8 - e^8)*e^(x^2) - 32*e^(x^2 + 8) - 72*gam
ma(-1, -3*log(x)) - 144*gamma(-2, -3*log(x)))*e^(-8)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=\frac {{\left (x^{2} e^{\left (x^{2} + x + 8\right )} \log \left (x\right )^{2} - 4 \, x^{2} e^{\left (x^{2} + 8\right )} \log \left (x\right )^{2} - x^{3}\right )} e^{\left (-8\right )}}{\log \left (x\right )^{2}} \]

[In]

integrate((((2*x^3+x^2+2*x)*exp(4)^2*exp(x)+(-8*x^3-8*x)*exp(4)^2)*exp(x^2)*log(x)^3-3*x^2*log(x)+2*x^2)/exp(4
)^2/log(x)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 11.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {2 x^2-3 x^2 \log (x)+e^{x^2} \left (e^8 \left (-8 x-8 x^3\right )+e^{8+x} \left (2 x+x^2+2 x^3\right )\right ) \log ^3(x)}{e^8 \log ^3(x)} \, dx=x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x-4\,x^2\,{\mathrm {e}}^{x^2}-\frac {x^3\,{\mathrm {e}}^{-8}}{{\ln \left (x\right )}^2} \]

[In]

int(-(exp(-8)*(3*x^2*log(x) - 2*x^2 + exp(x^2)*log(x)^3*(exp(8)*(8*x + 8*x^3) - exp(8)*exp(x)*(2*x + x^2 + 2*x
^3))))/log(x)^3,x)

[Out]

x^2*exp(x^2)*exp(x) - 4*x^2*exp(x^2) - (x^3*exp(-8))/log(x)^2