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

Optimal. Leaf size=28 \[ 2+e^{x^2} \left (-4+e^x\right ) x^2-\frac {x^3}{e^8 \log ^2(x)} \]

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Rubi [C]  time = 0.50, antiderivative size = 150, normalized size of antiderivative = 5.36, number of steps used = 14, number of rules used = 8, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.127, Rules used = {12, 6742, 2288, 2306, 2309, 2178, 2366, 6482} \begin {gather*} -\frac {9 \text {Ei}(3 \log (x))}{2 e^8}+\frac {9 (2-3 \log (x)) \text {Ei}(3 \log (x))}{2 e^8}+\frac {27 \log (x) \text {Ei}(3 \log (x))}{e^8}-\frac {9 (3 \log (x)+1) \text {Ei}(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 ) \end {gather*}

Antiderivative was successfully verified.

[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 2178

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

Rule 2288

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 2306

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 2309

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 2366

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 6482

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 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.19, size = 28, normalized size = 1.00 \begin {gather*} \frac {x^2 \left (e^{8+x^2} \left (-4+e^x\right )-\frac {x}{\log ^2(x)}\right )}{e^8} \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.57, size = 38, normalized size = 1.36 \begin {gather*} -\frac {{\left ({\left (4 \, x^{2} e^{8} - x^{2} e^{\left (x + 8\right )}\right )} e^{\left (x^{2}\right )} \log \relax (x)^{2} + x^{3}\right )} e^{\left (-8\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.23, size = 43, normalized size = 1.54 \begin {gather*} \frac {{\left (x^{2} e^{\left (x^{2} + x + 8\right )} \log \relax (x)^{2} - 4 \, x^{2} e^{\left (x^{2} + 8\right )} \log \relax (x)^{2} - x^{3}\right )} e^{\left (-8\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.04, size = 25, normalized size = 0.89

method	result	size
risch	\(x^{2} {\mathrm e}^{x^{2}} \left ({\mathrm e}^{x}-4\right )-\frac {x^{3} {\mathrm e}^{-8}}{\ln \relax (x )^{2}}\)	\(25\)
default	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{2} {\mathrm e}^{x^{2}+x}-\frac {x^{3}}{\ln \relax (x )^{2}}-4 \,{\mathrm e}^{8} {\mathrm e}^{x^{2}} x^{2}\right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [C]  time = 0.43, size = 281, normalized size = 10.04 \begin {gather*} \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 \relax (x)\right ) - 144 \, \Gamma \left (-2, -3 \, \log \relax (x)\right )\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.71, size = 31, normalized size = 1.11 \begin {gather*} x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x-4\,x^2\,{\mathrm {e}}^{x^2}-\frac {x^3\,{\mathrm {e}}^{-8}}{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.47, size = 27, normalized size = 0.96 \begin {gather*} - \frac {x^{3}}{e^{8} \log {\relax (x )}^{2}} + \left (x^{2} e^{x} - 4 x^{2}\right ) e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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