3.19.88 \(\int \frac {-4 x^{12}+13 x^{12} \log (x)}{5 \log (3) \log ^5(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {3+\frac {x^{13}}{\log ^4(x)}+\log ^2(\log (4))}{5 \log (3)} \]

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Rubi [C]  time = 0.47, antiderivative size = 188, normalized size of antiderivative = 8.17, number of steps used = 22, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {12, 2561, 2306, 2309, 2178, 2366, 14, 6482} \begin {gather*} -\frac {28561 (4-13 \log (x)) \text {Ei}(13 \log (x))}{120 \log (3)}-\frac {371293 \log (x) \text {Ei}(13 \log (x))}{120 \log (3)}+\frac {28561 \text {Ei}(13 \log (x))}{30 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {169 x^{13}}{60 \log (3) \log ^2(x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {2197 x^{13}}{40 \log (3) \log (x)}+\frac {28561 x^{13}}{120 \log (3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^12 + 13*x^12*Log[x])/(5*Log[3]*Log[x]^5),x]

[Out]

(28561*x^13)/(120*Log[3]) + (28561*ExpIntegralEi[13*Log[x]])/(30*Log[3]) - (28561*ExpIntegralEi[13*Log[x]]*(4
- 13*Log[x]))/(120*Log[3]) + (x^13*(4 - 13*Log[x]))/(20*Log[3]*Log[x]^4) - (13*x^13)/(60*Log[3]*Log[x]^3) + (1
3*x^13*(4 - 13*Log[x]))/(60*Log[3]*Log[x]^3) - (169*x^13)/(60*Log[3]*Log[x]^2) + (169*x^13*(4 - 13*Log[x]))/(1
20*Log[3]*Log[x]^2) - (2197*x^13)/(40*Log[3]*Log[x]) + (2197*x^13*(4 - 13*Log[x]))/(120*Log[3]*Log[x]) - (3712
93*ExpIntegralEi[13*Log[x]]*Log[x])/(120*Log[3])

Rule 12

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

Rule 14

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]]

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 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 2561

Int[(u_.)*((a_.)*(x_)^(m_.) + Log[(c_.)*(x_)^(n_.)]^(q_.)*(b_.)*(x_)^(r_.))^(p_.), x_Symbol] :> Int[u*x^(p*r)*
(a*x^(m - r) + b*Log[c*x^n]^q)^p, x] /; FreeQ[{a, b, c, m, n, p, q, r}, x] && IntegerQ[p]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-4 x^{12}+13 x^{12} \log (x)}{\log ^5(x)} \, dx}{5 \log (3)}\\ &=\frac {\int \frac {x^{12} (-4+13 \log (x))}{\log ^5(x)} \, dx}{5 \log (3)}\\ &=-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {13 \int \frac {28561 \text {Ei}(13 \log (x))-\frac {x^{13} \left (6+26 \log (x)+169 \log ^2(x)+2197 \log ^3(x)\right )}{\log ^4(x)}}{24 x} \, dx}{5 \log (3)}\\ &=-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {13 \int \frac {28561 \text {Ei}(13 \log (x))-\frac {x^{13} \left (6+26 \log (x)+169 \log ^2(x)+2197 \log ^3(x)\right )}{\log ^4(x)}}{x} \, dx}{120 \log (3)}\\ &=-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {13 \int \left (\frac {28561 \text {Ei}(13 \log (x))}{x}-\frac {6 x^{12}}{\log ^4(x)}-\frac {26 x^{12}}{\log ^3(x)}-\frac {169 x^{12}}{\log ^2(x)}-\frac {2197 x^{12}}{\log (x)}\right ) \, dx}{120 \log (3)}\\ &=-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}+\frac {13 \int \frac {x^{12}}{\log ^4(x)} \, dx}{20 \log (3)}+\frac {169 \int \frac {x^{12}}{\log ^3(x)} \, dx}{60 \log (3)}+\frac {2197 \int \frac {x^{12}}{\log ^2(x)} \, dx}{120 \log (3)}+\frac {28561 \int \frac {x^{12}}{\log (x)} \, dx}{120 \log (3)}-\frac {371293 \int \frac {\text {Ei}(13 \log (x))}{x} \, dx}{120 \log (3)}\\ &=-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {169 x^{13}}{120 \log (3) \log ^2(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {2197 x^{13}}{120 \log (3) \log (x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}+\frac {169 \int \frac {x^{12}}{\log ^3(x)} \, dx}{60 \log (3)}+\frac {2197 \int \frac {x^{12}}{\log ^2(x)} \, dx}{120 \log (3)}+\frac {28561 \int \frac {x^{12}}{\log (x)} \, dx}{120 \log (3)}+\frac {28561 \operatorname {Subst}\left (\int \frac {e^{13 x}}{x} \, dx,x,\log (x)\right )}{120 \log (3)}-\frac {371293 \operatorname {Subst}(\int \text {Ei}(13 x) \, dx,x,\log (x))}{120 \log (3)}\\ &=\frac {28561 x^{13}}{120 \log (3)}+\frac {28561 \text {Ei}(13 \log (x))}{120 \log (3)}-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {169 x^{13}}{60 \log (3) \log ^2(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {2197 x^{13}}{60 \log (3) \log (x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {371293 \text {Ei}(13 \log (x)) \log (x)}{120 \log (3)}+\frac {2197 \int \frac {x^{12}}{\log ^2(x)} \, dx}{120 \log (3)}+\frac {28561 \int \frac {x^{12}}{\log (x)} \, dx}{120 \log (3)}+\frac {28561 \operatorname {Subst}\left (\int \frac {e^{13 x}}{x} \, dx,x,\log (x)\right )}{120 \log (3)}\\ &=\frac {28561 x^{13}}{120 \log (3)}+\frac {28561 \text {Ei}(13 \log (x))}{60 \log (3)}-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {169 x^{13}}{60 \log (3) \log ^2(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {2197 x^{13}}{40 \log (3) \log (x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {371293 \text {Ei}(13 \log (x)) \log (x)}{120 \log (3)}+\frac {28561 \int \frac {x^{12}}{\log (x)} \, dx}{120 \log (3)}+\frac {28561 \operatorname {Subst}\left (\int \frac {e^{13 x}}{x} \, dx,x,\log (x)\right )}{120 \log (3)}\\ &=\frac {28561 x^{13}}{120 \log (3)}+\frac {28561 \text {Ei}(13 \log (x))}{40 \log (3)}-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {169 x^{13}}{60 \log (3) \log ^2(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {2197 x^{13}}{40 \log (3) \log (x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {371293 \text {Ei}(13 \log (x)) \log (x)}{120 \log (3)}+\frac {28561 \operatorname {Subst}\left (\int \frac {e^{13 x}}{x} \, dx,x,\log (x)\right )}{120 \log (3)}\\ &=\frac {28561 x^{13}}{120 \log (3)}+\frac {28561 \text {Ei}(13 \log (x))}{30 \log (3)}-\frac {28561 \text {Ei}(13 \log (x)) (4-13 \log (x))}{120 \log (3)}+\frac {x^{13} (4-13 \log (x))}{20 \log (3) \log ^4(x)}-\frac {13 x^{13}}{60 \log (3) \log ^3(x)}+\frac {13 x^{13} (4-13 \log (x))}{60 \log (3) \log ^3(x)}-\frac {169 x^{13}}{60 \log (3) \log ^2(x)}+\frac {169 x^{13} (4-13 \log (x))}{120 \log (3) \log ^2(x)}-\frac {2197 x^{13}}{40 \log (3) \log (x)}+\frac {2197 x^{13} (4-13 \log (x))}{120 \log (3) \log (x)}-\frac {371293 \text {Ei}(13 \log (x)) \log (x)}{120 \log (3)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 0.65 \begin {gather*} \frac {x^{13}}{5 \log (3) \log ^4(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^12 + 13*x^12*Log[x])/(5*Log[3]*Log[x]^5),x]

[Out]

x^13/(5*Log[3]*Log[x]^4)

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fricas [A]  time = 0.62, size = 13, normalized size = 0.57 \begin {gather*} \frac {x^{13}}{5 \, \log \relax (3) \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(13*x^12*log(x)-4*x^12)/log(3)/log(x)^5,x, algorithm="fricas")

[Out]

1/5*x^13/(log(3)*log(x)^4)

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giac [A]  time = 0.20, size = 13, normalized size = 0.57 \begin {gather*} \frac {x^{13}}{5 \, \log \relax (3) \log \relax (x)^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(13*x^12*log(x)-4*x^12)/log(3)/log(x)^5,x, algorithm="giac")

[Out]

1/5*x^13/(log(3)*log(x)^4)

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maple [A]  time = 0.02, size = 14, normalized size = 0.61

method	result	size
default	\(\frac {x^{13}}{5 \ln \relax (3) \ln \relax (x )^{4}}\)	\(14\)
risch	\(\frac {x^{13}}{5 \ln \relax (3) \ln \relax (x )^{4}}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(13*x^12*ln(x)-4*x^12)/ln(3)/ln(x)^5,x,method=_RETURNVERBOSE)

[Out]

1/5/ln(3)*x^13/ln(x)^4

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maxima [C]  time = 0.53, size = 21, normalized size = 0.91 \begin {gather*} \frac {28561 \, {\left (\Gamma \left (-3, -13 \, \log \relax (x)\right ) + 4 \, \Gamma \left (-4, -13 \, \log \relax (x)\right )\right )}}{5 \, \log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(13*x^12*log(x)-4*x^12)/log(3)/log(x)^5,x, algorithm="maxima")

[Out]

28561/5*(gamma(-3, -13*log(x)) + 4*gamma(-4, -13*log(x)))/log(3)

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mupad [B]  time = 1.24, size = 13, normalized size = 0.57 \begin {gather*} \frac {x^{13}}{5\,\ln \relax (3)\,{\ln \relax (x)}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((13*x^12*log(x))/5 - (4*x^12)/5)/(log(3)*log(x)^5),x)

[Out]

x^13/(5*log(3)*log(x)^4)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.52 \begin {gather*} \frac {x^{13}}{5 \log {\relax (3 )} \log {\relax (x )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(13*x**12*ln(x)-4*x**12)/ln(3)/ln(x)**5,x)

[Out]

x**13/(5*log(3)*log(x)**4)

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