3.10.22 \(\int \frac {-63+70 x-6 x^2-x^3+(63-140 x+18 x^2+4 x^3) \log (9 x)}{7 \log ^2(9 x)} \, dx\) [922]

Optimal. Leaf size=21 \[ \frac {(-1+x) x \left (-9+x+\frac {x^2}{7}\right )}{\log (9 x)} \]

[Out]

(-1+x)/ln(9*x)*(1/7*x^2+x-9)*x

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(21)=42\).
time = 0.20, antiderivative size = 47, normalized size of antiderivative = 2.24, number of steps used = 25, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {12, 6874, 2403, 2334, 2335, 2343, 2346, 2209} \begin {gather*} \frac {x^4}{7 \log (9 x)}+\frac {6 x^3}{7 \log (9 x)}-\frac {10 x^2}{\log (9 x)}+\frac {9 x}{\log (9 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-63 + 70*x - 6*x^2 - x^3 + (63 - 140*x + 18*x^2 + 4*x^3)*Log[9*x])/(7*Log[9*x]^2),x]

[Out]

(9*x)/Log[9*x] - (10*x^2)/Log[9*x] + (6*x^3)/(7*Log[9*x]) + x^4/(7*Log[9*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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, 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 2403

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rule 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{7} \int \frac {-63+70 x-6 x^2-x^3+\left (63-140 x+18 x^2+4 x^3\right ) \log (9 x)}{\log ^2(9 x)} \, dx\\ &=\frac {1}{7} \int \left (\frac {-63+70 x-6 x^2-x^3}{\log ^2(9 x)}+\frac {63-140 x+18 x^2+4 x^3}{\log (9 x)}\right ) \, dx\\ &=\frac {1}{7} \int \frac {-63+70 x-6 x^2-x^3}{\log ^2(9 x)} \, dx+\frac {1}{7} \int \frac {63-140 x+18 x^2+4 x^3}{\log (9 x)} \, dx\\ &=\frac {1}{7} \int \left (-\frac {63}{\log ^2(9 x)}+\frac {70 x}{\log ^2(9 x)}-\frac {6 x^2}{\log ^2(9 x)}-\frac {x^3}{\log ^2(9 x)}\right ) \, dx+\frac {1}{7} \int \left (\frac {63}{\log (9 x)}-\frac {140 x}{\log (9 x)}+\frac {18 x^2}{\log (9 x)}+\frac {4 x^3}{\log (9 x)}\right ) \, dx\\ &=-\left (\frac {1}{7} \int \frac {x^3}{\log ^2(9 x)} \, dx\right )+\frac {4}{7} \int \frac {x^3}{\log (9 x)} \, dx-\frac {6}{7} \int \frac {x^2}{\log ^2(9 x)} \, dx+\frac {18}{7} \int \frac {x^2}{\log (9 x)} \, dx-9 \int \frac {1}{\log ^2(9 x)} \, dx+9 \int \frac {1}{\log (9 x)} \, dx+10 \int \frac {x}{\log ^2(9 x)} \, dx-20 \int \frac {x}{\log (9 x)} \, dx\\ &=\frac {9 x}{\log (9 x)}-\frac {10 x^2}{\log (9 x)}+\frac {6 x^3}{7 \log (9 x)}+\frac {x^4}{7 \log (9 x)}+\text {li}(9 x)+\frac {4 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (9 x)\right )}{45927}+\frac {2}{567} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (9 x)\right )-\frac {20}{81} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (9 x)\right )-\frac {4}{7} \int \frac {x^3}{\log (9 x)} \, dx-\frac {18}{7} \int \frac {x^2}{\log (9 x)} \, dx-9 \int \frac {1}{\log (9 x)} \, dx+20 \int \frac {x}{\log (9 x)} \, dx\\ &=-\frac {20}{81} \text {Ei}(2 \log (9 x))+\frac {2}{567} \text {Ei}(3 \log (9 x))+\frac {4 \text {Ei}(4 \log (9 x))}{45927}+\frac {9 x}{\log (9 x)}-\frac {10 x^2}{\log (9 x)}+\frac {6 x^3}{7 \log (9 x)}+\frac {x^4}{7 \log (9 x)}-\frac {4 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (9 x)\right )}{45927}-\frac {2}{567} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (9 x)\right )+\frac {20}{81} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (9 x)\right )\\ &=\frac {9 x}{\log (9 x)}-\frac {10 x^2}{\log (9 x)}+\frac {6 x^3}{7 \log (9 x)}+\frac {x^4}{7 \log (9 x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 24, normalized size = 1.14 \begin {gather*} \frac {x \left (63-70 x+6 x^2+x^3\right )}{7 \log (9 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-63 + 70*x - 6*x^2 - x^3 + (63 - 140*x + 18*x^2 + 4*x^3)*Log[9*x])/(7*Log[9*x]^2),x]

[Out]

(x*(63 - 70*x + 6*x^2 + x^3))/(7*Log[9*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(43\) vs. \(2(19)=38\).
time = 0.09, size = 44, normalized size = 2.10

method	result	size
risch	\(\frac {x \left (x^{3}+6 x^{2}-70 x +63\right )}{7 \ln \left (9 x \right )}\)	\(23\)
norman	\(\frac {9 x -10 x^{2}+\frac {6}{7} x^{3}+\frac {1}{7} x^{4}}{\ln \left (9 x \right )}\)	\(27\)
derivativedivides	\(\frac {x^{4}}{7 \ln \left (9 x \right )}+\frac {6 x^{3}}{7 \ln \left (9 x \right )}-\frac {10 x^{2}}{\ln \left (9 x \right )}+\frac {9 x}{\ln \left (9 x \right )}\)	\(44\)
default	\(\frac {x^{4}}{7 \ln \left (9 x \right )}+\frac {6 x^{3}}{7 \ln \left (9 x \right )}-\frac {10 x^{2}}{\ln \left (9 x \right )}+\frac {9 x}{\ln \left (9 x \right )}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/7*((4*x^3+18*x^2-140*x+63)*ln(9*x)-x^3-6*x^2+70*x-63)/ln(9*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/7*x^4/ln(9*x)+6/7/ln(9*x)*x^3-10/ln(9*x)*x^2+9/ln(9*x)*x

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.29, size = 73, normalized size = 3.48 \begin {gather*} \frac {4}{45927} \, {\rm Ei}\left (4 \, \log \left (9 \, x\right )\right ) + \frac {2}{567} \, {\rm Ei}\left (3 \, \log \left (9 \, x\right )\right ) - \frac {20}{81} \, {\rm Ei}\left (2 \, \log \left (9 \, x\right )\right ) + {\rm Ei}\left (\log \left (9 \, x\right )\right ) - \Gamma \left (-1, -\log \left (9 \, x\right )\right ) + \frac {20}{81} \, \Gamma \left (-1, -2 \, \log \left (9 \, x\right )\right ) - \frac {2}{567} \, \Gamma \left (-1, -3 \, \log \left (9 \, x\right )\right ) - \frac {4}{45927} \, \Gamma \left (-1, -4 \, \log \left (9 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*((4*x^3+18*x^2-140*x+63)*log(9*x)-x^3-6*x^2+70*x-63)/log(9*x)^2,x, algorithm="maxima")

[Out]

4/45927*Ei(4*log(9*x)) + 2/567*Ei(3*log(9*x)) - 20/81*Ei(2*log(9*x)) + Ei(log(9*x)) - gamma(-1, -log(9*x)) + 2
0/81*gamma(-1, -2*log(9*x)) - 2/567*gamma(-1, -3*log(9*x)) - 4/45927*gamma(-1, -4*log(9*x))

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Fricas [A]
time = 0.35, size = 25, normalized size = 1.19 \begin {gather*} \frac {x^{4} + 6 \, x^{3} - 70 \, x^{2} + 63 \, x}{7 \, \log \left (9 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*((4*x^3+18*x^2-140*x+63)*log(9*x)-x^3-6*x^2+70*x-63)/log(9*x)^2,x, algorithm="fricas")

[Out]

1/7*(x^4 + 6*x^3 - 70*x^2 + 63*x)/log(9*x)

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Sympy [A]
time = 0.04, size = 22, normalized size = 1.05 \begin {gather*} \frac {x^{4} + 6 x^{3} - 70 x^{2} + 63 x}{7 \log {\left (9 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*((4*x**3+18*x**2-140*x+63)*ln(9*x)-x**3-6*x**2+70*x-63)/ln(9*x)**2,x)

[Out]

(x**4 + 6*x**3 - 70*x**2 + 63*x)/(7*log(9*x))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).
time = 0.39, size = 43, normalized size = 2.05 \begin {gather*} \frac {x^{4}}{7 \, \log \left (9 \, x\right )} + \frac {6 \, x^{3}}{7 \, \log \left (9 \, x\right )} - \frac {10 \, x^{2}}{\log \left (9 \, x\right )} + \frac {9 \, x}{\log \left (9 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*((4*x^3+18*x^2-140*x+63)*log(9*x)-x^3-6*x^2+70*x-63)/log(9*x)^2,x, algorithm="giac")

[Out]

1/7*x^4/log(9*x) + 6/7*x^3/log(9*x) - 10*x^2/log(9*x) + 9*x/log(9*x)

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Mupad [B]
time = 0.69, size = 22, normalized size = 1.05 \begin {gather*} \frac {x\,\left (x^3+6\,x^2-70\,x+63\right )}{7\,\ln \left (9\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((6*x^2)/7 - (log(9*x)*(18*x^2 - 140*x + 4*x^3 + 63))/7 - 10*x + x^3/7 + 9)/log(9*x)^2,x)

[Out]

(x*(6*x^2 - 70*x + x^3 + 63))/(7*log(9*x))

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