3.91.38 \(\int \frac {33 x+20 x^3+3 x^5+(-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5) \log (22-x)}{(-726 x+33 x^2-440 x^3+20 x^4-66 x^5+3 x^6) \log (22-x)} \, dx\)

Optimal. Leaf size=22 \[ \log \left (\left (x+\frac {2 x}{3 \left (3+x^2\right )}\right ) \log (22-x)\right ) \]

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Rubi [A]  time = 0.77, antiderivative size = 26, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 7, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {6741, 6742, 1663, 1628, 2390, 2302, 29} \begin {gather*} -\log \left (x^2+3\right )+\log \left (3 x^2+11\right )+\log (x)+\log (\log (22-x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(33*x + 20*x^3 + 3*x^5 + (-726 + 33*x - 352*x^2 + 16*x^3 - 66*x^4 + 3*x^5)*Log[22 - x])/((-726*x + 33*x^2
- 440*x^3 + 20*x^4 - 66*x^5 + 3*x^6)*Log[22 - x]),x]

[Out]

Log[x] - Log[3 + x^2] + Log[11 + 3*x^2] + Log[Log[22 - x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)
*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && Inte
gerQ[(m - 1)/2]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 6741

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

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 {-33 x-20 x^3-3 x^5-\left (-726+33 x-352 x^2+16 x^3-66 x^4+3 x^5\right ) \log (22-x)}{x \left (726-33 x+440 x^2-20 x^3+66 x^4-3 x^5\right ) \log (22-x)} \, dx\\ &=\int \left (\frac {33+16 x^2+3 x^4}{x \left (33+20 x^2+3 x^4\right )}+\frac {1}{(-22+x) \log (22-x)}\right ) \, dx\\ &=\int \frac {33+16 x^2+3 x^4}{x \left (33+20 x^2+3 x^4\right )} \, dx+\int \frac {1}{(-22+x) \log (22-x)} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {33+16 x+3 x^2}{x \left (33+20 x+3 x^2\right )} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,22-x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {2}{3+x}+\frac {6}{11+3 x}\right ) \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log (22-x)\right )\\ &=\log (x)-\log \left (3+x^2\right )+\log \left (11+3 x^2\right )+\log (\log (22-x))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.18 \begin {gather*} \log (x)-\log \left (3+x^2\right )+\log \left (11+3 x^2\right )+\log (\log (22-x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(33*x + 20*x^3 + 3*x^5 + (-726 + 33*x - 352*x^2 + 16*x^3 - 66*x^4 + 3*x^5)*Log[22 - x])/((-726*x + 3
3*x^2 - 440*x^3 + 20*x^4 - 66*x^5 + 3*x^6)*Log[22 - x]),x]

[Out]

Log[x] - Log[3 + x^2] + Log[11 + 3*x^2] + Log[Log[22 - x]]

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fricas [A]  time = 0.54, size = 26, normalized size = 1.18 \begin {gather*} \log \left (3 \, x^{3} + 11 \, x\right ) - \log \left (x^{2} + 3\right ) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+33*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33
*x^2-726*x)/log(22-x),x, algorithm="fricas")

[Out]

log(3*x^3 + 11*x) - log(x^2 + 3) + log(log(-x + 22))

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giac [A]  time = 0.16, size = 26, normalized size = 1.18 \begin {gather*} \log \left (3 \, x^{2} + 11\right ) - \log \left (x^{2} + 3\right ) + \log \relax (x) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+33*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33
*x^2-726*x)/log(22-x),x, algorithm="giac")

[Out]

log(3*x^2 + 11) - log(x^2 + 3) + log(x) + log(log(-x + 22))

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maple [A]  time = 0.11, size = 27, normalized size = 1.23

method	result	size
norman	\(-\ln \left (x^{2}+3\right )+\ln \relax (x )+\ln \left (\ln \left (22-x \right )\right )+\ln \left (3 x^{2}+11\right )\)	\(27\)
risch	\(-\ln \left (x^{2}+3\right )+\ln \left (3 x^{3}+11 x \right )+\ln \left (\ln \left (22-x \right )\right )\)	\(27\)
derivativedivides	\(\ln \left (\ln \left (22-x \right )\right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )\)	\(43\)
default	\(\ln \left (\ln \left (22-x \right )\right )-\ln \left (\left (22-x \right )^{2}-481+44 x \right )+\ln \left (3 \left (22-x \right )^{2}-1441+132 x \right )+\ln \left (-x \right )\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*ln(22-x)+3*x^5+20*x^3+33*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33*x^2-72
6*x)/ln(22-x),x,method=_RETURNVERBOSE)

[Out]

-ln(x^2+3)+ln(x)+ln(ln(22-x))+ln(3*x^2+11)

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maxima [A]  time = 0.50, size = 26, normalized size = 1.18 \begin {gather*} \log \left (3 \, x^{2} + 11\right ) - \log \left (x^{2} + 3\right ) + \log \relax (x) + \log \left (\log \left (-x + 22\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^5-66*x^4+16*x^3-352*x^2+33*x-726)*log(22-x)+3*x^5+20*x^3+33*x)/(3*x^6-66*x^5+20*x^4-440*x^3+33
*x^2-726*x)/log(22-x),x, algorithm="maxima")

[Out]

log(3*x^2 + 11) - log(x^2 + 3) + log(x) + log(log(-x + 22))

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mupad [B]  time = 5.47, size = 24, normalized size = 1.09 \begin {gather*} \ln \left (\ln \left (22-x\right )\right )+\ln \left (x^3+\frac {11\,x}{3}\right )-\ln \left (x^2+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(33*x + log(22 - x)*(33*x - 352*x^2 + 16*x^3 - 66*x^4 + 3*x^5 - 726) + 20*x^3 + 3*x^5)/(log(22 - x)*(726*
x - 33*x^2 + 440*x^3 - 20*x^4 + 66*x^5 - 3*x^6)),x)

[Out]

log(log(22 - x)) + log((11*x)/3 + x^3) - log(x^2 + 3)

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sympy [A]  time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} - \log {\left (x^{2} + 3 \right )} + \log {\left (3 x^{3} + 11 x \right )} + \log {\left (\log {\left (22 - x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**5-66*x**4+16*x**3-352*x**2+33*x-726)*ln(22-x)+3*x**5+20*x**3+33*x)/(3*x**6-66*x**5+20*x**4-44
0*x**3+33*x**2-726*x)/ln(22-x),x)

[Out]

-log(x**2 + 3) + log(3*x**3 + 11*x) + log(log(22 - x))

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