3.56.3 \(\int \frac {-36 x+13 x^2-x^3+(640-340 x+40 x^2) \log (x)+(320-160 x+20 x^2) \log (4 x^2)}{16 x-8 x^2+x^3} \, dx\)

Optimal. Leaf size=27 \[ 5-x-5 \log (x) \left (\frac {x}{4-x}-4 \log \left (4 x^2\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.34, antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 8, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1594, 27, 6742, 43, 2357, 2314, 31, 2301} \begin {gather*} 5 \log ^2\left (4 x^2\right )-x+20 \log ^2(x)-\frac {5 x \log (x)}{4-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-36*x + 13*x^2 - x^3 + (640 - 340*x + 40*x^2)*Log[x] + (320 - 160*x + 20*x^2)*Log[4*x^2])/(16*x - 8*x^2 +
x^3),x]

[Out]

-x - (5*x*Log[x])/(4 - x) + 20*Log[x]^2 + 5*Log[4*x^2]^2

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1594

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

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

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

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 {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{x \left (16-8 x+x^2\right )} \, dx\\ &=\int \frac {-36 x+13 x^2-x^3+\left (640-340 x+40 x^2\right ) \log (x)+\left (320-160 x+20 x^2\right ) \log \left (4 x^2\right )}{(-4+x)^2 x} \, dx\\ &=\int \left (-\frac {36}{(-4+x)^2}+\frac {13 x}{(-4+x)^2}-\frac {x^2}{(-4+x)^2}+\frac {20 \left (32-17 x+2 x^2\right ) \log (x)}{(-4+x)^2 x}+\frac {20 \log \left (4 x^2\right )}{x}\right ) \, dx\\ &=-\frac {36}{4-x}+13 \int \frac {x}{(-4+x)^2} \, dx+20 \int \frac {\left (32-17 x+2 x^2\right ) \log (x)}{(-4+x)^2 x} \, dx+20 \int \frac {\log \left (4 x^2\right )}{x} \, dx-\int \frac {x^2}{(-4+x)^2} \, dx\\ &=-\frac {36}{4-x}+5 \log ^2\left (4 x^2\right )+13 \int \left (\frac {4}{(-4+x)^2}+\frac {1}{-4+x}\right ) \, dx+20 \int \left (-\frac {\log (x)}{(-4+x)^2}+\frac {2 \log (x)}{x}\right ) \, dx-\int \left (1+\frac {16}{(-4+x)^2}+\frac {8}{-4+x}\right ) \, dx\\ &=-x+5 \log (4-x)+5 \log ^2\left (4 x^2\right )-20 \int \frac {\log (x)}{(-4+x)^2} \, dx+40 \int \frac {\log (x)}{x} \, dx\\ &=-x+5 \log (4-x)-\frac {5 x \log (x)}{4-x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right )-5 \int \frac {1}{-4+x} \, dx\\ &=-x-\frac {5 x \log (x)}{4-x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 30, normalized size = 1.11 \begin {gather*} -x+\frac {5 x \log (x)}{-4+x}+20 \log ^2(x)+5 \log ^2\left (4 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-36*x + 13*x^2 - x^3 + (640 - 340*x + 40*x^2)*Log[x] + (320 - 160*x + 20*x^2)*Log[4*x^2])/(16*x - 8
*x^2 + x^3),x]

[Out]

-x + (5*x*Log[x])/(-4 + x) + 20*Log[x]^2 + 5*Log[4*x^2]^2

________________________________________________________________________________________

fricas [A]  time = 0.60, size = 37, normalized size = 1.37 \begin {gather*} \frac {40 \, {\left (x - 4\right )} \log \relax (x)^{2} - x^{2} + 5 \, {\left (8 \, {\left (x - 4\right )} \log \relax (2) + x\right )} \log \relax (x) + 4 \, x}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-160*x+320)*log(4*x^2)+(40*x^2-340*x+640)*log(x)-x^3+13*x^2-36*x)/(x^3-8*x^2+16*x),x, algori
thm="fricas")

[Out]

(40*(x - 4)*log(x)^2 - x^2 + 5*(8*(x - 4)*log(2) + x)*log(x) + 4*x)/(x - 4)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {x^{3} - 13 \, x^{2} - 20 \, {\left (x^{2} - 8 \, x + 16\right )} \log \left (4 \, x^{2}\right ) - 20 \, {\left (2 \, x^{2} - 17 \, x + 32\right )} \log \relax (x) + 36 \, x}{x^{3} - 8 \, x^{2} + 16 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-160*x+320)*log(4*x^2)+(40*x^2-340*x+640)*log(x)-x^3+13*x^2-36*x)/(x^3-8*x^2+16*x),x, algori
thm="giac")

[Out]

integrate(-(x^3 - 13*x^2 - 20*(x^2 - 8*x + 16)*log(4*x^2) - 20*(2*x^2 - 17*x + 32)*log(x) + 36*x)/(x^3 - 8*x^2
+ 16*x), x)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 35, normalized size = 1.30

method	result	size
default	\(-x +40 \ln \relax (2) \ln \relax (x )+5 \ln \left (x^{2}\right )^{2}+\frac {5 \ln \relax (x ) x}{x -4}+20 \ln \relax (x )^{2}\)	\(35\)
norman	\(\frac {-80 \ln \relax (x )^{2}-20 \ln \left (4 x^{2}\right )^{2}+4 x +20 \ln \relax (x )+5 x \ln \left (4 x^{2}\right )^{2}+20 x \ln \relax (x )^{2}-x^{2}}{x -4}+5 \ln \relax (x )\)	\(59\)
risch	\(40 \ln \relax (x )^{2}+\frac {20 \ln \relax (x )}{x -4}-x -10 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+20 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-10 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3}+40 \ln \relax (2) \ln \relax (x )+5 \ln \relax (x )\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^2-160*x+320)*ln(4*x^2)+(40*x^2-340*x+640)*ln(x)-x^3+13*x^2-36*x)/(x^3-8*x^2+16*x),x,method=_RETURNV
ERBOSE)

[Out]

-x+40*ln(2)*ln(x)+5*ln(x^2)^2+5*ln(x)*x/(x-4)+20*ln(x)^2

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 33, normalized size = 1.22 \begin {gather*} 5 \, {\left (8 \, \log \relax (2) + 1\right )} \log \relax (x) - x + \frac {20 \, {\left (2 \, {\left (x - 4\right )} \log \relax (x)^{2} + \log \relax (x)\right )}}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^2-160*x+320)*log(4*x^2)+(40*x^2-340*x+640)*log(x)-x^3+13*x^2-36*x)/(x^3-8*x^2+16*x),x, algori
thm="maxima")

[Out]

5*(8*log(2) + 1)*log(x) - x + 20*(2*(x - 4)*log(x)^2 + log(x))/(x - 4)

________________________________________________________________________________________

mupad [B]  time = 3.93, size = 42, normalized size = 1.56 \begin {gather*} 5\,\ln \relax (x)-x+20\,\ln \left (x^2\right )\,\ln \relax (x)+40\,\ln \relax (2)\,\ln \relax (x)-\frac {20\,x^2\,\ln \relax (x)}{4\,x^2-x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(4*x^2)*(20*x^2 - 160*x + 320) - 36*x + log(x)*(40*x^2 - 340*x + 640) + 13*x^2 - x^3)/(16*x - 8*x^2 +
x^3),x)

[Out]

5*log(x) - x + 20*log(x^2)*log(x) + 40*log(2)*log(x) - (20*x^2*log(x))/(4*x^2 - x^3)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 27, normalized size = 1.00 \begin {gather*} - x + 40 \log {\relax (x )}^{2} + 5 \left (1 + 8 \log {\relax (2 )}\right ) \log {\relax (x )} + \frac {20 \log {\relax (x )}}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**2-160*x+320)*ln(4*x**2)+(40*x**2-340*x+640)*ln(x)-x**3+13*x**2-36*x)/(x**3-8*x**2+16*x),x)

[Out]

-x + 40*log(x)**2 + 5*(1 + 8*log(2))*log(x) + 20*log(x)/(x - 4)

________________________________________________________________________________________