3.11.66 \(\int \frac {20+4 x+(5-6 x+x^2) \log (\frac {-5+x}{(-1+x) (i \pi +\log (3))})}{5-6 x+x^2} \, dx\) [1066]

Optimal. Leaf size=27 \[ (5+x) \log \left (\frac {1+\frac {4}{1-x}}{i \pi +\log (3)}\right ) \]

[Out]

ln((4/(1-x)+1)/(ln(3)+I*Pi))*(5+x)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 48, normalized size of antiderivative = 1.78, number of steps used = 7, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {6860, 646, 31, 2536} \begin {gather*} -10 \log (1-x)+10 \log (5-x)-(5-x) \log \left (\frac {5-x}{(1-x) (\log (3)+i \pi )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 + 4*x + (5 - 6*x + x^2)*Log[(-5 + x)/((-1 + x)*(I*Pi + Log[3]))])/(5 - 6*x + x^2),x]

[Out]

-10*Log[1 - x] + 10*Log[5 - x] - (5 - x)*Log[(5 - x)/((1 - x)*(I*Pi + Log[3]))]

Rule 31

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

Rule 646

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 2536

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.), x_Symbol] :> Simp[
(a + b*x)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])^p/b), x] - Dist[B*n*p*((b*c - a*d)/b), Int[(A + B*Log[e*((
a + b*x)^n/(c + d*x)^n)])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && EqQ[n + mn, 0] &&
NeQ[b*c - a*d, 0] && IGtQ[p, 0]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 (5+x)}{5-6 x+x^2}+\log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right )\right ) \, dx\\ &=4 \int \frac {5+x}{5-6 x+x^2} \, dx+\int \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right ) \, dx\\ &=-\left ((5-x) \log \left (\frac {5-x}{(1-x) (i \pi +\log (3))}\right )\right )-4 \int \frac {1}{-1+x} \, dx-6 \int \frac {1}{-1+x} \, dx+10 \int \frac {1}{-5+x} \, dx\\ &=-10 \log (1-x)+10 \log (5-x)-(5-x) \log \left (\frac {5-x}{(1-x) (i \pi +\log (3))}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 41, normalized size = 1.52 \begin {gather*} -10 \log (1-x)+10 \log (5-x)+(-5+x) \log \left (\frac {-5+x}{(-1+x) (i \pi +\log (3))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + 4*x + (5 - 6*x + x^2)*Log[(-5 + x)/((-1 + x)*(I*Pi + Log[3]))])/(5 - 6*x + x^2),x]

[Out]

-10*Log[1 - x] + 10*Log[5 - x] + (-5 + x)*Log[(-5 + x)/((-1 + x)*(I*Pi + Log[3]))]

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (26 ) = 52\).
time = 0.86, size = 998, normalized size = 36.96

method	result	size
risch	\(\ln \left (\frac {x -5}{\left (x -1\right ) \left (\ln \left (3\right )+i \pi \right )}\right ) x +5 \ln \left (x -5\right )-5 \ln \left (x -1\right )\)	\(35\)
norman	\(\ln \left (\frac {x -5}{\left (x -1\right ) \left (\ln \left (3\right )+i \pi \right )}\right ) x +5 \ln \left (\frac {x -5}{\left (x -1\right ) \left (\ln \left (3\right )+i \pi \right )}\right )\)	\(44\)
derivativedivides	\(\text {Expression too large to display}\)	\(998\)
default	\(\text {Expression too large to display}\)	\(998\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-6*x+5)*ln((x-5)/(x-1)/(ln(3)+I*Pi))+20+4*x)/(x^2-6*x+5),x,method=_RETURNVERBOSE)

[Out]

-4/(ln(3)+I*Pi)^2*(-I*Pi-ln(3))*(2*ln(I+(-I*ln(3)+Pi)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi)))/(-I*ln(3)+Pi)*Pi
*ln(3)+I*ln(I+(-I*ln(3)+Pi)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi)))/(-I*ln(3)+Pi)*Pi^2-I*ln(I+(-I*ln(3)+Pi)*(-
4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi)))/(-I*ln(3)+Pi)*ln(3)^2-2*ln(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))*(-4/(x-
1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))/(-I*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*(-4/(x-1)/(ln(3)+I*Pi)+1/(
ln(3)+I*Pi))+I)*Pi*ln(3)-I*ln(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))/(-I
*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+I)*Pi^2+I*ln(-4/(x-1)/
(ln(3)+I*Pi)+1/(ln(3)+I*Pi))*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))/(-I*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+
I*Pi))+Pi*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+I)*ln(3)^2-3/(-I*ln(3)+Pi)^2*ln((-I*ln(3)+Pi)/(Pi^2+ln(3)^2)^
(1/2)*(-I*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+I))*Pi^2*ln(3
)+1/(-I*ln(3)+Pi)^2*ln((-I*ln(3)+Pi)/(Pi^2+ln(3)^2)^(1/2)*(-I*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*
(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+I))*ln(3)^3-I/(-I*ln(3)+Pi)^2*ln((-I*ln(3)+Pi)/(Pi^2+ln(3)^2)^(1/2)*(-I
*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+I))*Pi^3+3*I/(-I*ln(3)
+Pi)^2*ln((-I*ln(3)+Pi)/(Pi^2+ln(3)^2)^(1/2)*(-I*ln(3)*(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))+Pi*(-4/(x-1)/(ln
(3)+I*Pi)+1/(ln(3)+I*Pi))+I))*ln(3)^2*Pi+5/2*I*ln(-4/(x-1)/(ln(3)+I*Pi)+1/(ln(3)+I*Pi))*Pi+5/2*ln(-4/(x-1)/(ln
(3)+I*Pi)+1/(ln(3)+I*Pi))*ln(3))

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (24) = 48\).
time = 0.51, size = 141, normalized size = 5.22 \begin {gather*} -x \log \left (i \, \pi + \log \left (3\right )\right ) - \frac {1}{4} \, {\left (4 \, x + 5 \, \log \left (i \, \pi + \log \left (3\right )\right ) - 5 \, \log \left (x - 5\right ) - 4\right )} \log \left (x - 1\right ) - \frac {5}{4} \, \log \left (x - 1\right )^{2} + \frac {1}{4} \, {\left (4 \, x + 5 i - 20\right )} \log \left (x - 5\right ) + \frac {5}{4} \, \log \left (x - 1\right ) \log \left (x - 5\right ) - \frac {5}{4} \, \log \left (x - 5\right )^{2} - \frac {5}{4} \, {\left (\log \left (x - 1\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {x}{-i \, \pi + i \, \pi x + x \log \left (3\right ) - \log \left (3\right )} - \frac {5}{-i \, \pi + i \, \pi x + x \log \left (3\right ) - \log \left (3\right )}\right ) - 6 \, \log \left (x - 1\right ) + 10 \, \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5),x, algorithm="maxima")

[Out]

-x*log(I*pi + log(3)) - 1/4*(4*x + 5*log(I*pi + log(3)) - 5*log(x - 5) - 4)*log(x - 1) - 5/4*log(x - 1)^2 + 1/
4*(4*x + 5*I - 20)*log(x - 5) + 5/4*log(x - 1)*log(x - 5) - 5/4*log(x - 5)^2 - 5/4*(log(x - 1) - log(x - 5))*l
og(x/(-I*pi + I*pi*x + x*log(3) - log(3)) - 5/(-I*pi + I*pi*x + x*log(3) - log(3))) - 6*log(x - 1) + 10*log(x
- 5)

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 25, normalized size = 0.93 \begin {gather*} {\left (x + 5\right )} \log \left (\frac {x - 5}{-i \, \pi + i \, \pi x + {\left (x - 1\right )} \log \left (3\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5),x, algorithm="fricas")

[Out]

(x + 5)*log((x - 5)/(-I*pi + I*pi*x + (x - 1)*log(3)))

________________________________________________________________________________________

Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (17) = 34\).
time = 0.10, size = 53, normalized size = 1.96 \begin {gather*} x \log {\left (\frac {x}{x \log {\left (3 \right )} + i \pi x - \log {\left (3 \right )} - i \pi } - \frac {5}{x \log {\left (3 \right )} + i \pi x - \log {\left (3 \right )} - i \pi } \right )} + 5 \log {\left (x - 5 \right )} - 5 \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-6*x+5)*ln((-5+x)/(-1+x)/(ln(3)+I*pi))+20+4*x)/(x**2-6*x+5),x)

[Out]

x*log(x/(x*log(3) + I*pi*x - log(3) - I*pi) - 5/(x*log(3) + I*pi*x - log(3) - I*pi)) + 5*log(x - 5) - 5*log(x
- 1)

________________________________________________________________________________________

Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (24) = 48\).
time = 0.43, size = 164, normalized size = 6.07 \begin {gather*} 4 \, {\left (-i \, \pi - \log \left (3\right )\right )} {\left (\frac {\log \left (-\frac {-i \, x + 5 i}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )}\right )}{-i \, \pi - \frac {\pi ^{2} {\left (i \, x - 5 i\right )}}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \frac {2 \, \pi {\left (x - 5\right )} \log \left (3\right )}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \frac {{\left (-i \, x + 5 i\right )} \log \left (3\right )^{2}}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )} - \log \left (3\right )} + \frac {3 \, \log \left (-\frac {-i \, x + 5 i}{\pi - \pi x + i \, x \log \left (3\right ) - i \, \log \left (3\right )}\right )}{-2 i \, \pi - 2 \, \log \left (3\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-6*x+5)*log((-5+x)/(-1+x)/(log(3)+I*pi))+20+4*x)/(x^2-6*x+5),x, algorithm="giac")

[Out]

4*(-I*pi - log(3))*(log(-(-I*x + 5*I)/(pi - pi*x + I*x*log(3) - I*log(3)))/(-I*pi - pi^2*(I*x - 5*I)/(pi - pi*
x + I*x*log(3) - I*log(3)) - 2*pi*(x - 5)*log(3)/(pi - pi*x + I*x*log(3) - I*log(3)) - (-I*x + 5*I)*log(3)^2/(
pi - pi*x + I*x*log(3) - I*log(3)) - log(3)) + 3*log(-(-I*x + 5*I)/(pi - pi*x + I*x*log(3) - I*log(3)))/(-2*I*
pi - 2*log(3)))

________________________________________________________________________________________

Mupad [B]
time = 1.46, size = 30, normalized size = 1.11 \begin {gather*} -10\,\mathrm {atanh}\left (\frac {x}{2}-\frac {3}{2}\right )+x\,\ln \left (\frac {x-5}{\left (\ln \left (3\right )+\Pi \,1{}\mathrm {i}\right )\,\left (x-1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log((x - 5)/((Pi*1i + log(3))*(x - 1)))*(x^2 - 6*x + 5) + 20)/(x^2 - 6*x + 5),x)

[Out]

x*log((x - 5)/((Pi*1i + log(3))*(x - 1))) - 10*atanh(x/2 - 3/2)

________________________________________________________________________________________