\(\int \frac {-13 x-x^2+x^3+(390+19 x-63 x^2+4 x^3) \log (\frac {15}{-15+x})}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx\) [2619]

Optimal result

Integrand size = 68, antiderivative size = 26 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=\frac {\log \left (\frac {5}{-5+\frac {x}{3}}\right )}{x^2 \left (13+x-x^2\right )} \] Output:

ln(5/(-5+1/3*x))/x^2/(-x^2+x+13)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=-\frac {\log \left (\frac {15}{-15+x}\right )}{x^2 \left (-13-x+x^2\right )} \] Input:

Integrate[(-13*x - x^2 + x^3 + (390 + 19*x - 63*x^2 + 4*x^3)*Log[15/(-15 + 
 x)])/(-2535*x^3 - 221*x^4 + 401*x^5 + 5*x^6 - 17*x^7 + x^8),x]
 

Output:

-(Log[15/(-15 + x)]/(x^2*(-13 - x + x^2)))
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 2.83 (sec) , antiderivative size = 889, normalized size of antiderivative = 34.19, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2026, 2463, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3-x^2+\left (4 x^3-63 x^2+19 x+390\right ) \log \left (\frac {15}{x-15}\right )-13 x}{x^8-17 x^7+5 x^6+401 x^5-221 x^4-2535 x^3} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {x^3-x^2+\left (4 x^3-63 x^2+19 x+390\right ) \log \left (\frac {15}{x-15}\right )-13 x}{x^3 \left (x^5-17 x^4+5 x^3+401 x^2-221 x-2535\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(-x-14) \left (x^3-x^2+\left (4 x^3-63 x^2+19 x+390\right ) \log \left (\frac {15}{x-15}\right )-13 x\right )}{38809 x^3 \left (x^2-x-13\right )}+\frac {(-x-14) \left (x^3-x^2+\left (4 x^3-63 x^2+19 x+390\right ) \log \left (\frac {15}{x-15}\right )-13 x\right )}{197 x^3 \left (x^2-x-13\right )^2}+\frac {x^3-x^2+\left (4 x^3-63 x^2+19 x+390\right ) \log \left (\frac {15}{x-15}\right )-13 x}{38809 (x-15) x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (53+365 \sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{3529058}+\frac {27 \left (1-\sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{8957 \left (29+\sqrt {53}\right )}-\frac {80 \log \left (-2 x-\sqrt {53}+1\right )}{8957 \left (29+\sqrt {53}\right )}+\frac {27 \left (1+\sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{8957 \left (29-\sqrt {53}\right )}-\frac {80 \log \left (-2 x+\sqrt {53}+1\right )}{8957 \left (29-\sqrt {53}\right )}+\frac {\left (53-365 \sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{3529058}+\frac {\left (27+\sqrt {53}\right ) \log \left (-\frac {-2 x-\sqrt {53}+1}{29+\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{66586}-\frac {\left (1431-341 \sqrt {53}\right ) \log \left (-\frac {-2 x-\sqrt {53}+1}{29+\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{3529058}-\frac {\log \left (-\frac {-2 x-\sqrt {53}+1}{29+\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{169 \sqrt {53}}-\frac {\left (1431+341 \sqrt {53}\right ) \log \left (-\frac {-2 x+\sqrt {53}+1}{29-\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{3529058}+\frac {\left (27-\sqrt {53}\right ) \log \left (-\frac {-2 x+\sqrt {53}+1}{29-\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{66586}+\frac {\log \left (-\frac {-2 x+\sqrt {53}+1}{29-\sqrt {53}}\right ) \log \left (-\frac {15}{15-x}\right )}{169 \sqrt {53}}+\frac {27 \left (1-\sqrt {53}\right ) \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x-\sqrt {53}+1\right )}-\frac {80 \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x-\sqrt {53}+1\right )}+\frac {27 \left (1+\sqrt {53}\right ) \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x+\sqrt {53}+1\right )}-\frac {80 \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x+\sqrt {53}+1\right )}-\frac {\log \left (-\frac {15}{15-x}\right )}{169 x}+\frac {\log \left (-\frac {15}{15-x}\right )}{13 x^2}-\frac {27 \left (1-\sqrt {53}\right ) \log (15-x)}{8957 \left (29+\sqrt {53}\right )}+\frac {80 \log (15-x)}{8957 \left (29+\sqrt {53}\right )}-\frac {27 \left (1+\sqrt {53}\right ) \log (15-x)}{8957 \left (29-\sqrt {53}\right )}+\frac {80 \log (15-x)}{8957 \left (29-\sqrt {53}\right )}-\frac {\log (15-x)}{33293}+\frac {\left (1431+341 \sqrt {53}\right ) \operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29-\sqrt {53}}\right )}{3529058}-\frac {\left (27-\sqrt {53}\right ) \operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29-\sqrt {53}}\right )}{66586}-\frac {\operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29-\sqrt {53}}\right )}{169 \sqrt {53}}-\frac {\left (27+\sqrt {53}\right ) \operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29+\sqrt {53}}\right )}{66586}+\frac {\left (1431-341 \sqrt {53}\right ) \operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29+\sqrt {53}}\right )}{3529058}+\frac {\operatorname {PolyLog}\left (2,\frac {2 (15-x)}{29+\sqrt {53}}\right )}{169 \sqrt {53}}\)

Input:

Int[(-13*x - x^2 + x^3 + (390 + 19*x - 63*x^2 + 4*x^3)*Log[15/(-15 + x)])/ 
(-2535*x^3 - 221*x^4 + 401*x^5 + 5*x^6 - 17*x^7 + x^8),x]
 

Output:

(-80*Log[1 - Sqrt[53] - 2*x])/(8957*(29 + Sqrt[53])) + (27*(1 - Sqrt[53])* 
Log[1 - Sqrt[53] - 2*x])/(8957*(29 + Sqrt[53])) + ((53 + 365*Sqrt[53])*Log 
[1 - Sqrt[53] - 2*x])/3529058 + ((53 - 365*Sqrt[53])*Log[1 + Sqrt[53] - 2* 
x])/3529058 - (80*Log[1 + Sqrt[53] - 2*x])/(8957*(29 - Sqrt[53])) + (27*(1 
 + Sqrt[53])*Log[1 + Sqrt[53] - 2*x])/(8957*(29 - Sqrt[53])) - (80*Log[-15 
/(15 - x)])/(8957*(1 - Sqrt[53] - 2*x)) + (27*(1 - Sqrt[53])*Log[-15/(15 - 
 x)])/(8957*(1 - Sqrt[53] - 2*x)) - (80*Log[-15/(15 - x)])/(8957*(1 + Sqrt 
[53] - 2*x)) + (27*(1 + Sqrt[53])*Log[-15/(15 - x)])/(8957*(1 + Sqrt[53] - 
 2*x)) + Log[-15/(15 - x)]/(13*x^2) - Log[-15/(15 - x)]/(169*x) - (Log[-(( 
1 - Sqrt[53] - 2*x)/(29 + Sqrt[53]))]*Log[-15/(15 - x)])/(169*Sqrt[53]) - 
((1431 - 341*Sqrt[53])*Log[-((1 - Sqrt[53] - 2*x)/(29 + Sqrt[53]))]*Log[-1 
5/(15 - x)])/3529058 + ((27 + Sqrt[53])*Log[-((1 - Sqrt[53] - 2*x)/(29 + S 
qrt[53]))]*Log[-15/(15 - x)])/66586 + (Log[-((1 + Sqrt[53] - 2*x)/(29 - Sq 
rt[53]))]*Log[-15/(15 - x)])/(169*Sqrt[53]) + ((27 - Sqrt[53])*Log[-((1 + 
Sqrt[53] - 2*x)/(29 - Sqrt[53]))]*Log[-15/(15 - x)])/66586 - ((1431 + 341* 
Sqrt[53])*Log[-((1 + Sqrt[53] - 2*x)/(29 - Sqrt[53]))]*Log[-15/(15 - x)])/ 
3529058 - Log[15 - x]/33293 + (80*Log[15 - x])/(8957*(29 - Sqrt[53])) - (2 
7*(1 + Sqrt[53])*Log[15 - x])/(8957*(29 - Sqrt[53])) + (80*Log[15 - x])/(8 
957*(29 + Sqrt[53])) - (27*(1 - Sqrt[53])*Log[15 - x])/(8957*(29 + Sqrt[53 
])) - PolyLog[2, (2*(15 - x))/(29 - Sqrt[53])]/(169*Sqrt[53]) - ((27 - ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

method	result	size
norman	\(-\frac {\ln \left (\frac {15}{x -15}\right )}{x^{2} \left (x^{2}-x -13\right )}\)	\(24\)
risch	\(-\frac {\ln \left (\frac {15}{x -15}\right )}{x^{2} \left (x^{2}-x -13\right )}\)	\(24\)
parallelrisch	\(-\frac {\ln \left (\frac {15}{x -15}\right )}{x^{2} \left (x^{2}-x -13\right )}\)	\(24\)
orering	\(-\frac {x \left (9 x^{3}-127 x^{2}+25 x +780\right ) \left (\left (4 x^{3}-63 x^{2}+19 x +390\right ) \ln \left (\frac {15}{x -15}\right )+x^{3}-x^{2}-13 x \right )}{\left (16 x^{3}-189 x^{2}+38 x +390\right ) \left (x^{8}-17 x^{7}+5 x^{6}+401 x^{5}-221 x^{4}-2535 x^{3}\right )}-\frac {\left (x^{2}-x -13\right ) \left (x -15\right ) x^{2} \left (\frac {\left (12 x^{2}-126 x +19\right ) \ln \left (\frac {15}{x -15}\right )-\frac {15 \left (4 x^{3}-63 x^{2}+19 x +390\right ) \left (\frac {x}{15}-1\right )}{\left (x -15\right )^{2}}+3 x^{2}-2 x -13}{x^{8}-17 x^{7}+5 x^{6}+401 x^{5}-221 x^{4}-2535 x^{3}}-\frac {\left (\left (4 x^{3}-63 x^{2}+19 x +390\right ) \ln \left (\frac {15}{x -15}\right )+x^{3}-x^{2}-13 x \right ) \left (8 x^{7}-119 x^{6}+30 x^{5}+2005 x^{4}-884 x^{3}-7605 x^{2}\right )}{\left (x^{8}-17 x^{7}+5 x^{6}+401 x^{5}-221 x^{4}-2535 x^{3}\right )^{2}}\right )}{16 x^{3}-189 x^{2}+38 x +390}\)	\(326\)
derivativedivides	\(\frac {11 \ln \left (\frac {15}{x -15}\right )}{2535 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )}-\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{1764529}+\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{1764529}-\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {15}{x -15}+2\right )}{195 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )^{2}}+\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {7446600 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}-\frac {7446600 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}+\frac {1096200 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}-\frac {1096200 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}+37800 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}-37800 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}+\frac {2349225}{\left (x -15\right )^{2}}+\frac {2349225}{x -15}\right )}{\frac {78212747925}{\left (x -15\right )^{2}}+\frac {11513551725}{x -15}[phone]}\)	\(374\)
default	\(\frac {11 \ln \left (\frac {15}{x -15}\right )}{2535 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )}-\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{1764529}+\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{1764529}-\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {15}{x -15}+2\right )}{195 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )^{2}}+\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {7446600 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}-\frac {7446600 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}+\frac {1096200 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}-\frac {1096200 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}+37800 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}-37800 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}+\frac {2349225}{\left (x -15\right )^{2}}+\frac {2349225}{x -15}\right )}{\frac {78212747925}{\left (x -15\right )^{2}}+\frac {11513551725}{x -15}[phone]}\)	\(374\)
parts	\(-\frac {1}{195 x}-\frac {2 \ln \left (x \right )}{38025}+\frac {\ln \left (x -15\right )}{44325}+\frac {\ln \left (x^{2}-x -13\right )}{66586}+\frac {365 \sqrt {53}\, \operatorname {arctanh}\left (\frac {\left (-1+2 x \right ) \sqrt {53}}{53}\right )}{1764529}-\frac {\ln \left (\frac {44325}{\left (x -15\right )^{2}}+\frac {6525}{x -15}+225\right )}{66586}+\frac {365 \sqrt {53}\, \operatorname {arctanh}\left (\frac {\left (\frac {5910}{x -15}+435\right ) \sqrt {53}}{795}\right )}{1764529}+\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {7446600 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}-\frac {7446600 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{\left (x -15\right )^{2}}+\frac {1096200 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}-\frac {1096200 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}}{x -15}+37800 \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right ) \sqrt {53}-37800 \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right ) \sqrt {53}+\frac {2349225}{\left (x -15\right )^{2}}+\frac {2349225}{x -15}\right )}{\frac {78212747925}{\left (x -15\right )^{2}}+\frac {11513551725}{x -15}[phone]}-\frac {1}{2925 \left (1+\frac {15}{x -15}\right )}+\frac {2 \ln \left (1+\frac {15}{x -15}\right )}{38025}-\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {15}{x -15}+2\right )}{195 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )^{2}}+\frac {11 \ln \left (\frac {15}{x -15}\right )}{2535 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )}-\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{1764529}+\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{1764529}\)	\(480\)

Input:

int(((4*x^3-63*x^2+19*x+390)*ln(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^6 
+401*x^5-221*x^4-2535*x^3),x,method=_RETURNVERBOSE)
 

Output:

-ln(15/(x-15))/x^2/(x^2-x-13)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=-\frac {\log \left (\frac {15}{x - 15}\right )}{x^{4} - x^{3} - 13 \, x^{2}} \] Input:

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^ 
7+5*x^6+401*x^5-221*x^4-2535*x^3),x, algorithm="fricas")
 

Output:

-log(15/(x - 15))/(x^4 - x^3 - 13*x^2)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=- \frac {\log {\left (\frac {15}{x - 15} \right )}}{x^{4} - x^{3} - 13 x^{2}} \] Input:

integrate(((4*x**3-63*x**2+19*x+390)*ln(15/(x-15))+x**3-x**2-13*x)/(x**8-1 
7*x**7+5*x**6+401*x**5-221*x**4-2535*x**3),x)
 

Output:

-log(15/(x - 15))/(x**4 - x**3 - 13*x**2)
 

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=-\frac {\log \left (5\right ) + \log \left (3\right ) - \log \left (x - 15\right )}{x^{4} - x^{3} - 13 \, x^{2}} \] Input:

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^ 
7+5*x^6+401*x^5-221*x^4-2535*x^3),x, algorithm="maxima")
 

Output:

-(log(5) + log(3) - log(x - 15))/(x^4 - x^3 - 13*x^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (23) = 46\).

Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.08 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=-\frac {1}{7490925} \, {\left (\frac {197 \, {\left (\frac {165}{x - 15} - 2\right )}}{\frac {30}{x - 15} + \frac {225}{{\left (x - 15\right )}^{2}} + 1} - \frac {225 \, {\left (\frac {168}{x - 15} - 1\right )}}{\frac {29}{x - 15} + \frac {197}{{\left (x - 15\right )}^{2}} + 1}\right )} \log \left (\frac {15}{x - 15}\right ) - \frac {1}{44325} \, \log \left (\frac {15}{x - 15}\right ) \] Input:

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^ 
7+5*x^6+401*x^5-221*x^4-2535*x^3),x, algorithm="giac")
 

Output:

-1/7490925*(197*(165/(x - 15) - 2)/(30/(x - 15) + 225/(x - 15)^2 + 1) - 22 
5*(168/(x - 15) - 1)/(29/(x - 15) + 197/(x - 15)^2 + 1))*log(15/(x - 15)) 
- 1/44325*log(15/(x - 15))
 

Mupad [B] (verification not implemented)

Time = 4.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=\frac {\ln \left (15\right )+\ln \left (\frac {1}{x-15}\right )}{-x^4+x^3+13\,x^2} \] Input:

int((13*x - log(15/(x - 15))*(19*x - 63*x^2 + 4*x^3 + 390) + x^2 - x^3)/(2 
535*x^3 + 221*x^4 - 401*x^5 - 5*x^6 + 17*x^7 - x^8),x)
 

Output:

(log(15) + log(1/(x - 15)))/(13*x^2 + x^3 - x^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-13 x-x^2+x^3+\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx=\frac {-195 \,\mathrm {log}\left (\frac {15}{x -15}\right )+x^{4}-x^{3}-13 x^{2}}{195 x^{2} \left (x^{2}-x -13\right )} \] Input:

int(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^ 
6+401*x^5-221*x^4-2535*x^3),x)
 

Output:

( - 195*log(15/(x - 15)) + x**4 - x**3 - 13*x**2)/(195*x**2*(x**2 - x - 13 
))