\(\int \frac {16-48 \log (\frac {-32-7 x}{4+x})+48 \log ^2(\frac {-32-7 x}{4+x})-16 \log ^3(\frac {-32-7 x}{4+x})}{128+60 x+7 x^2} \, dx\) [699]

Optimal result

Integrand size = 61, antiderivative size = 16 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=261+\left (-1+\log \left (-8+\frac {x}{4+x}\right )\right )^4 \] Output:

(ln(x/(4+x)-8)-1)^4+261
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(16)=32\).

Time = 1.55 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.81 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=-16 \left (\frac {1}{4} \log \left (-\frac {32+7 x}{4+x}\right )-\frac {3}{8} \log ^2\left (-\frac {32+7 x}{4+x}\right )+\frac {1}{4} \log ^3\left (-\frac {32+7 x}{4+x}\right )-\frac {1}{16} \log ^4\left (-\frac {32+7 x}{4+x}\right )\right ) \] Input:

Integrate[(16 - 48*Log[(-32 - 7*x)/(4 + x)] + 48*Log[(-32 - 7*x)/(4 + x)]^ 
2 - 16*Log[(-32 - 7*x)/(4 + x)]^3)/(128 + 60*x + 7*x^2),x]
 

Output:

-16*(Log[-((32 + 7*x)/(4 + x))]/4 - (3*Log[-((32 + 7*x)/(4 + x))]^2)/8 + L 
og[-((32 + 7*x)/(4 + x))]^3/4 - Log[-((32 + 7*x)/(4 + x))]^4/16)
 

Rubi [A] (warning: unable to verify)

Time = 0.43 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {7239, 27, 2975, 2962, 2739, 15}

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.

\(\displaystyle \int \frac {-16 \log ^3\left (\frac {-7 x-32}{x+4}\right )+48 \log ^2\left (\frac {-7 x-32}{x+4}\right )-48 \log \left (\frac {-7 x-32}{x+4}\right )+16}{7 x^2+60 x+128} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {16 \left (1-\log \left (-\frac {7 x+32}{x+4}\right )\right )^3}{7 x^2+60 x+128}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 16 \int \frac {\left (1-\log \left (-\frac {7 x+32}{x+4}\right )\right )^3}{7 x^2+60 x+128}dx\)

\(\Big \downarrow \) 2975

\(\displaystyle 16 \int \frac {\left (1-\log \left (-\frac {7 x+32}{x+4}\right )\right )^3}{(x+4) (7 x+32)}dx\)

\(\Big \downarrow \) 2962

\(\displaystyle -4 \int \frac {(x+4) \left (1-\log \left (-\frac {7 x+32}{x+4}\right )\right )^3}{7 x+32}d\frac {7 x+32}{x+4}\)

\(\Big \downarrow \) 2739

\(\displaystyle 4 \int \frac {(7 x+32)^3}{(x+4)^3}d\left (1-\log \left (-\frac {7 x+32}{x+4}\right )\right )\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {(7 x+32)^4}{(x+4)^4}\)

Input:

Int[(16 - 48*Log[(-32 - 7*x)/(4 + x)] + 48*Log[(-32 - 7*x)/(4 + x)]^2 - 16 
*Log[(-32 - 7*x)/(4 + x)]^3)/(128 + 60*x + 7*x^2),x]
 

Output:

(32 + 7*x)^4/(4 + x)^4
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2739

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

rule 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

rule 2975

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(m_.), x_Symbol] :> Sim 
p[h^m/(b^m*d^m)   Int[(a + b*x)^m*(c + d*x)^m*(A + B*Log[e*((a + b*x)^n/(c 
+ d*x)^n)])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, n, p}, x] && 
EqQ[n + mn, 0] && IGtQ[n, 0] && EqQ[b*d*f - a*c*h, 0] && EqQ[b*d*g - h*(b*c 
 + a*d), 0] && IntegerQ[m]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(16)=32\).

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.38

method	result	size
derivativedivides	\(\ln \left (-7-\frac {4}{4+x}\right )^{4}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )\)	\(54\)
default	\(\ln \left (-7-\frac {4}{4+x}\right )^{4}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )\)	\(54\)
risch	\(4 \ln \left (4+x \right )-4 \ln \left (7 x +32\right )+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+\ln \left (-7-\frac {4}{4+x}\right )^{4}\)	\(56\)
parts	\(4 \ln \left (4+x \right )-4 \ln \left (7 x +32\right )+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+\ln \left (-7-\frac {4}{4+x}\right )^{4}\)	\(56\)
norman	\(\ln \left (\frac {-7 x -32}{4+x}\right )^{4}-4 \ln \left (\frac {-7 x -32}{4+x}\right )+6 \ln \left (\frac {-7 x -32}{4+x}\right )^{2}-4 \ln \left (\frac {-7 x -32}{4+x}\right )^{3}\)	\(62\)

Input:

int((-16*ln((-7*x-32)/(4+x))^3+48*ln((-7*x-32)/(4+x))^2-48*ln((-7*x-32)/(4 
+x))+16)/(7*x^2+60*x+128),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (16) = 32\).

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 4.06 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=\log \left (-\frac {7 \, x + 32}{x + 4}\right )^{4} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{3} + 6 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{2} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right ) \] Input:

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7 
*x-32)/(4+x))+16)/(7*x^2+60*x+128),x, algorithm="fricas")
 

Output:

log(-(7*x + 32)/(x + 4))^4 - 4*log(-(7*x + 32)/(x + 4))^3 + 6*log(-(7*x + 
32)/(x + 4))^2 - 4*log(-(7*x + 32)/(x + 4))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).

Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.62 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=\log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{4} - 4 \log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{3} + 6 \log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{2} + 4 \log {\left (x + 4 \right )} - 4 \log {\left (x + \frac {32}{7} \right )} \] Input:

integrate((-16*ln((-7*x-32)/(4+x))**3+48*ln((-7*x-32)/(4+x))**2-48*ln((-7* 
x-32)/(4+x))+16)/(7*x**2+60*x+128),x)
 

Output:

log((-7*x - 32)/(x + 4))**4 - 4*log((-7*x - 32)/(x + 4))**3 + 6*log((-7*x 
- 32)/(x + 4))**2 + 4*log(x + 4) - 4*log(x + 32/7)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (16) = 32\).

Time = 0.06 (sec) , antiderivative size = 411, normalized size of antiderivative = 25.69 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=-\log \left (7 \, x + 32\right )^{4} + 4 \, \log \left (7 \, x + 32\right )^{3} \log \left (x + 4\right ) - 6 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right )^{2} + 4 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{3} - \log \left (x + 4\right )^{4} + 4 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{3} - 4 \, \log \left (7 \, x + 32\right )^{3} + 12 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right ) - 12 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{2} + 4 \, \log \left (x + 4\right )^{3} - 6 \, {\left (\log \left (7 \, x + 32\right )^{2} - 2 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) + \log \left (x + 4\right )^{2}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{2} - 12 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{2} - 6 \, \log \left (7 \, x + 32\right )^{2} + 12 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) - 6 \, \log \left (x + 4\right )^{2} + 4 \, {\left (\log \left (7 \, x + 32\right )^{3} - 3 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right ) + 3 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{2} - \log \left (x + 4\right )^{3}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) + 12 \, {\left (\log \left (7 \, x + 32\right )^{2} - 2 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) + \log \left (x + 4\right )^{2}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) + 12 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) - 4 \, \log \left (7 \, x + 32\right ) + 4 \, \log \left (x + 4\right ) \] Input:

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7 
*x-32)/(4+x))+16)/(7*x^2+60*x+128),x, algorithm="maxima")
 

Output:

-log(7*x + 32)^4 + 4*log(7*x + 32)^3*log(x + 4) - 6*log(7*x + 32)^2*log(x 
+ 4)^2 + 4*log(7*x + 32)*log(x + 4)^3 - log(x + 4)^4 + 4*(log(7*x + 32) - 
log(x + 4))*log(-7*x/(x + 4) - 32/(x + 4))^3 - 4*log(7*x + 32)^3 + 12*log( 
7*x + 32)^2*log(x + 4) - 12*log(7*x + 32)*log(x + 4)^2 + 4*log(x + 4)^3 - 
6*(log(7*x + 32)^2 - 2*log(7*x + 32)*log(x + 4) + log(x + 4)^2)*log(-7*x/( 
x + 4) - 32/(x + 4))^2 - 12*(log(7*x + 32) - log(x + 4))*log(-7*x/(x + 4) 
- 32/(x + 4))^2 - 6*log(7*x + 32)^2 + 12*log(7*x + 32)*log(x + 4) - 6*log( 
x + 4)^2 + 4*(log(7*x + 32)^3 - 3*log(7*x + 32)^2*log(x + 4) + 3*log(7*x + 
 32)*log(x + 4)^2 - log(x + 4)^3)*log(-7*x/(x + 4) - 32/(x + 4)) + 12*(log 
(7*x + 32)^2 - 2*log(7*x + 32)*log(x + 4) + log(x + 4)^2)*log(-7*x/(x + 4) 
 - 32/(x + 4)) + 12*(log(7*x + 32) - log(x + 4))*log(-7*x/(x + 4) - 32/(x 
+ 4)) - 4*log(7*x + 32) + 4*log(x + 4)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (16) = 32\).

Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 4.06 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=\log \left (-\frac {7 \, x + 32}{x + 4}\right )^{4} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{3} + 6 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{2} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right ) \] Input:

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7 
*x-32)/(4+x))+16)/(7*x^2+60*x+128),x, algorithm="giac")
 

Output:

log(-(7*x + 32)/(x + 4))^4 - 4*log(-(7*x + 32)/(x + 4))^3 + 6*log(-(7*x + 
32)/(x + 4))^2 - 4*log(-(7*x + 32)/(x + 4))
 

Mupad [B] (verification not implemented)

Time = 4.70 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.81 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx={\ln \left (-\frac {7\,x+32}{x+4}\right )}^4-4\,{\ln \left (-\frac {7\,x+32}{x+4}\right )}^3+6\,{\ln \left (-\frac {7\,x+32}{x+4}\right )}^2+\mathrm {atan}\left (\frac {x\,7{}\mathrm {i}}{2}+15{}\mathrm {i}\right )\,8{}\mathrm {i} \] Input:

int(-(48*log(-(7*x + 32)/(x + 4)) - 48*log(-(7*x + 32)/(x + 4))^2 + 16*log 
(-(7*x + 32)/(x + 4))^3 - 16)/(60*x + 7*x^2 + 128),x)
 

Output:

atan((x*7i)/2 + 15i)*8i + 6*log(-(7*x + 32)/(x + 4))^2 - 4*log(-(7*x + 32) 
/(x + 4))^3 + log(-(7*x + 32)/(x + 4))^4
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.81 \[ \int \frac {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx=-4 \,\mathrm {log}\left (7 x +32\right )+4 \,\mathrm {log}\left (x +4\right )+\mathrm {log}\left (\frac {-7 x -32}{x +4}\right )^{4}-4 \mathrm {log}\left (\frac {-7 x -32}{x +4}\right )^{3}+6 \mathrm {log}\left (\frac {-7 x -32}{x +4}\right )^{2} \] Input:

int((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7*x-32) 
/(4+x))+16)/(7*x^2+60*x+128),x)
 

Output:

 - 4*log(7*x + 32) + 4*log(x + 4) + log(( - 7*x - 32)/(x + 4))**4 - 4*log( 
( - 7*x - 32)/(x + 4))**3 + 6*log(( - 7*x - 32)/(x + 4))**2