\(\int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx\) [40]

Optimal result

Integrand size = 19, antiderivative size = 265 \[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\frac {(e+f x)^3 \log \left (1-e^{-2 (c+d x)}\right )}{3 f}-\frac {(e+f x)^3 \log \left (1-\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{3 f}+\frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{-2 (c+d x)}\right )}{2 d}+\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{2 d}-\frac {f (e+f x) \operatorname {PolyLog}\left (3,e^{-2 (c+d x)}\right )}{2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{2 d^2}-\frac {f^2 \operatorname {PolyLog}\left (4,e^{-2 (c+d x)}\right )}{4 d^3}+\frac {f^2 \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{4 d^3} \] Output:

1/3*(f*x+e)^3*ln(1-exp(-2*d*x-2*c))/f-1/3*(f*x+e)^3*ln(1-(a-b)/(a+b)/exp(2 
*d*x+2*c))/f+1/3*(f*x+e)^3*ln(a+b*coth(d*x+c))/f-1/2*(f*x+e)^2*polylog(2,e 
xp(-2*d*x-2*c))/d+1/2*(f*x+e)^2*polylog(2,(a-b)/(a+b)/exp(2*d*x+2*c))/d-1/ 
2*f*(f*x+e)*polylog(3,exp(-2*d*x-2*c))/d^2+1/2*f*(f*x+e)*polylog(3,(a-b)/( 
a+b)/exp(2*d*x+2*c))/d^2-1/4*f^2*polylog(4,exp(-2*d*x-2*c))/d^3+1/4*f^2*po 
lylog(4,(a-b)/(a+b)/exp(2*d*x+2*c))/d^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(652\) vs. \(2(265)=530\).

Time = 4.57 (sec) , antiderivative size = 652, normalized size of antiderivative = 2.46 \[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\frac {12 d^3 e^2 x \log \left (1-e^{-c-d x}\right )+12 d^3 e f x^2 \log \left (1-e^{-c-d x}\right )+4 d^3 f^2 x^3 \log \left (1-e^{-c-d x}\right )+12 d^3 e^2 x \log \left (1+e^{-c-d x}\right )+12 d^3 e f x^2 \log \left (1+e^{-c-d x}\right )+4 d^3 f^2 x^3 \log \left (1+e^{-c-d x}\right )-12 d^3 e^2 x \log \left (1+\frac {(-a+b) e^{-2 (c+d x)}}{a+b}\right )-12 d^3 e f x^2 \log \left (1+\frac {(-a+b) e^{-2 (c+d x)}}{a+b}\right )-4 d^3 f^2 x^3 \log \left (1+\frac {(-a+b) e^{-2 (c+d x)}}{a+b}\right )+12 d^3 e^2 x \log (a+b \coth (c+d x))+12 d^3 e f x^2 \log (a+b \coth (c+d x))+4 d^3 f^2 x^3 \log (a+b \coth (c+d x))-12 d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-12 d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 d^2 e^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )+12 d^2 e f x \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )+6 d^2 f^2 x^2 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )-24 d e f \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )-24 d f^2 x \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )-24 d e f \operatorname {PolyLog}\left (3,e^{-c-d x}\right )-24 d f^2 x \operatorname {PolyLog}\left (3,e^{-c-d x}\right )+6 d e f \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )+6 d f^2 x \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )-24 f^2 \operatorname {PolyLog}\left (4,-e^{-c-d x}\right )-24 f^2 \operatorname {PolyLog}\left (4,e^{-c-d x}\right )+3 f^2 \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{12 d^3} \] Input:

Integrate[(e + f*x)^2*Log[a + b*Coth[c + d*x]],x]
 

Output:

(12*d^3*e^2*x*Log[1 - E^(-c - d*x)] + 12*d^3*e*f*x^2*Log[1 - E^(-c - d*x)] 
 + 4*d^3*f^2*x^3*Log[1 - E^(-c - d*x)] + 12*d^3*e^2*x*Log[1 + E^(-c - d*x) 
] + 12*d^3*e*f*x^2*Log[1 + E^(-c - d*x)] + 4*d^3*f^2*x^3*Log[1 + E^(-c - d 
*x)] - 12*d^3*e^2*x*Log[1 + (-a + b)/((a + b)*E^(2*(c + d*x)))] - 12*d^3*e 
*f*x^2*Log[1 + (-a + b)/((a + b)*E^(2*(c + d*x)))] - 4*d^3*f^2*x^3*Log[1 + 
 (-a + b)/((a + b)*E^(2*(c + d*x)))] + 12*d^3*e^2*x*Log[a + b*Coth[c + d*x 
]] + 12*d^3*e*f*x^2*Log[a + b*Coth[c + d*x]] + 4*d^3*f^2*x^3*Log[a + b*Cot 
h[c + d*x]] - 12*d^2*(e + f*x)^2*PolyLog[2, -E^(-c - d*x)] - 12*d^2*(e + f 
*x)^2*PolyLog[2, E^(-c - d*x)] + 6*d^2*e^2*PolyLog[2, (a - b)/((a + b)*E^( 
2*(c + d*x)))] + 12*d^2*e*f*x*PolyLog[2, (a - b)/((a + b)*E^(2*(c + d*x))) 
] + 6*d^2*f^2*x^2*PolyLog[2, (a - b)/((a + b)*E^(2*(c + d*x)))] - 24*d*e*f 
*PolyLog[3, -E^(-c - d*x)] - 24*d*f^2*x*PolyLog[3, -E^(-c - d*x)] - 24*d*e 
*f*PolyLog[3, E^(-c - d*x)] - 24*d*f^2*x*PolyLog[3, E^(-c - d*x)] + 6*d*e* 
f*PolyLog[3, (a - b)/((a + b)*E^(2*(c + d*x)))] + 6*d*f^2*x*PolyLog[3, (a 
- b)/((a + b)*E^(2*(c + d*x)))] - 24*f^2*PolyLog[4, -E^(-c - d*x)] - 24*f^ 
2*PolyLog[4, E^(-c - d*x)] + 3*f^2*PolyLog[4, (a - b)/((a + b)*E^(2*(c + d 
*x)))])/(12*d^3)
 

Rubi [F]

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.

\(\displaystyle \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx\)

\(\Big \downarrow \) 3031

\(\displaystyle \frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}-\frac {\int -\frac {b d (e+f x)^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{3 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b d (e+f x)^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{3 f}+\frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{3 f}+\frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {b d \int \left (\frac {\text {csch}^2(c+d x) e^3}{a+b \coth (c+d x)}+\frac {3 f x \text {csch}^2(c+d x) e^2}{a+b \coth (c+d x)}+\frac {3 f^2 x^2 \text {csch}^2(c+d x) e}{a+b \coth (c+d x)}+\frac {f^3 x^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}\right )dx}{3 f}+\frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b d \left (3 e^2 f \int \frac {x \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx+3 e f^2 \int \frac {x^2 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx+f^3 \int \frac {x^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx-\frac {e^3 \log (a+b \coth (c+d x))}{b d}\right )}{3 f}+\frac {(e+f x)^3 \log (a+b \coth (c+d x))}{3 f}\)

Input:

Int[(e + f*x)^2*Log[a + b*Coth[c + d*x]],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2009

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

rule 3031

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1) 
*(Log[u]/(b*(m + 1))), x] - Simp[1/(b*(m + 1))   Int[SimplifyIntegrand[(a + 
 b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && InverseFunc 
tionFreeQ[u, x] && NeQ[m, -1]
 

rule 7293

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 8.65 (sec) , antiderivative size = 4416, normalized size of antiderivative = 16.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(4416\)

Input:

int((f*x+e)^2*ln(a+b*coth(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

-1/4/d^3*f^2*a/(a+b)*polylog(4,(a+b)*exp(2*d*x+2*c)/(a-b))-1/4/d^3*f^2*b/( 
a+b)*polylog(4,(a+b)*exp(2*d*x+2*c)/(a-b))-1/3/f*a*e^3/(a+b)*ln(a*exp(2*d* 
x+2*c)+exp(2*d*x+2*c)*b-a+b)-1/3/f*b*e^3/(a+b)*ln(a*exp(2*d*x+2*c)+exp(2*d 
*x+2*c)*b-a+b)-1/d*a*e^2/(a+b)*dilog((-exp(d*x+c)*a-b*exp(d*x+c)+((a+b)*(a 
-b))^(1/2))/((a+b)*(a-b))^(1/2))-1/d*a*e^2/(a+b)*dilog((exp(d*x+c)*a+b*exp 
(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a-b))^(1/2))-1/d*b*e^2/(a+b)*dilog((- 
exp(d*x+c)*a-b*exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a-b))^(1/2))-1/d*b* 
e^2/(a+b)*dilog((exp(d*x+c)*a+b*exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a- 
b))^(1/2))-1/3*f^2*b/(a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(a-b))*x^3-1/3*f^2*a/ 
(a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(a-b))*x^3-b*e^2/(a+b)*ln((exp(d*x+c)*a+b* 
exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a-b))^(1/2))*x-b*e^2/(a+b)*ln((-ex 
p(d*x+c)*a-b*exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a-b))^(1/2))*x-a*e^2/ 
(a+b)*ln((-exp(d*x+c)*a-b*exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+b)*(a-b))^(1 
/2))*x-a*e^2/(a+b)*ln((exp(d*x+c)*a+b*exp(d*x+c)+((a+b)*(a-b))^(1/2))/((a+ 
b)*(a-b))^(1/2))*x-ln(exp(2*d*x+2*c)-1)*f*x^2*e+2*f/d*e*ln(-exp(d*x+c)+1)* 
x*c-f^2/d^2*ln(-exp(d*x+c)+1)*x*c^2+2*f/d^2*c*e*dilog(exp(d*x+c))-2*f/d^2* 
c*e*dilog(exp(d*x+c)+1)+f*c^2/d^2*e*ln(exp(d*x+c)-1)+f/d^2*e*ln(-exp(d*x+c 
)+1)*c^2+2*f/d*e*polylog(2,exp(d*x+c))*x+2*f/d^2*e*polylog(2,exp(d*x+c))*c 
+2*f/d*e*polylog(2,-exp(d*x+c))*x+2*f/d^2*e*polylog(2,-exp(d*x+c))*c-f^2/d 
^3*polylog(2,exp(d*x+c))*c^2-2*f^2/d^2*polylog(3,exp(d*x+c))*x+f^2/d*po...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1000 vs. \(2 (245) = 490\).

Time = 0.12 (sec) , antiderivative size = 1000, normalized size of antiderivative = 3.77 \[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^2*log(a+b*coth(d*x+c)),x, algorithm="fricas")
 

Output:

-1/3*(6*f^2*polylog(4, sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c 
))) + 6*f^2*polylog(4, -sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + 
c))) - 6*f^2*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 6*f^2*polylog(4, 
-cosh(d*x + c) - sinh(d*x + c)) + 3*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)* 
dilog(sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c))) + 3*(d^2*f^2* 
x^2 + 2*d^2*e*f*x + d^2*e^2)*dilog(-sqrt((a + b)/(a - b))*(cosh(d*x + c) + 
 sinh(d*x + c))) - 3*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*dilog(cosh(d*x 
+ c) + sinh(d*x + c)) - 3*(d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2)*dilog(-cos 
h(d*x + c) - sinh(d*x + c)) - (3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*log(2* 
(a + b)*cosh(d*x + c) + 2*(a + b)*sinh(d*x + c) + 2*(a - b)*sqrt((a + b)/( 
a - b))) - (3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*log(2*(a + b)*cosh(d*x + 
c) + 2*(a + b)*sinh(d*x + c) - 2*(a - b)*sqrt((a + b)/(a - b))) + (d^3*f^2 
*x^3 + 3*d^3*e*f*x^2 + 3*d^3*e^2*x + 3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)* 
log(sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c)) + 1) + (d^3*f^2* 
x^3 + 3*d^3*e*f*x^2 + 3*d^3*e^2*x + 3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*l 
og(-sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c)) + 1) - (d^3*f^2* 
x^3 + 3*d^3*e*f*x^2 + 3*d^3*e^2*x)*log((b*cosh(d*x + c) + a*sinh(d*x + c)) 
/sinh(d*x + c)) - (d^3*f^2*x^3 + 3*d^3*e*f*x^2 + 3*d^3*e^2*x)*log(cosh(d*x 
 + c) + sinh(d*x + c) + 1) + (3*c*d^2*e^2 - 3*c^2*d*e*f + c^3*f^2)*log(cos 
h(d*x + c) + sinh(d*x + c) - 1) - (d^3*f^2*x^3 + 3*d^3*e*f*x^2 + 3*d^3*...
 

Sympy [F]

\[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\int \left (e + f x\right )^{2} \log {\left (a + b \coth {\left (c + d x \right )} \right )}\, dx \] Input:

integrate((f*x+e)**2*ln(a+b*coth(d*x+c)),x)
 

Output:

Integral((e + f*x)**2*log(a + b*coth(c + d*x)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (245) = 490\).

Time = 0.23 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.55 \[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)^2*log(a+b*coth(d*x+c)),x, algorithm="maxima")
 

Output:

-1/18*b*d*(9*(2*d*x*log(-(a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + 
dilog((a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b)))*e^2/(b*d^2) - 18*(d*x*lo 
g(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e^2/(b*d^2) - 18*(d*x*log(-e^(d* 
x + c) + 1) + dilog(e^(d*x + c)))*e^2/(b*d^2) + 9*(2*d^2*x^2*log(-(a*e^(2* 
c) + b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + 2*d*x*dilog((a*e^(2*c) + b*e^(2*c 
))*e^(2*d*x)/(a - b)) - polylog(3, (a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - 
b)))*e*f/(b*d^3) - 18*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x 
+ c)) - 2*polylog(3, -e^(d*x + c)))*e*f/(b*d^3) - 18*(d^2*x^2*log(-e^(d*x 
+ c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*e*f/(b*d 
^3) + 2*(4*d^3*x^3*log(-(a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + 6 
*d^2*x^2*dilog((a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b)) - 6*d*x*polylog( 
3, (a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b)) + 3*polylog(4, (a*e^(2*c) + 
b*e^(2*c))*e^(2*d*x)/(a - b)))*f^2/(b*d^4) - 6*(d^3*x^3*log(e^(d*x + c) + 
1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*po 
lylog(4, -e^(d*x + c)))*f^2/(b*d^4) - 6*(d^3*x^3*log(-e^(d*x + c) + 1) + 3 
*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, 
 e^(d*x + c)))*f^2/(b*d^4)) + 1/3*(f^2*x^3 + 3*e*f*x^2 + 3*e^2*x)*log(b*co 
th(d*x + c) + a)
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\int { {\left (f x + e\right )}^{2} \log \left (b \coth \left (d x + c\right ) + a\right ) \,d x } \] Input:

integrate((f*x+e)^2*log(a+b*coth(d*x+c)),x, algorithm="giac")
 

Output:

integrate((f*x + e)^2*log(b*coth(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\int \ln \left (a+b\,\mathrm {coth}\left (c+d\,x\right )\right )\,{\left (e+f\,x\right )}^2 \,d x \] Input:

int(log(a + b*coth(c + d*x))*(e + f*x)^2,x)
 

Output:

int(log(a + b*coth(c + d*x))*(e + f*x)^2, x)
 

Reduce [F]

\[ \int (e+f x)^2 \log (a+b \coth (c+d x)) \, dx=\left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right )d x \right ) e^{2}+\left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right ) x^{2}d x \right ) f^{2}+2 \left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right ) x d x \right ) e f \] Input:

int((f*x+e)^2*log(a+b*coth(d*x+c)),x)
 

Output:

int(log(coth(c + d*x)*b + a),x)*e**2 + int(log(coth(c + d*x)*b + a)*x**2,x 
)*f**2 + 2*int(log(coth(c + d*x)*b + a)*x,x)*e*f