\(\int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx\) [39]

Optimal result

Integrand size = 19, antiderivative size = 332 \[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\frac {(e+f x)^4 \log \left (1-e^{-2 (c+d x)}\right )}{4 f}-\frac {(e+f x)^4 \log \left (1-\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{4 f}+\frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}-\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,e^{-2 (c+d x)}\right )}{2 d}+\frac {(e+f x)^3 \operatorname {PolyLog}\left (2,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{2 d}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (3,e^{-2 (c+d x)}\right )}{4 d^2}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (3,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{4 d^2}-\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (4,e^{-2 (c+d x)}\right )}{4 d^3}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (4,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{4 d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (5,e^{-2 (c+d x)}\right )}{8 d^4}+\frac {3 f^3 \operatorname {PolyLog}\left (5,\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )}{8 d^4} \] Output:

1/4*(f*x+e)^4*ln(1-exp(-2*d*x-2*c))/f-1/4*(f*x+e)^4*ln(1-(a-b)/(a+b)/exp(2 
*d*x+2*c))/f+1/4*(f*x+e)^4*ln(a+b*coth(d*x+c))/f-1/2*(f*x+e)^3*polylog(2,e 
xp(-2*d*x-2*c))/d+1/2*(f*x+e)^3*polylog(2,(a-b)/(a+b)/exp(2*d*x+2*c))/d-3/ 
4*f*(f*x+e)^2*polylog(3,exp(-2*d*x-2*c))/d^2+3/4*f*(f*x+e)^2*polylog(3,(a- 
b)/(a+b)/exp(2*d*x+2*c))/d^2-3/4*f^2*(f*x+e)*polylog(4,exp(-2*d*x-2*c))/d^ 
3+3/4*f^2*(f*x+e)*polylog(4,(a-b)/(a+b)/exp(2*d*x+2*c))/d^3-3/8*f^3*polylo 
g(5,exp(-2*d*x-2*c))/d^4+3/8*f^3*polylog(5,(a-b)/(a+b)/exp(2*d*x+2*c))/d^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1046\) vs. \(2(332)=664\).

Time = 6.29 (sec) , antiderivative size = 1046, normalized size of antiderivative = 3.15 \[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx =\text {Too large to display} \] Input:

Integrate[(e + f*x)^3*Log[a + b*Coth[c + d*x]],x]
 

Output:

(8*d^4*e^3*x*Log[1 - E^(-c - d*x)] + 12*d^4*e^2*f*x^2*Log[1 - E^(-c - d*x) 
] + 8*d^4*e*f^2*x^3*Log[1 - E^(-c - d*x)] + 2*d^4*f^3*x^4*Log[1 - E^(-c - 
d*x)] + 8*d^4*e^3*x*Log[1 + E^(-c - d*x)] + 12*d^4*e^2*f*x^2*Log[1 + E^(-c 
 - d*x)] + 8*d^4*e*f^2*x^3*Log[1 + E^(-c - d*x)] + 2*d^4*f^3*x^4*Log[1 + E 
^(-c - d*x)] - 8*d^4*e^3*x*Log[1 + (-a + b)/((a + b)*E^(2*(c + d*x)))] - 1 
2*d^4*e^2*f*x^2*Log[1 + (-a + b)/((a + b)*E^(2*(c + d*x)))] - 8*d^4*e*f^2* 
x^3*Log[1 + (-a + b)/((a + b)*E^(2*(c + d*x)))] - 2*d^4*f^3*x^4*Log[1 + (- 
a + b)/((a + b)*E^(2*(c + d*x)))] + 8*d^4*e^3*x*Log[a + b*Coth[c + d*x]] + 
 12*d^4*e^2*f*x^2*Log[a + b*Coth[c + d*x]] + 8*d^4*e*f^2*x^3*Log[a + b*Cot 
h[c + d*x]] + 2*d^4*f^3*x^4*Log[a + b*Coth[c + d*x]] - 8*d^3*(e + f*x)^3*P 
olyLog[2, -E^(-c - d*x)] - 8*d^3*(e + f*x)^3*PolyLog[2, E^(-c - d*x)] + 4* 
d^3*e^3*PolyLog[2, (a - b)/((a + b)*E^(2*(c + d*x)))] + 12*d^3*e^2*f*x*Pol 
yLog[2, (a - b)/((a + b)*E^(2*(c + d*x)))] + 12*d^3*e*f^2*x^2*PolyLog[2, ( 
a - b)/((a + b)*E^(2*(c + d*x)))] + 4*d^3*f^3*x^3*PolyLog[2, (a - b)/((a + 
 b)*E^(2*(c + d*x)))] - 24*d^2*e^2*f*PolyLog[3, -E^(-c - d*x)] - 48*d^2*e* 
f^2*x*PolyLog[3, -E^(-c - d*x)] - 24*d^2*f^3*x^2*PolyLog[3, -E^(-c - d*x)] 
 - 24*d^2*e^2*f*PolyLog[3, E^(-c - d*x)] - 48*d^2*e*f^2*x*PolyLog[3, E^(-c 
 - d*x)] - 24*d^2*f^3*x^2*PolyLog[3, E^(-c - d*x)] + 6*d^2*e^2*f*PolyLog[3 
, (a - b)/((a + b)*E^(2*(c + d*x)))] + 12*d^2*e*f^2*x*PolyLog[3, (a - b)/( 
(a + b)*E^(2*(c + d*x)))] + 6*d^2*f^3*x^2*PolyLog[3, (a - b)/((a + b)*E...
 

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.

\(\displaystyle \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx\)

\(\Big \downarrow \) 3031

\(\displaystyle \frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}-\frac {\int -\frac {b d (e+f x)^4 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{4 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b d (e+f x)^4 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{4 f}+\frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b d \int \frac {(e+f x)^4 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx}{4 f}+\frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}\)

\(\Big \downarrow \) 7293

\(\displaystyle \frac {b d \int \left (\frac {\text {csch}^2(c+d x) e^4}{a+b \coth (c+d x)}+\frac {4 f x \text {csch}^2(c+d x) e^3}{a+b \coth (c+d x)}+\frac {6 f^2 x^2 \text {csch}^2(c+d x) e^2}{a+b \coth (c+d x)}+\frac {4 f^3 x^3 \text {csch}^2(c+d x) e}{a+b \coth (c+d x)}+\frac {f^4 x^4 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}\right )dx}{4 f}+\frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b d \left (4 e^3 f \int \frac {x \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx+6 e^2 f^2 \int \frac {x^2 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx+4 e f^3 \int \frac {x^3 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx+f^4 \int \frac {x^4 \text {csch}^2(c+d x)}{a+b \coth (c+d x)}dx-\frac {e^4 \log (a+b \coth (c+d x))}{b d}\right )}{4 f}+\frac {(e+f x)^4 \log (a+b \coth (c+d x))}{4 f}\)

Input:

Int[(e + f*x)^3*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 = 11.25 (sec) , antiderivative size = 6300, normalized size of antiderivative = 18.98

method	result	size
risch	\(\text {Expression too large to display}\)	\(6300\)

Input:

int((f*x+e)^3*ln(a+b*coth(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1439 vs. \(2 (308) = 616\).

Time = 0.12 (sec) , antiderivative size = 1439, normalized size of antiderivative = 4.33 \[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^3*log(a+b*coth(d*x+c)),x, algorithm="fricas")
 

Output:

1/4*(24*f^3*polylog(5, sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c 
))) + 24*f^3*polylog(5, -sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + 
 c))) - 24*f^3*polylog(5, cosh(d*x + c) + sinh(d*x + c)) - 24*f^3*polylog( 
5, -cosh(d*x + c) - sinh(d*x + c)) - 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3* 
d^3*e^2*f*x + d^3*e^3)*dilog(sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d 
*x + c))) - 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*di 
log(-sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c))) + 4*(d^3*f^3*x 
^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*dilog(cosh(d*x + c) + sinh 
(d*x + c)) + 4*(d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*d 
ilog(-cosh(d*x + c) - sinh(d*x + c)) + (4*c*d^3*e^3 - 6*c^2*d^2*e^2*f + 4* 
c^3*d*e*f^2 - c^4*f^3)*log(2*(a + b)*cosh(d*x + c) + 2*(a + b)*sinh(d*x + 
c) + 2*(a - b)*sqrt((a + b)/(a - b))) + (4*c*d^3*e^3 - 6*c^2*d^2*e^2*f + 4 
*c^3*d*e*f^2 - c^4*f^3)*log(2*(a + b)*cosh(d*x + c) + 2*(a + b)*sinh(d*x + 
 c) - 2*(a - b)*sqrt((a + b)/(a - b))) - (d^4*f^3*x^4 + 4*d^4*e*f^2*x^3 + 
6*d^4*e^2*f*x^2 + 4*d^4*e^3*x + 4*c*d^3*e^3 - 6*c^2*d^2*e^2*f + 4*c^3*d*e* 
f^2 - c^4*f^3)*log(sqrt((a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c)) + 
 1) - (d^4*f^3*x^4 + 4*d^4*e*f^2*x^3 + 6*d^4*e^2*f*x^2 + 4*d^4*e^3*x + 4*c 
*d^3*e^3 - 6*c^2*d^2*e^2*f + 4*c^3*d*e*f^2 - c^4*f^3)*log(-sqrt((a + b)/(a 
 - b))*(cosh(d*x + c) + sinh(d*x + c)) + 1) + (d^4*f^3*x^4 + 4*d^4*e*f^2*x 
^3 + 6*d^4*e^2*f*x^2 + 4*d^4*e^3*x)*log((b*cosh(d*x + c) + a*sinh(d*x +...
 

Sympy [F]

\[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\int \left (e + f x\right )^{3} \log {\left (a + b \coth {\left (c + d x \right )} \right )}\, dx \] Input:

integrate((f*x+e)**3*ln(a+b*coth(d*x+c)),x)
 

Output:

Integral((e + f*x)**3*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. 1051 vs. \(2 (308) = 616\).

Time = 0.24 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.17 \[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)^3*log(a+b*coth(d*x+c)),x, algorithm="maxima")
 

Output:

-1/12*b*d*(6*(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^3/(b*d^2) - 12*(d*x*lo 
g(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e^3/(b*d^2) - 12*(d*x*log(-e^(d* 
x + c) + 1) + dilog(e^(d*x + c)))*e^3/(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^2*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^2*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^2* 
f/(b*d^3) + 4*(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*po 
lylog(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)))*e*f^2/(b*d^4) - 12*(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)))*e*f^2/(b*d^4) - 12*(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)))*e*f^2/(b*d^4) + 3*(2*d^4*x^4*log(-(a*e^(2*c) + 
b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + 4*d^3*x^3*dilog((a*e^(2*c) + b*e^(2*c) 
)*e^(2*d*x)/(a - b)) - 6*d^2*x^2*polylog(3, (a*e^(2*c) + b*e^(2*c))*e^(2*d 
*x)/(a - b)) + 6*d*x*polylog(4, (a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - ...
                                                                                    
                                                                                    
 

Giac [F]

\[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\int { {\left (f x + e\right )}^{3} \log \left (b \coth \left (d x + c\right ) + a\right ) \,d x } \] Input:

integrate((f*x+e)^3*log(a+b*coth(d*x+c)),x, algorithm="giac")
 

Output:

integrate((f*x + e)^3*log(b*coth(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\int \ln \left (a+b\,\mathrm {coth}\left (c+d\,x\right )\right )\,{\left (e+f\,x\right )}^3 \,d x \] Input:

int(log(a + b*coth(c + d*x))*(e + f*x)^3,x)
 

Output:

int(log(a + b*coth(c + d*x))*(e + f*x)^3, x)
 

Reduce [F]

\[ \int (e+f x)^3 \log (a+b \coth (c+d x)) \, dx=\left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right )d x \right ) e^{3}+\left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right ) x^{3}d x \right ) f^{3}+3 \left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right ) x^{2}d x \right ) e \,f^{2}+3 \left (\int \mathrm {log}\left (\coth \left (d x +c \right ) b +a \right ) x d x \right ) e^{2} f \] Input:

int((f*x+e)^3*log(a+b*coth(d*x+c)),x)
 

Output:

int(log(coth(c + d*x)*b + a),x)*e**3 + int(log(coth(c + d*x)*b + a)*x**3,x 
)*f**3 + 3*int(log(coth(c + d*x)*b + a)*x**2,x)*e*f**2 + 3*int(log(coth(c 
+ d*x)*b + a)*x,x)*e**2*f