\(\int (e+f x) \log (a+b \tanh (c+d x)) \, dx\) [35]

Optimal result

Integrand size = 17, antiderivative size = 201 \[ \int (e+f x) \log (a+b \tanh (c+d x)) \, dx=\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{2 f}-\frac {(e+f x)^2 \log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a-b}\right )}{2 f}+\frac {(e+f x)^2 \log (a+b \tanh (c+d x))}{2 f}+\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a-b}\right )}{2 d}-\frac {f \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {f \operatorname {PolyLog}\left (3,-\frac {(a+b) e^{2 (c+d x)}}{a-b}\right )}{4 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.38 \[ \int (e+f x) \log (a+b \tanh (c+d x)) \, dx=\frac {2 d^2 f x^2 \log \left (1+e^{-2 (c+d x)}\right )-2 d^2 f x^2 \log \left (1+\frac {(a-b) e^{-2 (c+d x)}}{a+b}\right )+2 d^2 f x^2 \log (a+b \tanh (c+d x))-2 d e \log \left (-\frac {b (-1+\tanh (c+d x))}{a+b}\right ) \log (a+b \tanh (c+d x))+2 d e \log \left (-\frac {b (1+\tanh (c+d x))}{a-b}\right ) \log (a+b \tanh (c+d x))-2 d f x \operatorname {PolyLog}\left (2,-e^{-2 (c+d x)}\right )+2 d f x \operatorname {PolyLog}\left (2,\frac {(-a+b) e^{-2 (c+d x)}}{a+b}\right )+2 d e \operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a-b}\right )-2 d e \operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a+b}\right )-f \operatorname {PolyLog}\left (3,-e^{-2 (c+d x)}\right )+f \operatorname {PolyLog}\left (3,\frac {(-a+b) e^{-2 (c+d x)}}{a+b}\right )}{4 d^2} \] Input:

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

Output:

(2*d^2*f*x^2*Log[1 + E^(-2*(c + d*x))] - 2*d^2*f*x^2*Log[1 + (a - b)/((a + 
 b)*E^(2*(c + d*x)))] + 2*d^2*f*x^2*Log[a + b*Tanh[c + d*x]] - 2*d*e*Log[- 
((b*(-1 + Tanh[c + d*x]))/(a + b))]*Log[a + b*Tanh[c + d*x]] + 2*d*e*Log[- 
((b*(1 + Tanh[c + d*x]))/(a - b))]*Log[a + b*Tanh[c + d*x]] - 2*d*f*x*Poly 
Log[2, -E^(-2*(c + d*x))] + 2*d*f*x*PolyLog[2, (-a + b)/((a + b)*E^(2*(c + 
 d*x)))] + 2*d*e*PolyLog[2, (a + b*Tanh[c + d*x])/(a - b)] - 2*d*e*PolyLog 
[2, (a + b*Tanh[c + d*x])/(a + b)] - f*PolyLog[3, -E^(-2*(c + d*x))] + f*P 
olyLog[3, (-a + b)/((a + b)*E^(2*(c + d*x)))])/(4*d^2)
 

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.

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

\(\Big \downarrow \) 3031

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

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 = 6.29 (sec) , antiderivative size = 2640, normalized size of antiderivative = 13.13

method	result	size
risch	\(\text {Expression too large to display}\)	\(2640\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 686, normalized size of antiderivative = 3.41 \[ \int (e+f x) \log (a+b \tanh (c+d x)) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.53 \[ \int (e+f x) \log (a+b \tanh (c+d x)) \, dx=-\frac {1}{4} \, b d {\left (\frac {2 \, {\left (2 \, d x \log \left (\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b}\right )\right )} e}{b d^{2}} - \frac {2 \, {\left (2 \, d x \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, d x + 2 \, c\right )}\right )\right )} e}{b d^{2}} + \frac {{\left (2 \, d^{2} x^{2} \log \left (\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b} + 1\right ) + 2 \, d x {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b}\right ) - {\rm Li}_{3}(-\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b})\right )} f}{b d^{3}} - \frac {{\left (2 \, d^{2} x^{2} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (2 \, d x + 2 \, c\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, d x + 2 \, c\right )})\right )} f}{b d^{3}}\right )} + \frac {1}{2} \, {\left (f x^{2} + 2 \, e x\right )} \log \left (b \tanh \left (d x + c\right ) + a\right ) \] Input:

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

Output:

-1/4*b*d*(2*(2*d*x*log((a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + di 
log(-(a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b)))*e/(b*d^2) - 2*(2*d*x*log( 
e^(2*d*x + 2*c) + 1) + dilog(-e^(2*d*x + 2*c)))*e/(b*d^2) + (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)))*f/(b*d^3) - (2*d^2*x^2*log(e^(2*d*x + 2*c) + 1) + 2*d*x*di 
log(-e^(2*d*x + 2*c)) - polylog(3, -e^(2*d*x + 2*c)))*f/(b*d^3)) + 1/2*(f* 
x^2 + 2*e*x)*log(b*tanh(d*x + c) + a)
 

Giac [F]

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

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

Output:

integrate((f*x + e)*log(b*tanh(d*x + c) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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