\(\int \log (a+b \tanh (c+d x)) \, dx\) [13]

Optimal result

Integrand size = 11, antiderivative size = 126 \[ \int \log (a+b \tanh (c+d x)) \, dx=-\frac {\log \left (\frac {b (1-\tanh (c+d x))}{a+b}\right ) \log (a+b \tanh (c+d x))}{2 d}+\frac {\log \left (-\frac {b (1+\tanh (c+d x))}{a-b}\right ) \log (a+b \tanh (c+d x))}{2 d}+\frac {\operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a-b}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a+b}\right )}{2 d} \] Output:

-1/2*ln(b*(1-tanh(d*x+c))/(a+b))*ln(a+b*tanh(d*x+c))/d+1/2*ln(-b*(1+tanh(d 
*x+c))/(a-b))*ln(a+b*tanh(d*x+c))/d+1/2*polylog(2,(a+b*tanh(d*x+c))/(a-b)) 
/d-1/2*polylog(2,(a+b*tanh(d*x+c))/(a+b))/d
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \log (a+b \tanh (c+d x)) \, dx=-\frac {\log \left (\frac {b (1-\tanh (c+d x))}{a+b}\right ) \log (a+b \tanh (c+d x))}{2 d}+\frac {\log \left (-\frac {b (1+\tanh (c+d x))}{a-b}\right ) \log (a+b \tanh (c+d x))}{2 d}+\frac {\operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a-b}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,\frac {a+b \tanh (c+d x)}{a+b}\right )}{2 d} \] Input:

Integrate[Log[a + b*Tanh[c + d*x]],x]
 

Output:

-1/2*(Log[(b*(1 - Tanh[c + d*x]))/(a + b)]*Log[a + b*Tanh[c + d*x]])/d + ( 
Log[-((b*(1 + Tanh[c + d*x]))/(a - b))]*Log[a + b*Tanh[c + d*x]])/(2*d) + 
PolyLog[2, (a + b*Tanh[c + d*x])/(a - b)]/(2*d) - PolyLog[2, (a + b*Tanh[c 
 + d*x])/(a + b)]/(2*d)
 

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.

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

\(\Big \downarrow \) 3028

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7299

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

Input:

Int[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 3028

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, 
 x]/u), x], x] /; InverseFunctionFreeQ[u, x]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [A] (verified)

Time = 9.42 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.99

method	result	size
derivativedivides	\(\frac {-\frac {\left (\operatorname {dilog}\left (\frac {b \tanh \left (d x +c \right )-b}{-a -b}\right )+\ln \left (a +b \tanh \left (d x +c \right )\right ) \ln \left (\frac {b \tanh \left (d x +c \right )-b}{-a -b}\right )\right ) b}{2}+\frac {\left (\operatorname {dilog}\left (\frac {b \tanh \left (d x +c \right )+b}{-a +b}\right )+\ln \left (a +b \tanh \left (d x +c \right )\right ) \ln \left (\frac {b \tanh \left (d x +c \right )+b}{-a +b}\right )\right ) b}{2}}{d b}\)	\(125\)
default	\(\frac {-\frac {\left (\operatorname {dilog}\left (\frac {b \tanh \left (d x +c \right )-b}{-a -b}\right )+\ln \left (a +b \tanh \left (d x +c \right )\right ) \ln \left (\frac {b \tanh \left (d x +c \right )-b}{-a -b}\right )\right ) b}{2}+\frac {\left (\operatorname {dilog}\left (\frac {b \tanh \left (d x +c \right )+b}{-a +b}\right )+\ln \left (a +b \tanh \left (d x +c \right )\right ) \ln \left (\frac {b \tanh \left (d x +c \right )+b}{-a +b}\right )\right ) b}{2}}{d b}\)	\(125\)
risch	\(\text {Expression too large to display}\)	\(1166\)

Input:

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

Output:

1/d/b*(-1/2*(dilog((b*tanh(d*x+c)-b)/(-a-b))+ln(a+b*tanh(d*x+c))*ln((b*tan 
h(d*x+c)-b)/(-a-b)))*b+1/2*(dilog((b*tanh(d*x+c)+b)/(-a+b))+ln(a+b*tanh(d* 
x+c))*ln((b*tanh(d*x+c)+b)/(-a+b)))*b)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 391, normalized size of antiderivative = 3.10 \[ \int \log (a+b \tanh (c+d x)) \, dx=\frac {d x \log \left (\frac {a \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )}\right ) + c \log \left (2 \, {\left (a + b\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a + b\right )} \sinh \left (d x + c\right ) + 2 \, {\left (a - b\right )} \sqrt {-\frac {a + b}{a - b}}\right ) + c \log \left (2 \, {\left (a + b\right )} \cosh \left (d x + c\right ) + 2 \, {\left (a + b\right )} \sinh \left (d x + c\right ) - 2 \, {\left (a - b\right )} \sqrt {-\frac {a + b}{a - b}}\right ) - {\left (d x + c\right )} \log \left (\sqrt {-\frac {a + b}{a - b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + 1\right ) - {\left (d x + c\right )} \log \left (-\sqrt {-\frac {a + b}{a - b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )} + 1\right ) - c \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + i\right ) - c \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - i\right ) + {\left (d x + c\right )} \log \left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right ) + 1\right ) + {\left (d x + c\right )} \log \left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right ) + 1\right ) - {\rm Li}_2\left (\sqrt {-\frac {a + b}{a - b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) - {\rm Li}_2\left (-\sqrt {-\frac {a + b}{a - b}} {\left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right )}\right ) + {\rm Li}_2\left (i \, \cosh \left (d x + c\right ) + i \, \sinh \left (d x + c\right )\right ) + {\rm Li}_2\left (-i \, \cosh \left (d x + c\right ) - i \, \sinh \left (d x + c\right )\right )}{d} \] Input:

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

Output:

(d*x*log((a*cosh(d*x + c) + b*sinh(d*x + c))/cosh(d*x + c)) + c*log(2*(a + 
 b)*cosh(d*x + c) + 2*(a + b)*sinh(d*x + c) + 2*(a - b)*sqrt(-(a + b)/(a - 
 b))) + c*log(2*(a + b)*cosh(d*x + c) + 2*(a + b)*sinh(d*x + c) - 2*(a - b 
)*sqrt(-(a + b)/(a - b))) - (d*x + c)*log(sqrt(-(a + b)/(a - b))*(cosh(d*x 
 + c) + sinh(d*x + c)) + 1) - (d*x + c)*log(-sqrt(-(a + b)/(a - b))*(cosh( 
d*x + c) + sinh(d*x + c)) + 1) - c*log(cosh(d*x + c) + sinh(d*x + c) + I) 
- c*log(cosh(d*x + c) + sinh(d*x + c) - I) + (d*x + c)*log(I*cosh(d*x + c) 
 + I*sinh(d*x + c) + 1) + (d*x + c)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) 
 + 1) - dilog(sqrt(-(a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c))) - di 
log(-sqrt(-(a + b)/(a - b))*(cosh(d*x + c) + sinh(d*x + c))) + dilog(I*cos 
h(d*x + c) + I*sinh(d*x + c)) + dilog(-I*cosh(d*x + c) - I*sinh(d*x + c))) 
/d
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.99 \[ \int \log (a+b \tanh (c+d x)) \, dx=-\frac {1}{2} \, b d {\left (\frac {2 \, d x \log \left (\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (a e^{\left (2 \, c\right )} + b e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}{a - b}\right )}{b d^{2}} - \frac {2 \, d x \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (2 \, d x + 2 \, c\right )}\right )}{b d^{2}}\right )} + x \log \left (b \tanh \left (d x + c\right ) + a\right ) \] Input:

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

Output:

-1/2*b*d*((2*d*x*log((a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b) + 1) + dilo 
g(-(a*e^(2*c) + b*e^(2*c))*e^(2*d*x)/(a - b)))/(b*d^2) - (2*d*x*log(e^(2*d 
*x + 2*c) + 1) + dilog(-e^(2*d*x + 2*c)))/(b*d^2)) + x*log(b*tanh(d*x + c) 
 + a)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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