3.2.38 \(\int \coth (c-b x) \coth (a+b x) \, dx\) [138]

3.2.38.1 Optimal result

Integrand size = 14, antiderivative size = 36 \[ \int \coth (c-b x) \coth (a+b x) \, dx=-x-\frac {\coth (a+c) \log (\sinh (c-b x))}{b}+\frac {\coth (a+c) \log (\sinh (a+b x))}{b} \]

output

-x-coth(a+c)*ln(-sinh(b*x-c))/b+coth(a+c)*ln(sinh(b*x+a))/b
 

3.2.38.2 Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.89 \[ \int \coth (c-b x) \coth (a+b x) \, dx=-x+\frac {\coth (a+c) (-\log (\sinh (c-b x))+\log (-\sinh (a+b x)))}{b} \]

input

Integrate[Coth[c - b*x]*Coth[a + b*x],x]
 

output

-x + (Coth[a + c]*(-Log[Sinh[c - b*x]] + Log[-Sinh[a + b*x]]))/b
 

3.2.38.3 Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6181, 6179, 3042, 26, 3956}

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.

\(\displaystyle \int \coth (a+b x) \coth (c-b x) \, dx\)

\(\Big \downarrow \) 6181

\(\displaystyle \cosh (a+c) \int \text {csch}(c-b x) \text {csch}(a+b x)dx-x\)

\(\Big \downarrow \) 6179

\(\displaystyle \cosh (a+c) (\text {csch}(a+c) \int \coth (c-b x)dx+\text {csch}(a+c) \int \coth (a+b x)dx)-x\)

\(\Big \downarrow \) 3042

\(\displaystyle -x+\cosh (a+c) \left (\text {csch}(a+c) \int -i \tan \left (i c-i b x+\frac {\pi }{2}\right )dx+\text {csch}(a+c) \int -i \tan \left (i a+i b x+\frac {\pi }{2}\right )dx\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -x+\cosh (a+c) \left (-i \text {csch}(a+c) \int \tan \left (\frac {1}{2} (2 i c+\pi )-i b x\right )dx-i \text {csch}(a+c) \int \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx\right )\)

\(\Big \downarrow \) 3956

\(\displaystyle -x+\cosh (a+c) \left (\frac {\text {csch}(a+c) \log (-i \sinh (a+b x))}{b}-\frac {\text {csch}(a+c) \log (-i \sinh (c-b x))}{b}\right )\)

input

Int[Coth[c - b*x]*Coth[a + b*x],x]
 

output

-x + Cosh[a + c]*(-((Csch[a + c]*Log[(-I)*Sinh[c - b*x]])/b) + (Csch[a + c 
]*Log[(-I)*Sinh[a + b*x]])/b)
 

3.2.38.3.1 Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 6179

Int[Csch[(a_.) + (b_.)*(x_)]*Csch[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[Csc 
h[(b*c - a*d)/b]   Int[Coth[a + b*x], x], x] - Simp[Csch[(b*c - a*d)/d]   I 
nt[Coth[c + d*x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && 
NeQ[b*c - a*d, 0]
 

rule 6181

Int[Coth[(a_.) + (b_.)*(x_)]*Coth[(c_) + (d_.)*(x_)], x_Symbol] :> Simp[b*( 
x/d), x] + Simp[Cosh[(b*c - a*d)/d]   Int[Csch[a + b*x]*Csch[c + d*x], x], 
x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2 - d^2, 0] && NeQ[b*c - a*d, 0]
 

3.2.38.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(152\) vs. \(2(39)=78\).

Time = 0.25 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.25

method	result	size
risch	\(-x -\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}-\frac {\ln \left (-{\mathrm e}^{2 a +2 c}+{\mathrm e}^{2 b x +2 a}\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right ) {\mathrm e}^{2 a +2 c}}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-1\right )}{b \left ({\mathrm e}^{2 a +2 c}-1\right )}\)	\(153\)

input

int(-coth(b*x-c)*coth(b*x+a),x,method=_RETURNVERBOSE)
 

output

-x-1/b/(exp(2*a+2*c)-1)*ln(-exp(2*a+2*c)+exp(2*b*x+2*a))*exp(2*a+2*c)-1/b/ 
(exp(2*a+2*c)-1)*ln(-exp(2*a+2*c)+exp(2*b*x+2*a))+1/b/(exp(2*a+2*c)-1)*ln( 
exp(2*b*x+2*a)-1)*exp(2*a+2*c)+1/b/(exp(2*a+2*c)-1)*ln(exp(2*b*x+2*a)-1)
 

3.2.38.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (39) = 78\).

Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 6.00 \[ \int \coth (c-b x) \coth (a+b x) \, dx=-\frac {b x \cosh \left (a + c\right )^{2} - 2 \, b x \cosh \left (a + c\right ) \sinh \left (a + c\right ) + b x \sinh \left (a + c\right )^{2} - b x - {\left (\cosh \left (a + c\right )^{2} - 2 \, \cosh \left (a + c\right ) \sinh \left (a + c\right ) + \sinh \left (a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, {\left (\cosh \left (a + c\right ) \sinh \left (b x + a\right ) - \cosh \left (b x + a\right ) \sinh \left (a + c\right )\right )}}{\cosh \left (b x + a\right ) \cosh \left (a + c\right ) - {\left (\cosh \left (a + c\right ) + \sinh \left (a + c\right )\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right ) \sinh \left (a + c\right )}\right ) + {\left (\cosh \left (a + c\right )^{2} - 2 \, \cosh \left (a + c\right ) \sinh \left (a + c\right ) + \sinh \left (a + c\right )^{2} + 1\right )} \log \left (\frac {2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b \cosh \left (a + c\right )^{2} - 2 \, b \cosh \left (a + c\right ) \sinh \left (a + c\right ) + b \sinh \left (a + c\right )^{2} - b} \]

input

integrate(-coth(b*x-c)*coth(b*x+a),x, algorithm="fricas")
 

output

-(b*x*cosh(a + c)^2 - 2*b*x*cosh(a + c)*sinh(a + c) + b*x*sinh(a + c)^2 - 
b*x - (cosh(a + c)^2 - 2*cosh(a + c)*sinh(a + c) + sinh(a + c)^2 + 1)*log( 
2*(cosh(a + c)*sinh(b*x + a) - cosh(b*x + a)*sinh(a + c))/(cosh(b*x + a)*c 
osh(a + c) - (cosh(a + c) + sinh(a + c))*sinh(b*x + a) + cosh(b*x + a)*sin 
h(a + c))) + (cosh(a + c)^2 - 2*cosh(a + c)*sinh(a + c) + sinh(a + c)^2 + 
1)*log(2*sinh(b*x + a)/(cosh(b*x + a) - sinh(b*x + a))))/(b*cosh(a + c)^2 
- 2*b*cosh(a + c)*sinh(a + c) + b*sinh(a + c)^2 - b)
 

3.2.38.6 Sympy [F]

\[ \int \coth (c-b x) \coth (a+b x) \, dx=- \int \coth {\left (a + b x \right )} \coth {\left (b x - c \right )}\, dx \]

input

integrate(-coth(b*x-c)*coth(b*x+a),x)
 

output

-Integral(coth(a + b*x)*coth(b*x - c), x)
 

3.2.38.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (39) = 78\).

Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 4.44 \[ \int \coth (c-b x) \coth (a+b x) \, dx=-x - \frac {a}{b} + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x - a\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x - a\right )} - 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} - \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x + c\right )} + 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} - \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left (e^{\left (-b x + c\right )} - 1\right )}{b {\left (e^{\left (2 \, a + 2 \, c\right )} - 1\right )}} \]

input

integrate(-coth(b*x-c)*coth(b*x+a),x, algorithm="maxima")
 

output

-x - a/b + (e^(2*a + 2*c) + 1)*log(e^(-b*x - a) + 1)/(b*(e^(2*a + 2*c) - 1 
)) + (e^(2*a + 2*c) + 1)*log(e^(-b*x - a) - 1)/(b*(e^(2*a + 2*c) - 1)) - ( 
e^(2*a + 2*c) + 1)*log(e^(-b*x + c) + 1)/(b*(e^(2*a + 2*c) - 1)) - (e^(2*a 
 + 2*c) + 1)*log(e^(-b*x + c) - 1)/(b*(e^(2*a + 2*c) - 1))
 

3.2.38.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (39) = 78\).

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.50 \[ \int \coth (c-b x) \coth (a+b x) \, dx=-\frac {b x + \frac {{\left (e^{\left (2 \, a + 2 \, c\right )} + 1\right )} \log \left ({\left | e^{\left (2 \, b x\right )} - e^{\left (2 \, c\right )} \right |}\right )}{e^{\left (2 \, a + 2 \, c\right )} - 1} + \frac {{\left (e^{\left (2 \, a\right )} + e^{\left (4 \, a + 2 \, c\right )}\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{e^{\left (2 \, a\right )} - e^{\left (4 \, a + 2 \, c\right )}}}{b} \]

input

integrate(-coth(b*x-c)*coth(b*x+a),x, algorithm="giac")
 

output

-(b*x + (e^(2*a + 2*c) + 1)*log(abs(e^(2*b*x) - e^(2*c)))/(e^(2*a + 2*c) - 
 1) + (e^(2*a) + e^(4*a + 2*c))*log(abs(e^(2*b*x + 2*a) - 1))/(e^(2*a) - e 
^(4*a + 2*c)))/b
 

3.2.38.9 Mupad [B] (verification not implemented)

Time = 2.56 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.36 \[ \int \coth (c-b x) \coth (a+b x) \, dx=\frac {\mathrm {coth}\left (a+c\right )\,\ln \left (4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}+4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}-4\,{\mathrm {e}}^{6\,a}\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{2\,b\,x}\right )}{b}-\frac {\mathrm {coth}\left (a+c\right )\,\ln \left (4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-4\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,c}-4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{4\,c}+4\,{\mathrm {e}}^{4\,a}\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,b\,x}\right )}{b}-x \]

input

int(coth(a + b*x)*coth(c - b*x),x)
 

output

(coth(a + c)*log(4*exp(2*a)*exp(2*c) + 4*exp(4*a)*exp(4*c) - 4*exp(4*a)*ex 
p(2*c)*exp(2*b*x) - 4*exp(6*a)*exp(4*c)*exp(2*b*x)))/b - (coth(a + c)*log( 
4*exp(2*a)*exp(2*b*x) - 4*exp(2*a)*exp(2*c) - 4*exp(4*a)*exp(4*c) + 4*exp( 
4*a)*exp(2*c)*exp(2*b*x)))/b - x