\(\int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [451]

Optimal result

Integrand size = 30, antiderivative size = 243 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d}+\frac {b (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \] Output:

-f*arctanh(cosh(d*x+c))/a/d^2-(f*x+e)*csch(d*x+c)/a/d+b*(f*x+e)*ln(1+b*exp 
(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d+b*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2 
)^(1/2)))/a^2/d-b*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d+b*f*polylog(2,-b*exp( 
d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2+b*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^ 
2)^(1/2)))/a^2/d^2-1/2*b*f*polylog(2,exp(2*d*x+2*c))/a^2/d^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(561\) vs. \(2(243)=486\).

Time = 7.89 (sec) , antiderivative size = 561, normalized size of antiderivative = 2.31 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-2 b d e (c+d x)+2 b c f (c+d x)-b f (c+d x)^2-\frac {b d^2 (e+f x)^2}{f}+\frac {4 a b \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a b \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-a d e \coth \left (\frac {1}{2} (c+d x)\right )+a c f \coth \left (\frac {1}{2} (c+d x)\right )-a f (c+d x) \coth \left (\frac {1}{2} (c+d x)\right )-2 (-a f+b d (e+f x)) \log \left (1-e^{-c-d x}\right )-2 (a f+b d (e+f x)) \log \left (1+e^{-c-d x}\right )+2 b f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 b f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 b c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 b d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 b f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+a d e \tanh \left (\frac {1}{2} (c+d x)\right )-a c f \tanh \left (\frac {1}{2} (c+d x)\right )+a f (c+d x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \] Input:

Integrate[((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

(-2*b*d*e*(c + d*x) + 2*b*c*f*(c + d*x) - b*f*(c + d*x)^2 - (b*d^2*(e + f* 
x)^2)/f + (4*a*b*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 
- b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*b*Sqrt[-(a^2 + b^2)^2]*d*e*ArcTanh[(a 
 + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) - a*d*e*Coth[(c + d 
*x)/2] + a*c*f*Coth[(c + d*x)/2] - a*f*(c + d*x)*Coth[(c + d*x)/2] - 2*(-( 
a*f) + b*d*(e + f*x))*Log[1 - E^(-c - d*x)] - 2*(a*f + b*d*(e + f*x))*Log[ 
1 + E^(-c - d*x)] + 2*b*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 
+ b^2])] + 2*b*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] 
- 2*b*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*b*d*e*Log[2*a*E 
^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*b*f*PolyLog[2, -E^(-c - d*x)] + 
 2*b*f*PolyLog[2, E^(-c - d*x)] + 2*b*f*PolyLog[2, (b*E^(c + d*x))/(-a + S 
qrt[a^2 + b^2])] + 2*b*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2] 
))] + a*d*e*Tanh[(c + d*x)/2] - a*c*f*Tanh[(c + d*x)/2] + a*f*(c + d*x)*Ta 
nh[(c + d*x)/2])/(2*a^2*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.60 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.25, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {6121, 5975, 3042, 26, 4257, 6103, 3042, 26, 4201, 2620, 2715, 2838, 6095, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\)

\(\Big \downarrow \) 6121

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

\(\Big \downarrow \) 5975

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}(c+d x)}{d}+\frac {f \int i \csc (i c+i d x)dx}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)}dx}{a}+\frac {-\frac {(e+f x) \text {csch}(c+d x)}{d}+\frac {i f \int \csc (i c+i d x)dx}{d}}{a}\)

\(\Big \downarrow \) 4257

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

\(\Big \downarrow \) 6103

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)}dx}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 6095

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}-\sqrt {a^2+b^2}}dx+\int \frac {e^{c+d x} (e+f x)}{a+b e^{c+d x}+\sqrt {a^2+b^2}}dx-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {f \int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d}-\frac {f \int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cosh (c+d x))}{d^2}-\frac {(e+f x) \text {csch}(c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d}+\frac {(e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d}-\frac {(e+f x)^2}{2 b f}\right )}{a}-\frac {i \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{a}\right )}{a}\)

Input:

Int[((e + f*x)*Coth[c + d*x]*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]
 

Output:

(-((f*ArcTanh[Cosh[c + d*x]])/d^2) - ((e + f*x)*Csch[c + d*x])/d)/a - (b*( 
-((b*(-1/2*(e + f*x)^2/(b*f) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqr 
t[a^2 + b^2])])/(b*d) + ((e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + 
 b^2])])/(b*d) + (f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/ 
(b*d^2) + (f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) 
))/a) - (I*(((-1/2*I)*(e + f*x)^2)/f + (2*I)*(((e + f*x)*Log[1 + E^(2*c - 
I*Pi + 2*d*x)])/(2*d) + (f*PolyLog[2, -E^(2*c - I*Pi + 2*d*x)])/(4*d^2)))) 
/a))/a
 

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
 

rule 4257

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

rule 5975

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)) 
, x] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /; 
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6103

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Coth[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*Cosh[c + d*x]*(Coth[c + d*x 
]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 
 IGtQ[m, 0] && IGtQ[n, 0]
 

rule 6121

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + 
 (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> S 
imp[1/a   Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Simp[b/ 
a   Int[(e + f*x)^m*Csch[c + d*x]^(p - 1)*(Coth[c + d*x]^n/(a + b*Sinh[c + 
d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] 
&& IGtQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(527\) vs. \(2(229)=458\).

Time = 0.77 (sec) , antiderivative size = 528, normalized size of antiderivative = 2.17

method	result	size
risch	\(-\frac {2 \left (f x +e \right ) {\mathrm e}^{d x +c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}-\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}-\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}+\frac {b e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d}-\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}+\frac {b f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {b f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2}}+\frac {c b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {c b f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d^{2}}\)	\(528\)

Input:

int((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERB 
OSE)
 

Output:

-2/d*(f*x+e)/a*exp(d*x+c)/(exp(2*d*x+2*c)-1)-1/a^2/d*b*f*ln(exp(d*x+c)+1)* 
x+1/a^2/d*b*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x 
+1/a^2/d*b*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/ 
a/d^2*f*ln(exp(d*x+c)-1)-1/a/d^2*f*ln(exp(d*x+c)+1)-1/a^2/d^2*b*f*dilog(ex 
p(d*x+c)+1)+1/a^2/d^2*b*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2 
+b^2)^(1/2)))+1/a^2/d^2*b*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2 
+b^2)^(1/2)))+1/a^2/d^2*b*f*dilog(exp(d*x+c))-1/a^2/d*b*e*ln(exp(d*x+c)-1) 
+1/a^2/d*b*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/a^2/d*b*e*ln(exp(d*x+ 
c)+1)+1/a^2/d^2*b*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/ 
2)))*c+1/a^2/d^2*b*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2 
)))*c+1/a^2/d^2*c*b*f*ln(exp(d*x+c)-1)-1/a^2/d^2*c*b*f*ln(b*exp(2*d*x+2*c) 
+2*a*exp(d*x+c)-b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1221 vs. \(2 (226) = 452\).

Time = 0.12 (sec) , antiderivative size = 1221, normalized size of antiderivative = 5.02 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm=" 
fricas")
 

Output:

-(2*(a*d*f*x + a*d*e)*cosh(d*x + c) - (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d* 
x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*dilog((a*cosh(d*x + c) + 
 a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^ 
2) - b)/b + 1) - (b*f*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) 
+ b*f*sinh(d*x + c)^2 - b*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b 
*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (b*f 
*cosh(d*x + c)^2 + 2*b*f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 
 - b*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) + (b*f*cosh(d*x + c)^2 + 2*b* 
f*cosh(d*x + c)*sinh(d*x + c) + b*f*sinh(d*x + c)^2 - b*f)*dilog(-cosh(d*x 
 + c) - sinh(d*x + c)) + (b*d*e - b*c*f - (b*d*e - b*c*f)*cosh(d*x + c)^2 
- 2*(b*d*e - b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*e - b*c*f)*sinh(d*x 
 + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/ 
b^2) + 2*a) + (b*d*e - b*c*f - (b*d*e - b*c*f)*cosh(d*x + c)^2 - 2*(b*d*e 
- b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*e - b*c*f)*sinh(d*x + c)^2)*lo 
g(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) 
 + (b*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d*x + c)^2 - 2*(b*d*f*x + b*c 
*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x + b*c*f)*sinh(d*x + c)^2)*log(- 
(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*s 
qrt((a^2 + b^2)/b^2) - b)/b) + (b*d*f*x + b*c*f - (b*d*f*x + b*c*f)*cosh(d 
*x + c)^2 - 2*(b*d*f*x + b*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f*x ...
 

Sympy [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \coth {\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

Integral((e + f*x)*coth(c + d*x)*csch(c + d*x)/(a + b*sinh(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm=" 
maxima")
 

Output:

(2*b*d*integrate(1/2*x/(a^2*d*e^(d*x + c) + a^2*d), x) - 2*b*d*integrate(1 
/2*x/(a^2*d*e^(d*x + c) - a^2*d), x) + a*((d*x + c)/(a^2*d^2) - log(e^(d*x 
 + c) + 1)/(a^2*d^2)) - a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) - 1)/(a^2 
*d^2)) - 2*x*e^(d*x + c)/(a*d*e^(2*d*x + 2*c) - a*d) - 2*integrate((a*b*x* 
e^(d*x + c) - b^2*x)/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), 
x))*f + e*(2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) + b*log(-2*a*e^(-d* 
x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) 
 - b*log(e^(-d*x - c) - 1)/(a^2*d))
 

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm=" 
giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {coth}\left (c+d\,x\right )\,\left (e+f\,x\right )}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)
 

Output:

int((coth(c + d*x)*(e + f*x))/(sinh(c + d*x)*(a + b*sinh(c + d*x))), x)
 

Reduce [F]

\[ \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-8 e^{2 d x +5 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -3 e^{4 d x +4 c} b -4 e^{3 d x +3 c} a +3 e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{3} d^{2} f +8 e^{2 d x +4 c} \left (\int \frac {e^{2 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -3 e^{4 d x +4 c} b -4 e^{3 d x +3 c} a +3 e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b \,d^{2} f +e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2} f -e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{2} d e +e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2} f -e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{2} d e +e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2} d e -2 e^{2 d x +2 c} a^{2} d f x -2 e^{d x +c} a b d e +8 e^{3 c} \left (\int \frac {e^{3 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -3 e^{4 d x +4 c} b -4 e^{3 d x +3 c} a +3 e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{3} d^{2} f -8 e^{2 c} \left (\int \frac {e^{2 d x} x}{e^{6 d x +6 c} b +2 e^{5 d x +5 c} a -3 e^{4 d x +4 c} b -4 e^{3 d x +3 c} a +3 e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a^{2} b \,d^{2} f -\mathrm {log}\left (e^{d x +c}-1\right ) a^{2} f +\mathrm {log}\left (e^{d x +c}-1\right ) b^{2} d e -\mathrm {log}\left (e^{d x +c}+1\right ) a^{2} f +\mathrm {log}\left (e^{d x +c}+1\right ) b^{2} d e -\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) b^{2} d e}{a^{2} b \,d^{2} \left (e^{2 d x +2 c}-1\right )} \] Input:

int((f*x+e)*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)
 

Output:

( - 8*e**(5*c + 2*d*x)*int((e**(3*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c 
+ 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d 
*x)*b + 2*e**(c + d*x)*a - b),x)*a**3*d**2*f + 8*e**(4*c + 2*d*x)*int((e** 
(2*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x) 
*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x 
)*a**2*b*d**2*f + e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*a**2*f - e**(2*c 
+ 2*d*x)*log(e**(c + d*x) - 1)*b**2*d*e + e**(2*c + 2*d*x)*log(e**(c + d*x 
) + 1)*a**2*f - e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*b**2*d*e + e**(2*c 
+ 2*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*b**2*d*e - 2*e**(2 
*c + 2*d*x)*a**2*d*f*x - 2*e**(c + d*x)*a*b*d*e + 8*e**(3*c)*int((e**(3*d* 
x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 
4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a** 
3*d**2*f - 8*e**(2*c)*int((e**(2*d*x)*x)/(e**(6*c + 6*d*x)*b + 2*e**(5*c + 
 5*d*x)*a - 3*e**(4*c + 4*d*x)*b - 4*e**(3*c + 3*d*x)*a + 3*e**(2*c + 2*d* 
x)*b + 2*e**(c + d*x)*a - b),x)*a**2*b*d**2*f - log(e**(c + d*x) - 1)*a**2 
*f + log(e**(c + d*x) - 1)*b**2*d*e - log(e**(c + d*x) + 1)*a**2*f + log(e 
**(c + d*x) + 1)*b**2*d*e - log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b) 
*b**2*d*e)/(a**2*b*d**2*(e**(2*c + 2*d*x) - 1))