\(\int \frac {(e+f x) \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [19]

Optimal result

Integrand size = 24, antiderivative size = 212 \[ \int \frac {(e+f x) \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \cosh (c+d x)}{a d^2}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {b f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2}+\frac {(e+f x) \sinh (c+d x)}{a d} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.25 \[ \int \frac {(e+f x) \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {\text {csch}(c+d x) (b+a \sinh (c+d x)) \left (2 d e (b \log (b+a \sinh (c+d x))-a \sinh (c+d x))+f \left (2 a \cosh (c+d x)+b \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )-2 c \log \left (a-2 b e^{c+d x}-a e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {a e^{c+d x}}{-b+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )\right )-2 a d x \sinh (c+d x)\right )\right )}{2 a^2 d^2 (a+b \text {csch}(c+d x))} \] Input:

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

Output:

-1/2*(Csch[c + d*x]*(b + a*Sinh[c + d*x])*(2*d*e*(b*Log[b + a*Sinh[c + d*x 
]] - a*Sinh[c + d*x]) + f*(2*a*Cosh[c + d*x] + b*(2*c*(c + d*x) - (c + d*x 
)^2 + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*(c + 
d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] - 2*c*Log[a - 2*b*E^(c 
 + d*x) - a*E^(2*(c + d*x))] + 2*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 
 + b^2])] + 2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]) - 2*a* 
d*x*Sinh[c + d*x])))/(a^2*d^2*(a + b*Csch[c + d*x]))
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6128, 6113, 3042, 3777, 26, 3042, 26, 3118, 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.

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

\(\Big \downarrow \) 6128

\(\displaystyle \int \frac {(e+f x) \sinh (c+d x) \cosh (c+d x)}{a \sinh (c+d x)+b}dx\)

\(\Big \downarrow \) 6113

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \int -i \sin (i c+i d x)dx}{d}}{a}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b \int \frac {(e+f x) \cosh (c+d x)}{b+a \sinh (c+d x)}dx}{a}+\frac {\frac {(e+f x) \sinh (c+d x)}{d}+\frac {i f \int \sin (i c+i d x)dx}{d}}{a}\)

\(\Big \downarrow \) 3118

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle \frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{a}-\frac {b \left (-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}-\frac {f \int e^{-c-d x} \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^2}{2 a f}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {(e+f x) \sinh (c+d x)}{d}-\frac {f \cosh (c+d x)}{d^2}}{a}-\frac {b \left (\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a d^2}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a d}+\frac {(e+f x) \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a d}-\frac {(e+f x)^2}{2 a f}\right )}{a}\)

Input:

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

Output:

-((b*(-1/2*(e + f*x)^2/(a*f) + ((e + f*x)*Log[1 + (a*E^(c + d*x))/(b - Sqr 
t[a^2 + b^2])])/(a*d) + ((e + f*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + 
 b^2])])/(a*d) + (f*PolyLog[2, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/ 
(a*d^2) + (f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a*d^2) 
))/a) + (-((f*Cosh[c + d*x])/d^2) + ((e + f*x)*Sinh[c + d*x])/d)/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 3118

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

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && 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 6113

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

rule 6128

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) 
 + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F 
[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x] && H 
yperbolicQ[F] && IntegersQ[m, n]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(482\) vs. \(2(198)=396\).

Time = 0.96 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.28

method	result	size
risch	\(\frac {b f \,x^{2}}{2 a^{2}}-\frac {b e x}{a^{2}}+\frac {\left (d f x +d e -f \right ) {\mathrm e}^{d x +c}}{2 a \,d^{2}}-\frac {\left (d f x +d e +f \right ) {\mathrm e}^{-d x -c}}{2 a \,d^{2}}+\frac {2 b f c x}{d \,a^{2}}+\frac {2 b e \ln \left ({\mathrm e}^{d x +c}\right )}{d \,a^{2}}+\frac {b f \,c^{2}}{d^{2} a^{2}}-\frac {b f \ln \left (\frac {-{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}-\frac {b f \ln \left (\frac {{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{2}}+\frac {b c f \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d^{2} a^{2}}-\frac {2 b c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{2}}-\frac {b e \ln \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )}{d \,a^{2}}-\frac {b f \ln \left (\frac {-{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}-\frac {b f \ln \left (\frac {{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{2}}-\frac {b f \operatorname {dilog}\left (\frac {-{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}-b}{-b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}-\frac {b f \operatorname {dilog}\left (\frac {{\mathrm e}^{d x +c} a +\sqrt {a^{2}+b^{2}}+b}{b +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{2}}\)	\(483\)

Input:

int((f*x+e)*cosh(d*x+c)/(a+csch(d*x+c)*b),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (196) = 392\).

Time = 0.10 (sec) , antiderivative size = 692, normalized size of antiderivative = 3.26 \[ \int \frac {(e+f x) \cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(a*d*f*x + a*d*e - (a*d*f*x + a*d*e - a*f)*cosh(d*x + c)^2 - (a*d*f*x 
 + a*d*e - a*f)*sinh(d*x + c)^2 + a*f - (b*d^2*f*x^2 + 2*b*d^2*e*x + 4*b*c 
*d*e - 2*b*c^2*f)*cosh(d*x + c) + 2*(b*f*cosh(d*x + c) + b*f*sinh(d*x + c) 
)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x 
 + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 2*(b*f*cosh(d*x + c) + b*f*sinh 
(d*x + c))*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a 
*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 2*((b*d*e - b*c*f)*cos 
h(d*x + c) + (b*d*e - b*c*f)*sinh(d*x + c))*log(2*a*cosh(d*x + c) + 2*a*si 
nh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 2*((b*d*e - b*c*f)*cosh(d 
*x + c) + (b*d*e - b*c*f)*sinh(d*x + c))*log(2*a*cosh(d*x + c) + 2*a*sinh( 
d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 2*((b*d*f*x + b*c*f)*cosh(d* 
x + c) + (b*d*f*x + b*c*f)*sinh(d*x + c))*log(-(b*cosh(d*x + c) + b*sinh(d 
*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a 
) + 2*((b*d*f*x + b*c*f)*cosh(d*x + c) + (b*d*f*x + b*c*f)*sinh(d*x + c))* 
log(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + 
c))*sqrt((a^2 + b^2)/a^2) - a)/a) - (b*d^2*f*x^2 + 2*b*d^2*e*x + 4*b*c*d*e 
 - 2*b*c^2*f + 2*(a*d*f*x + a*d*e - a*f)*cosh(d*x + c))*sinh(d*x + c))/(a^ 
2*d^2*cosh(d*x + c) + a^2*d^2*sinh(d*x + c))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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