\(\int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [245]

Optimal result

Integrand size = 26, antiderivative size = 306 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x) \coth (c+d x)}{a d}+\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {f \log (\sinh (c+d x))}{a d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2} \] Output:

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

Mathematica [A] (verified)

Time = 5.24 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.14 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )-2 \left (a f (c+d x)+(a f-b d (e+f x)) \log \left (1-e^{-c-d x}\right )+(a f+b d (e+f x)) \log \left (1+e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right )-\frac {2 b^2 \left (-2 d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 c f \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2}}+a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \] Input:

Integrate[((e + f*x)*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

-1/2*(a*d*(e + f*x)*Coth[(c + d*x)/2] - 2*(a*f*(c + d*x) + (a*f - b*d*(e + 
 f*x))*Log[1 - E^(-c - d*x)] + (a*f + b*d*(e + f*x))*Log[1 + E^(-c - d*x)] 
 - b*f*PolyLog[2, -E^(-c - d*x)] + b*f*PolyLog[2, E^(-c - d*x)]) - (2*b^2* 
(-2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + 
b*E^(c + d*x))/Sqrt[a^2 + b^2]] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - 
 Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b 
^2])] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - f*PolyLog[2 
, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/Sqrt[a^2 + b^2] + a*d*(e + f 
*x)*Tanh[(c + d*x)/2])/(a^2*d^2)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.66 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.808, Rules used = {6109, 3042, 25, 4672, 26, 3042, 26, 3956, 6109, 3042, 26, 3803, 25, 2694, 27, 2620, 2715, 2838, 4670, 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.

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

\(\Big \downarrow \) 6109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3956

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

\(\Big \downarrow \) 6109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2694

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[((e + f*x)*Csch[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3803

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

rule 3956

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

rule 4670

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

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 6109

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ 
.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a   Int[(e + f*x)^m*Csch[ 
c + d*x]^n, x], x] - Simp[b/a   Int[(e + f*x)^m*(Csch[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(625\) vs. \(2(283)=566\).

Time = 0.44 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.05

method	result	size
risch	\(-\frac {2 \left (f x +e \right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {c b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}+\frac {b^{2} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {b^{2} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}-\frac {2 b^{2} e \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {2 c \,b^{2} f \,\operatorname {arctanh}\left (\frac {2 b \,{\mathrm e}^{d x +c}+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {a^{2}+b^{2}}}+\frac {b e \ln \left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d}-\frac {b e \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d}-\frac {2 f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {b f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{a^{2} d^{2}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}+\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {a^{2}+b^{2}}}\)	\(626\)

Input:

int((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1830 vs. \(2 (279) = 558\).

Time = 0.15 (sec) , antiderivative size = 1830, normalized size of antiderivative = 5.98 \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
 

Output:

-(2*(a^3 + a*b^2)*d*e - 2*(a^3 + a*b^2)*c*f + 2*((a^3 + a*b^2)*d*f*x + (a^ 
3 + a*b^2)*c*f)*cosh(d*x + c)^2 + 4*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c 
*f)*cosh(d*x + c)*sinh(d*x + c) + 2*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*c 
*f)*sinh(d*x + c)^2 - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*sinh( 
d*x + c) + b^3*f*sinh(d*x + c)^2 - b^3*f)*sqrt((a^2 + b^2)/b^2)*dilog((a*c 
osh(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^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + 
 c)*sinh(d*x + c) + b^3*f*sinh(d*x + c)^2 - b^3*f)*sqrt((a^2 + b^2)/b^2)*d 
ilog((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^3*d*e - b^3*c*f - (b^3*d*e - b^ 
3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*e - b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) 
 - (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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^3*d* 
e - b^3*c*f - (b^3*d*e - b^3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*e - b^3*c*f)* 
cosh(d*x + c)*sinh(d*x + c) - (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)*sqrt((a 
^2 + b^2)/b^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^3*d*f*x + b^3*c*f - (b^3*d*f*x + b^3*c*f)*cosh(d*x 
+ c)^2 - 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f*x 
+ b^3*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(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)/...
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \operatorname {csch}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*csch(d*x+c)**2/(a+b*sinh(d*x+c)),x)
 

Output:

Integral((e + f*x)*csch(c + d*x)**2/(a + b*sinh(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \operatorname {csch}\left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \] Input:

int((e + f*x)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))),x)
 

Output:

int((e + f*x)/(sinh(c + d*x)^2*(a + b*sinh(c + d*x))), x)
 

Reduce [F]

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

int((f*x+e)*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x)
 

Output:

(2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a 
**2 + b**2))*b**2*e*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/ 
sqrt(a**2 + b**2))*b**2*e*i + 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**4*d*f + 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**2*b**2*d*f - e**(2*c + 2*d*x)*log(e**(c + d*x) 
 - 1)*a**2*b*e - e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b**3*e + e**(2*c + 
 2*d*x)*log(e**(c + d*x) + 1)*a**2*b*e + e**(2*c + 2*d*x)*log(e**(c + d*x) 
 + 1)*b**3*e - 2*e**(2*c + 2*d*x)*a**3*e - 2*e**(2*c + 2*d*x)*a*b**2*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**4*d*f - 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**2*b**2*d*f + log(e* 
*(c + d*x) - 1)*a**2*b*e + log(e**(c + d*x) - 1)*b**3*e - log(e**(c + d*x) 
 + 1)*a**2*b*e - log(e**(c + d*x) + 1)*b**3*e)/(a**2*d*(e**(2*c + 2*d*x)*a 
**2 + e**(2*c + 2*d*x)*b**2 - a**2 - b**2))