\(\int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [478]

Optimal result

Integrand size = 32, antiderivative size = 298 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2} \]

[Out]

b*f*arctanh(cosh(d*x+c))/a^2/d^2-1/2*f*coth(d*x+c)/a/d^2+b*(f*x+e)*csch(d*x+c)/a^2/d-1/2*(f*x+e)*csch(d*x+c)^2
/a/d+b^2*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^3/d-b^2*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d-b^2*(f*x+
e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d+1/2*b^2*f*polylog(2,exp(2*d*x+2*c))/a^3/d^2-b^2*f*polylog(2,-b
*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^2-b^2*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^2

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {5706, 5560, 3852, 8, 3855, 5688, 3797, 2221, 2317, 2438, 5680} \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}-\frac {b^2 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d} \]

[In]

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

[Out]

(b*f*ArcTanh[Cosh[c + d*x]])/(a^2*d^2) - (f*Coth[c + d*x])/(2*a*d^2) + (b*(e + f*x)*Csch[c + d*x])/(a^2*d) - (
(e + f*x)*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) -
(b^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f*x)*Log[1 - E^(2*(c + d*x)
)])/(a^3*d) - (b^2*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b^2*f*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (b^2*f*PolyLog[2, E^(2*(c + d*x))])/(2*a^3*d^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

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]
- Dist[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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

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

Rule 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[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 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(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 5688

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[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 5706

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \coth (c+d x) \text {csch}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \coth (c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d}-\frac {b \int (e+f x) \coth (c+d x) \text {csch}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {f \int \text {csch}^2(c+d x) \, dx}{2 a d} \\ & = \frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d}+\frac {b^2 \int (e+f x) \coth (c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(i f) \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{2 a d^2}-\frac {(b f) \int \text {csch}(c+d x) \, dx}{a^2 d} \\ & = \frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 a d}-\frac {\left (2 b^2\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1-e^{2 (c+d x)}} \, dx}{a^3}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3}-\frac {b^3 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3} \\ & = \frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 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^3 d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 d}-\frac {\left (b^2 f\right ) \int \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^3 d} \\ & = \frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 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^3 d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2} \\ & = \frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \text {csch}^2(c+d x)}{2 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^3 d}-\frac {b^2 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(713\) vs. \(2(298)=596\).

Time = 8.40 (sec) , antiderivative size = 713, normalized size of antiderivative = 2.39 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {b \left (\frac {b (d e+d f x)^2}{2 f}+(b d e-a f+b d f x) \log \left (1-e^{-c-d x}\right )+(b d e+a f+b d f x) \log \left (1+e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right )}{a^3 d^2}-\frac {b^2 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a^3 d^2}+\frac {(d e-c f+f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )-a f \sinh \left (\frac {1}{2} (c+d x)\right )+2 b c f \sinh \left (\frac {1}{2} (c+d x)\right )-2 b f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^2 d^2} \]

[In]

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

[Out]

((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d*x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*
x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) + (b*((b*
(d*e + d*f*x)^2)/(2*f) + (b*d*e - a*f + b*d*f*x)*Log[1 - E^(-c - d*x)] + (b*d*e + a*f + b*d*f*x)*Log[1 + E^(-c
- d*x)] - b*f*PolyLog[2, -E^(-c - d*x)] - b*f*PolyLog[2, E^(-c - d*x)]))/(a^3*d^2) - (b^2*(-2*d*e*(c + d*x) +
2*c*f*(c + d*x) - f*(c + d*x)^2 + (4*a*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*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) + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])] - 2*c*f*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 +
E^(2*(c + d*x)))] + 2*f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))]))/(2*a^3*d^2) + ((d*e - c*f + f*(c + d*x))*Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c
+ d*x)/2]*(-2*b*d*e*Sinh[(c + d*x)/2] - a*f*Sinh[(c + d*x)/2] + 2*b*c*f*Sinh[(c + d*x)/2] - 2*b*f*(c + d*x)*Si
nh[(c + d*x)/2]))/(4*a^2*d^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(648\) vs. \(2(280)=560\).

Time = 2.16 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.18

method	result	size
risch	\(-\frac {-2 b d f x \,{\mathrm e}^{3 d x +3 c}+2 a d f x \,{\mathrm e}^{2 d x +2 c}-2 b d e \,{\mathrm e}^{3 d x +3 c}+2 a d e \,{\mathrm e}^{2 d x +2 c}+2 b d f x \,{\mathrm e}^{d x +c}+a f \,{\mathrm e}^{2 d x +2 c}+2 b d e \,{\mathrm e}^{d x +c}-a f}{a^{2} d^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}+\frac {b^{2} f \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{3}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3}}-\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{d^{2} a^{3}}-\frac {b^{2} f \ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3}}-\frac {b^{2} f \ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{d \,a^{3}}+\frac {b^{2} f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d \,a^{3}}-\frac {b^{2} c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{3}}+\frac {b^{2} c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{3}}+\frac {b^{2} e \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{3}}-\frac {b^{2} e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d \,a^{3}}-\frac {b^{2} f \operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3}}-\frac {b^{2} f \operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{d^{2} a^{3}}-\frac {b f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a^{2}}+\frac {b f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a^{2}}\)	\(649\)

[In]

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

[Out]

-(-2*b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-2*b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+2*b*d*f*x*e
xp(d*x+c)+a*f*exp(2*d*x+2*c)+2*b*d*e*exp(d*x+c)-a*f)/a^2/d^2/(exp(2*d*x+2*c)-1)^2-1/d^2*b^2/a^3*f*dilog(exp(d*
x+c))+1/d^2*b^2/a^3*f*dilog(exp(d*x+c)+1)-1/d^2*b^2/a^3*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/
2)))*c-1/d^2*b^2/a^3*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d*b^2/a^3*f*ln((b*exp(d*
x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d*b^2/a^3*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2
)^(1/2)))*x+1/d*b^2/a^3*f*ln(exp(d*x+c)+1)*x-1/d^2*b^2/a^3*c*f*ln(exp(d*x+c)-1)+1/d^2*b^2/a^3*c*f*ln(b*exp(2*d
*x+2*c)+2*a*exp(d*x+c)-b)+1/d*b^2/a^3*e*ln(exp(d*x+c)-1)+1/d*b^2/a^3*e*ln(exp(d*x+c)+1)-1/d*b^2/a^3*e*ln(b*exp
(2*d*x+2*c)+2*a*exp(d*x+c)-b)-1/d^2*b^2/a^3*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/
d^2*b^2/a^3*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2*b/a^2*f*ln(exp(d*x+c)-1)+1/d^2
*b/a^2*f*ln(exp(d*x+c)+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2899 vs. \(2 (277) = 554\).

Time = 0.32 (sec) , antiderivative size = 2899, normalized size of antiderivative = 9.73 \[ \int \frac {(e+f x) \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c)^3 + 2*(a*b*d*f*x + a*b*d*e)*sinh(d*x + c)^3 + a^2*f - (2*a^2*d*f*x + 2*
a^2*d*e + a^2*f)*cosh(d*x + c)^2 - (2*a^2*d*f*x + 2*a^2*d*e + a^2*f - 6*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c))*s
inh(d*x + c)^2 - 2*(a*b*d*f*x + a*b*d*e)*cosh(d*x + c) - (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d
*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh
(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*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^2*f*cosh(d*x + c)^4 + 4*b
^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh
(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*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^
2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2
+ b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c)
)*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + (b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*
x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(
d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x +
c)) - ((b^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e -
b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - b^2*c*f -
3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*e - b
^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e -
b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - b^2*c*f -
3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*e -
b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x +
c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*
x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c
)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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)/b^2) - b)/b) - (b^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 +
4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(
b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*lo
g(-(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) + (b
^2*d*f*x + (b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)*sinh(
d*x + c)^3 + (b^2*d*f*x + b^2*d*e + a*b*f)*sinh(d*x + c)^4 + b^2*d*e + a*b*f - 2*(b^2*d*f*x + b^2*d*e + a*b*f)
*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*d*e + a*b*f - 3*(b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 4*((b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*d*e + a*b*f)*cosh(d*x + c))*sinh(
d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^4 + 4*(b^2*d*e -
(b^2*c + a*b)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e - (b^2*c + a*b)*f)*sinh(d*x + c)^4 + b^2*d*e - 2*(b
^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - 3*(b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^2 - (b^2*
c + a*b)*f)*sinh(d*x + c)^2 - (b^2*c + a*b)*f + 4*((b^2*d*e - (b^2*c + a*b)*f)*cosh(d*x + c)^3 - (b^2*d*e - (b
^2*c + a*b)*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (b^2*d*f*x + (b^2*d*f*x
+ b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sin
h(d*x + c)^4 + b^2*c*f - 2*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2
*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh
(d*x + c))*sinh(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*(a*b*d*f*x + a*b*d*e - 3*(a*b*d*f*x + a*
b*d*e)*cosh(d*x + c)^2 + (2*a^2*d*f*x + 2*a^2*d*e + a^2*f)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^2*cosh(d*x + c
)^4 + 4*a^3*d^2*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d^2*sinh(d*x + c)^4 - 2*a^3*d^2*cosh(d*x + c)^2 + a^3*d^2
+ 2*(3*a^3*d^2*cosh(d*x + c)^2 - a^3*d^2)*sinh(d*x + c)^2 + 4*(a^3*d^2*cosh(d*x + c)^3 - a^3*d^2*cosh(d*x + c)
)*sinh(d*x + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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