\(\int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [461]

Optimal result

Integrand size = 32, antiderivative size = 324 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{2 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {5704, 5558, 3377, 2718, 5560, 3855, 5554, 2715, 8, 3797, 2221, 2317, 2438, 5684, 5680} \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {f \left (a^2+b^2\right ) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b (e+f x)^2}{2 a^2 f}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \]

[In]

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

[Out]

(b*(e + f*x)^2)/(2*a^2*f) - ((a^2 + b^2)*(e + f*x)^2)/(2*a^2*b*f) - (f*ArcTanh[Cosh[c + d*x]])/(a*d^2) - ((e +
f*x)*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*b*d)
+ ((a^2 + b^2)*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*d) - (b*(e + f*x)*Log[1 - E^(2
*(c + d*x))])/(a^2*d) + ((a^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^2) + ((a
^2 + b^2)*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^2) - (b*f*PolyLog[2, E^(2*(c + d*x)
)])/(2*a^2*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 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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

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 3855

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

Rule 5554

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

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

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 5684

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5704

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(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] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 7.16 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-\frac {b d^2 (e+f x)^2}{f}-a d (e+f x) \coth \left (\frac {1}{2} (c+d x)\right )-2 (-a f+b d (e+f x)) \log \left (1-e^{-c-d x}\right )-2 (a f+b d (e+f x)) \log \left (1+e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+2 b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+\frac {\left (a^2+b^2\right ) \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{b}+a d (e+f x) \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(937\) vs. \(2(306)=612\).

Time = 3.02 (sec) , antiderivative size = 938, normalized size of antiderivative = 2.90

method	result	size
risch	\(\frac {f b \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 {f b \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 {f b \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 b \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 b \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a^{2} d}+\frac {b c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a^{2} d^{2}}-\frac {b c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{a^{2} d^{2}}-\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 {f \,c^{2}}{d^{2} b}-\frac {2 e \ln \left ({\mathrm e}^{d x +c}\right )}{d b}+\frac {e \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d b}+\frac {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} b}+\frac {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} b}-\frac {f \,x^{2}}{2 b}+\frac {e x}{b}-\frac {c f \ln \left (b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}-b \right )}{d^{2} b}+\frac {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 b}+\frac {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 b}+\frac {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} b}+\frac {2 c f \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} b}-\frac {2 c f x}{d b}+\frac {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} b}-\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 {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 {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}+1\right )}{a^{2} d^{2}}+\frac {f b \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 {f b \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 {f b \operatorname {dilog}\left ({\mathrm e}^{d x +c}\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}}\)	\(938\)

[In]

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

[Out]

1/a^2/d^2*f*b*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*f*b*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*f*b*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*f*b*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*f*b*ln(exp(d*x+c)+1)*x+1/a^2/d
^2*b*c*f*ln(exp(d*x+c)-1)-1/a^2/d^2*b*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/d*(f*x+e)/a*exp(d*x+c)/(exp(
2*d*x+2*c)-1)-1/d^2/b*f*c^2-2/d/b*e*ln(exp(d*x+c))+1/d/b*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/d^2/b*f*dil
og((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2/b*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a
+(a^2+b^2)^(1/2)))-1/2*f*x^2/b+e*x/b-1/d^2/b*c*f*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/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/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/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+2/d^2/b*c*f*ln(exp(d*x+c))-2/d/b*c*f
*x+1/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*b*e*ln(exp(d*x+c)-1)-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^2*f*b*dilog(exp(d*x+c)+1)+1/a^2/
d^2*f*b*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*f*b*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*f*b*dilog(exp(d*x+c))+1/a/d^2*f*ln(exp(d*x+c)-1)-1/a/d^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. 1735 vs. \(2 (303) = 606\).

Time = 0.29 (sec) , antiderivative size = 1735, normalized size of antiderivative = 5.35 \[ \int \frac {(e+f x) \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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