\(\int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [460]

Optimal result

Integrand size = 34, antiderivative size = 518 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}-\frac {b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3} \]

[Out]

1/3*b*(f*x+e)^3/a^2/f-1/3*(a^2+b^2)*(f*x+e)^3/a^2/b/f-4*f*(f*x+e)*arctanh(exp(d*x+c))/a/d^2-(f*x+e)^2*csch(d*x
+c)/a/d-b*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a^2/d+(a^2+b^2)*(f*x+e)^2*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)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d-2*f^2*polylog(2,-exp(d*x+c))/a/d^3+2*f^
2*polylog(2,exp(d*x+c))/a/d^3-b*f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a^2/d^2+2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b
*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/b/d^2+2*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/
a^2/b/d^2+1/2*b*f^2*polylog(3,exp(2*d*x+2*c))/a^2/d^3-2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/
2)))/a^2/b/d^3-2*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/b/d^3

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5704, 5558, 3377, 2717, 5560, 4267, 2317, 2438, 5554, 3391, 3797, 2221, 2611, 2320, 6724, 5684, 5680} \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 f^2 \left (a^2+b^2\right ) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 f^2 \left (a^2+b^2\right ) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 f \left (a^2+b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 b d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}+\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac {b f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {b (e+f x)^3}{3 a^2 f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d} \]

[In]

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

[Out]

(b*(e + f*x)^3)/(3*a^2*f) - ((a^2 + b^2)*(e + f*x)^3)/(3*a^2*b*f) - (4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^
2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)])/(a^2*b*d) + ((a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*d) - (b*(e + f
*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2*PolyLog[2, E^(c +
d*x)])/(a*d^3) + (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^2)
+ (2*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^2) - (b*f*(e + f*x
)*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
]))])/(a^2*b*d^3) - (2*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^3) + (b*
f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3)

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[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 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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \cosh (c+d x) \coth ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = \frac {\int (e+f x)^2 \cosh (c+d x) \, dx}{a}+\frac {\int (e+f x)^2 \coth (c+d x) \text {csch}(c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \cosh ^2(c+d x) \coth (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2} \\ & = -\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {(e+f x)^2 \sinh (c+d x)}{a d}-\frac {\int (e+f x)^2 \cosh (c+d x) \, dx}{a}-\frac {b \int (e+f x)^2 \coth (c+d x) \, dx}{a^2}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(2 f) \int (e+f x) \text {csch}(c+d x) \, dx}{a d}-\frac {(2 f) \int (e+f x) \sinh (c+d x) \, dx}{a d} \\ & = \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \cosh (c+d x)}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)^2}{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)^2}{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)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac {(2 f) \int (e+f x) \sinh (c+d x) \, dx}{a d}+\frac {\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{a d^2}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^2} \\ & = \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac {2 f^2 \sinh (c+d x)}{a d^3}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d}-\frac {\left (2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 b d}-\frac {\left (2 \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 b d}-\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{a d^2} \\ & = \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \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 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {\left (b f^2\right ) \int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 b d^2}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 b d^2} \\ & = \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \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 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {\left (b f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 b d^3}-\frac {\left (2 \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 b d^3} \\ & = \frac {b (e+f x)^3}{3 a^2 f}-\frac {\left (a^2+b^2\right ) (e+f x)^3}{3 a^2 b f}-\frac {4 f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {\left (a^2+b^2\right ) (e+f x)^2 \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)^2 \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)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^2}+\frac {2 \left (a^2+b^2\right ) f (e+f x) \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 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 b d^3}-\frac {2 \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 b d^3}+\frac {b f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1806\) vs. \(2(518)=1036\).

Time = 10.06 (sec) , antiderivative size = 1806, normalized size of antiderivative = 3.49 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 d^2 e \left (-1+e^{2 c}\right ) f (b d e-2 a f) x+3 d^2 e \left (-1+e^{2 c}\right ) f (b d e+2 a f) x+2 b d^3 (e+f x)^3-6 d \left (-1+e^{2 c}\right ) f^2 (b d e-a f) x \log \left (1-e^{-c-d x}\right )-3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \log \left (1-e^{-c-d x}\right )-6 d \left (-1+e^{2 c}\right ) f^2 (b d e+a f) x \log \left (1+e^{-c-d x}\right )-3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \log \left (1+e^{-c-d x}\right )-3 d e \left (-1+e^{2 c}\right ) f (b d e-2 a f) \log \left (1-e^{c+d x}\right )-3 d e \left (-1+e^{2 c}\right ) f (b d e+2 a f) \log \left (1+e^{c+d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 (b d e+a f) \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-6 \left (-1+e^{2 c}\right ) f^2 (-b d e+a f) \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (3,e^{-c-d x}\right )}{3 a^2 d^3 \left (-1+e^{2 c}\right ) f}-\frac {\left (a^2+b^2\right ) \left (6 e^2 e^{2 c} x+6 e e^{2 c} f x^2+2 e^{2 c} f^2 x^3+\frac {6 a \sqrt {a^2+b^2} e^2 \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {3 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}-\frac {3 e^2 e^{2 c} \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}\right )}{3 a^2 b \left (-1+e^{2 c}\right )}+\frac {\left (-3 b e^2-6 b e f x-3 b f^2 x^2+3 a d e^2 x \cosh (c)+3 a d e f x^2 \cosh (c)+a d f^2 x^3 \cosh (c)\right ) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right )}{6 a b d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]

[In]

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

[Out]

(3*d^2*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*x + 3*d^2*e*(-1 + E^(2*c))*f*(b*d*e + 2*a*f)*x + 2*b*d^3*(e + f*x)^3
- 6*d*(-1 + E^(2*c))*f^2*(b*d*e - a*f)*x*Log[1 - E^(-c - d*x)] - 3*b*d^2*(-1 + E^(2*c))*f^3*x^2*Log[1 - E^(-c
- d*x)] - 6*d*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x*Log[1 + E^(-c - d*x)] - 3*b*d^2*(-1 + E^(2*c))*f^3*x^2*Log[1
+ E^(-c - d*x)] - 3*d*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*Log[1 - E^(c + d*x)] - 3*d*e*(-1 + E^(2*c))*f*(b*d*e
+ 2*a*f)*Log[1 + E^(c + d*x)] + 6*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*PolyLog[2, -E^(-c - d*x)] + 6*b*d*(-1 + E^
(2*c))*f^3*x*PolyLog[2, -E^(-c - d*x)] - 6*(-1 + E^(2*c))*f^2*(-(b*d*e) + a*f)*PolyLog[2, E^(-c - d*x)] + 6*b*
d*(-1 + E^(2*c))*f^3*x*PolyLog[2, E^(-c - d*x)] + 6*b*(-1 + E^(2*c))*f^3*PolyLog[3, -E^(-c - d*x)] + 6*b*(-1 +
E^(2*c))*f^3*PolyLog[3, E^(-c - d*x)])/(3*a^2*d^3*(-1 + E^(2*c))*f) - ((a^2 + b^2)*(6*e^2*E^(2*c)*x + 6*e*E^(
2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[
-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2
+ b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(
3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3
/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-
1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E
^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c +
d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^
2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(
2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*
x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2
+ b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*
E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2
*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f
^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c +
d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqr
t[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*a^2*b*(-1 + E^(2*c))) + ((-3*b*e^2 - 6*b*e*f*x - 3*b*f^2*x^2 + 3*a*d*e^2*x
*Cosh[c] + 3*a*d*e*f*x^2*Cosh[c] + a*d*f^2*x^3*Cosh[c])*Csch[c/2]*Sech[c/2])/(6*a*b*d) + (Csch[c/2]*Csch[c/2 +
(d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d) + (Sech[c/2]*Sech[c/2 +
(d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d)

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3506 vs. \(2 (486) = 972\).

Time = 0.32 (sec) , antiderivative size = 3506, normalized size of antiderivative = 6.77 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2
- (a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2)
*cosh(d*x + c)^2 - (a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f
+ 2*a^2*c^3*f^2)*sinh(d*x + c)^2 - 6*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2)*cosh(d*x + c) - 6*((a^2
+ b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)^2 - 2*((a^2 + b^
2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*sinh(d
*x + c)^2)*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) - 6*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*
x + c)^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d*f^2*x + (a
^2 + b^2)*d*e*f)*sinh(d*x + c)^2)*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) + 6*(b^2*d*f^2*x + b^2*d*e*f - a*b*f^2 - (b^2*d*f^2*x + b^2*d*e*f - a*b*f
^2)*cosh(d*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*e*f - a*b*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d*f^2*x + b^2*d
*e*f - a*b*f^2)*sinh(d*x + c)^2)*dilog(cosh(d*x + c) + sinh(d*x + c)) + 6*(b^2*d*f^2*x + b^2*d*e*f + a*b*f^2 -
(b^2*d*f^2*x + b^2*d*e*f + a*b*f^2)*cosh(d*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*e*f + a*b*f^2)*cosh(d*x + c)*sin
h(d*x + c) - (b^2*d*f^2*x + b^2*d*e*f + a*b*f^2)*sinh(d*x + c)^2)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 3*((
a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*
f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^
2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*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) - 3*((a^2 + b^2)*d^2*e^2
- 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2
*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c) - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*sinh(d*x + c)^2)*log(2*b*cos
h(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 3*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)
*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2 - ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x
+ 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^
2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d^2*f^2*x^2
+ 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*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) - 3*((a^2 + b^2)*d^2*f
^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2 - ((a^2 + b^2)*d^2*f^2*x^2 + 2*
(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*f^2*
x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a
^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*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)
+ 3*(b^2*d^2*f^2*x^2 + b^2*d^2*e^2 + 2*a*b*d*e*f - (b^2*d^2*f^2*x^2 + b^2*d^2*e^2 + 2*a*b*d*e*f + 2*(b^2*d^2*e
*f + a*b*d*f^2)*x)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^2*x^2 + b^2*d^2*e^2 + 2*a*b*d*e*f + 2*(b^2*d^2*e*f + a*b*d*f
^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d^2*f^2*x^2 + b^2*d^2*e^2 + 2*a*b*d*e*f + 2*(b^2*d^2*e*f + a*b*d*f^2
)*x)*sinh(d*x + c)^2 + 2*(b^2*d^2*e*f + a*b*d*f^2)*x)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 3*(b^2*d^2*e^2
- 2*(b^2*c + a*b)*d*e*f + (b^2*c^2 + 2*a*b*c)*f^2 - (b^2*d^2*e^2 - 2*(b^2*c + a*b)*d*e*f + (b^2*c^2 + 2*a*b*c)
*f^2)*cosh(d*x + c)^2 - 2*(b^2*d^2*e^2 - 2*(b^2*c + a*b)*d*e*f + (b^2*c^2 + 2*a*b*c)*f^2)*cosh(d*x + c)*sinh(d
*x + c) - (b^2*d^2*e^2 - 2*(b^2*c + a*b)*d*e*f + (b^2*c^2 + 2*a*b*c)*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) +
sinh(d*x + c) - 1) + 3*(b^2*d^2*f^2*x^2 + 2*b^2*c*d*e*f - (b^2*c^2 + 2*a*b*c)*f^2 - (b^2*d^2*f^2*x^2 + 2*b^2*
c*d*e*f - (b^2*c^2 + 2*a*b*c)*f^2 + 2*(b^2*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^2*x^2 + 2*b^
2*c*d*e*f - (b^2*c^2 + 2*a*b*c)*f^2 + 2*(b^2*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d^2*f^
2*x^2 + 2*b^2*c*d*e*f - (b^2*c^2 + 2*a*b*c)*f^2 + 2*(b^2*d^2*e*f - a*b*d*f^2)*x)*sinh(d*x + c)^2 + 2*(b^2*d^2*
e*f - a*b*d*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 6*((a^2 + b^2)*f^2*cosh(d*x + c)^2 + 2*(a^2 + b^
2)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^2*sinh(d*x + c)^2 - (a^2 + b^2)*f^2)*polylog(3, (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) - 6*((a^2 + b^2)*f^2*c
osh(d*x + c)^2 + 2*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^2*sinh(d*x + c)^2 - (a^2 + b^2)
*f^2)*polylog(3, (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) + 6*(b^2*f^2*cosh(d*x + c)^2 + 2*b^2*f^2*cosh(d*x + c)*sinh(d*x + c) + b^2*f^2*sinh(d*x + c)^2 - b^2*f^2
)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + 6*(b^2*f^2*cosh(d*x + c)^2 + 2*b^2*f^2*cosh(d*x + c)*sinh(d*x +
c) + b^2*f^2*sinh(d*x + c)^2 - b^2*f^2)*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(3*a*b*d^2*f^2*x^2 + 6*
a*b*d^2*e*f*x + 3*a*b*d^2*e^2 + (a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a
^2*c^2*d*e*f + 2*a^2*c^3*f^2)*cosh(d*x + c))*sinh(d*x + c))/(a^2*b*d^3*cosh(d*x + c)^2 + 2*a^2*b*d^3*cosh(d*x
+ c)*sinh(d*x + c) + a^2*b*d^3*sinh(d*x + c)^2 - a^2*b*d^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

e^2*((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)) - 1/3*(a*d*f
^2*x^3 + 3*a*d*e*f*x^2 - (a*d*f^2*x^3*e^(2*c) + 3*a*d*e*f*x^2*e^(2*c))*e^(2*d*x) + 6*(b*f^2*x^2*e^c + 2*b*e*f*
x*e^c)*e^(d*x))/(a*b*d*e^(2*d*x + 2*c) - a*b*d) - 2*e*f*log(e^(d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) -
1)/(a*d^2) - (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*b*f^2/(a
^2*d^3) - (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*b*f^2/(a^2*d^
3) - 2*(b*d*e*f + a*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d*e*f - a*f^2)*(d*x
*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^
2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(-2*((a^2*b*f^2 + b^3*f^2)*x^2
+ 2*(a^2*b*e*f + b^3*e*f)*x - ((a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*f*e^c)*x)*e^(d*x)
)/(a^2*b^2*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^2*b^2), x)

Giac [F(-1)]

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

[In]

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

[In]

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

[Out]

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