3.5.93 \(\int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [493]

Optimal. Leaf size=762 \[ \frac {f x}{2 a d}+\frac {2 b f x \text {ArcTan}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \text {ArcTan}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \text {ArcTan}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \text {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2} \]

[Out]

-1/2*f*polylog(2,exp(2*d*x+2*c))/a/d^2+1/2*f*x/a/d-1/2*(f*x+e)*coth(d*x+c)^2/a/d+1/2*f*polylog(2,-exp(2*d*x+2*
c))/a/d^2-1/2*f*coth(d*x+c)/a/d^2-(f*x+e)*ln(tanh(d*x+c))/a/d+f*x*ln(tanh(d*x+c))/a/d+b^4*(f*x+e)*ln(1+exp(2*d
*x+2*c))/a^3/(a^2+b^2)/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*(f*x+e)*ln(1+b
*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b
^2)/d^2-b^4*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2+2*b*f*x*arctan(exp(d*x+c))/a^2/d-
2*b^3*(f*x+e)*arctan(exp(d*x+c))/a^2/(a^2+b^2)/d-b*f*x*arctan(sinh(d*x+c))/a^2/d+I*b*f*polylog(2,I*exp(d*x+c))
/a^2/d^2+1/2*b^4*f*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^2-I*b*f*polylog(2,-I*exp(d*x+c))/a^2/d^2+I*b^3*f
*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2-I*b^3*f*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+1/2*b^2*f*polylo
g(2,exp(2*d*x+2*c))/a^3/d^2+b*(f*x+e)*arctan(sinh(d*x+c))/a^2/d-2*b^2*(f*x+e)*arctanh(exp(2*d*x+2*c))/a^3/d+b*
f*arctanh(cosh(d*x+c))/a^2/d^2+b*(f*x+e)*csch(d*x+c)/a^2/d-1/2*b^2*f*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+2*f*x*
arctanh(exp(2*d*x+2*c))/a/d

________________________________________________________________________________________

Rubi [A]
time = 0.82, antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 23, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.719, Rules used = {5708, 2700, 14, 5570, 3554, 8, 2628, 12, 4267, 2317, 2438, 2701, 327, 213, 5311, 4265, 3855, 5569, 5692, 5680, 2221, 6874, 3799} \begin {gather*} -\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {2 b^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a^2 d \left (a^2+b^2\right )}+\frac {b (e+f x) \text {ArcTan}(\sinh (c+d x))}{a^2 d}+\frac {2 b f x \text {ArcTan}\left (e^{c+d x}\right )}{a^2 d}-\frac {b f x \text {ArcTan}(\sinh (c+d x))}{a^2 d}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \left (a^2+b^2\right )}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2 \left (a^2+b^2\right )}+\frac {b^4 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 d^2 \left (a^2+b^2\right )}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \left (a^2+b^2\right )}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \left (a^2+b^2\right )}+\frac {b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a^3 d \left (a^2+b^2\right )}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac {f x \log (\tanh (c+d x))}{a d}+\frac {f x}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

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

Rule 8

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, 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 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && 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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*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 5311

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 + u^2)), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

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 5692

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

Rule 5708

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

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {f \int \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a}\\ &=\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x) \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(b f) \int \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2}+\frac {f \int \coth ^2(c+d x) \, dx}{2 a d}+\frac {f \int \log (\tanh (c+d x)) \, dx}{a d}\\ &=\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x) \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \int 1 \, dx}{2 a d}-\frac {f \int 2 d x \text {csch}(2 c+2 d x) \, dx}{a d}-\frac {(b f) \int \tan ^{-1}(\sinh (c+d x)) \, dx}{a^2 d}-\frac {(b f) \int \text {csch}(c+d x) \, dx}{a^2 d}\\ &=\frac {f x}{2 a d}+\frac {b^4 (e+f x)^2}{2 a^3 \left (a^2+b^2\right ) f}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^3 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {(2 f) \int x \text {csch}(2 c+2 d x) \, dx}{a}+\frac {(b f) \int d x \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (b^2 f\right ) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d}\\ &=\frac {f x}{2 a d}+\frac {b^4 (e+f x)^2}{2 a^3 \left (a^2+b^2\right ) f}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(b f) \int x \text {sech}(c+d x) \, dx}{a^2}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {f \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {f \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}\\ &=\frac {f x}{2 a d}+\frac {2 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {\left (b^4 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 \left (a^2+b^2\right ) d^2}+\frac {\left (b^4 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 \left (a^2+b^2\right ) d^2}-\frac {(i b f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {(i b f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {\left (i b^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (i b^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}\\ &=\frac {f x}{2 a d}+\frac {2 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}+\frac {(i b f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}+\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (b^4 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}\\ &=\frac {f x}{2 a d}+\frac {2 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}\\ &=\frac {f x}{2 a d}+\frac {2 b f x \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \tan ^{-1}(\sinh (c+d x))}{a^2 d}+\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \tanh ^{-1}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a^3 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 7.18, size = 913, normalized size = 1.20 \begin {gather*} \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 {e \log (\sinh (c+d x))}{a d}+\frac {b^2 e \log (\sinh (c+d x))}{a^3 d}+\frac {c f \log (\sinh (c+d x))}{a d^2}-\frac {b^2 c f \log (\sinh (c+d x))}{a^3 d^2}-\frac {b f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d^2}+\frac {i f \left (i (c+d x) \log \left (1-e^{-2 (c+d x)}\right )-\frac {1}{2} i \left (-(c+d x)^2+\text {PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )\right )}{a d^2}-\frac {i b^2 f \left (i (c+d x) \log \left (1-e^{-2 (c+d x)}\right )-\frac {1}{2} i \left (-(c+d x)^2+\text {PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )\right )}{a^3 d^2}-\frac {b^4 \left (-\frac {1}{2} f (c+d x)^2+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 )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {-a d e (c+d x)+a c f (c+d x)-\frac {1}{2} a f (c+d x)^2+2 b d e \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))-2 b c f \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))+2 b f (c+d x) \text {ArcTan}(\cosh (c+d x)+\sinh (c+d x))+a d e \log (1+\cosh (2 (c+d x))+\sinh (2 (c+d x)))-a c f \log (1+\cosh (2 (c+d x))+\sinh (2 (c+d x)))+a f (c+d x) \log (1+\cosh (2 (c+d x))+\sinh (2 (c+d x)))-i b f \text {PolyLog}(2,-i (\cosh (c+d x)+\sinh (c+d x)))+i b f \text {PolyLog}(2,i (\cosh (c+d x)+\sinh (c+d x)))+\frac {1}{2} a f \text {PolyLog}(2,-\cosh (2 (c+d x))-\sinh (2 (c+d x)))}{\left (a^2+b^2\right ) 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} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x])/(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) - (e*Log[
Sinh[c + d*x]])/(a*d) + (b^2*e*Log[Sinh[c + d*x]])/(a^3*d) + (c*f*Log[Sinh[c + d*x]])/(a*d^2) - (b^2*c*f*Log[S
inh[c + d*x]])/(a^3*d^2) - (b*f*Log[Tanh[(c + d*x)/2]])/(a^2*d^2) + (I*f*(I*(c + d*x)*Log[1 - E^(-2*(c + d*x))
] - (I/2)*(-(c + d*x)^2 + PolyLog[2, E^(-2*(c + d*x))])))/(a*d^2) - (I*b^2*f*(I*(c + d*x)*Log[1 - E^(-2*(c + d
*x))] - (I/2)*(-(c + d*x)^2 + PolyLog[2, E^(-2*(c + d*x))])))/(a^3*d^2) - (b^4*(-1/2*(f*(c + d*x)^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]
)] + d*e*Log[a + b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + 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]))]))/(a^3*(a^2 + b^2)*d^2) + (-(a*d*e*(c + d*x
)) + a*c*f*(c + d*x) - (a*f*(c + d*x)^2)/2 + 2*b*d*e*ArcTan[Cosh[c + d*x] + Sinh[c + d*x]] - 2*b*c*f*ArcTan[Co
sh[c + d*x] + Sinh[c + d*x]] + 2*b*f*(c + d*x)*ArcTan[Cosh[c + d*x] + Sinh[c + d*x]] + a*d*e*Log[1 + Cosh[2*(c
+ d*x)] + Sinh[2*(c + d*x)]] - a*c*f*Log[1 + Cosh[2*(c + d*x)] + Sinh[2*(c + d*x)]] + a*f*(c + d*x)*Log[1 + C
osh[2*(c + d*x)] + Sinh[2*(c + d*x)]] - I*b*f*PolyLog[2, (-I)*(Cosh[c + d*x] + Sinh[c + d*x])] + I*b*f*PolyLog
[2, I*(Cosh[c + d*x] + Sinh[c + d*x])] + (a*f*PolyLog[2, -Cosh[2*(c + d*x)] - Sinh[2*(c + d*x)]])/2)/((a^2 + b
^2)*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)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (714 ) = 1428\).
time = 8.66, size = 1478, normalized size = 1.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1478\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

4/d*e/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))+1/d/a^3*b^2*e*ln(exp(d*x+c)+1)+1/d/a^3*b^2*e*ln(exp(d*x+c)-1)+1/d^2
/a^2*b*f*ln(exp(d*x+c)+1)-1/d^2/a^2*b*f*ln(exp(d*x+c)-1)+4/d*a*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+4/d*a*f/(4
*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+4/d^2*a*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c-1/d/a^3*b^4*e/(a^2+b^2)*ln(b*exp
(2*d*x+2*c)+2*a*exp(d*x+c)-b)+4/d^2*a*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c+1/d^2/a^2*f*b^4/(a^2+b^2)^(3/2)*arc
tanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2/a^2*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/d^2/a^3*f*b^4/(a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2
/a^3*f*b^4/(a^2+b^2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-4/d^2*a*f*c/(4*a^2+4*b^2)*l
n(1+exp(2*d*x+2*c))-8/d^2*f*c/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-4*I/d^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))
*b+4*I/d^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b+1/d^2*b^2*f/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a
)/(a^2+b^2)^(1/2))+8/d*e/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-1/d^2/a^3*f*b^4/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b
^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2/a^3*f*b^4/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2
)^(1/2)))*c-1/d/a^3*f*b^4/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d/a^3*f*b^4
/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2/a^3*f*b^4*c/(a^2+b^2)*ln(b*exp(2*d
*x+2*c)+2*a*exp(d*x+c)-b)+4*I/d*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*x+4*I/d^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+
c))*b*c-4*I/d*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x-4*I/d^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c-1/d^2*f/a*
dilog(exp(d*x+c)+1)+1/d^2*f*dilog(exp(d*x+c))/a+1/d/a^3*b^2*f*ln(exp(d*x+c)+1)*x-1/d^2/a^3*b^2*f*c*ln(exp(d*x+
c)-1)-(-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*exp(d*x+c)+a*f*exp(2*d*x+2*c)+2*b*d*e*exp(d*x+c)-f*a)/d^2/a^2/(exp(2*d*x+2*c)-1)^2-1/d*e/a*ln(exp(d*x+c)-1
)-1/d*e/a*ln(exp(d*x+c)+1)-1/d*f/a*ln(exp(d*x+c)+1)*x+1/d^2*f*c/a*ln(exp(d*x+c)-1)-1/d^2/a^3*b^2*f*dilog(exp(d
*x+c))+1/d^2/a^3*b^2*f*dilog(exp(d*x+c)+1)+4/d^2*a*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+4/d^2*a*f/(4*a^2+4*b^
2)*dilog(1-I*exp(d*x+c))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(16*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^
3*d), x) - 16*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d), x) + 16*b^2*d*integrate(1/16*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) - l
og(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) + 16*integrate(-1/8*(a*b^4*x*
e^(d*x + c) - b^5*x)/(a^5*b + a^3*b^3 - (a^5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^c
)*e^(d*x)), x) + 16*integrate(1/8*(b*x*e^(d*x + c) - a*x)/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)),
x))*f - (b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/(
(a^2 + b^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 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) + (a^2 - b^2)*log(e^(-d*x - c) + 1)/(a
^3*d) + (a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6634 vs. \(2 (704) = 1408\).
time = 0.48, size = 6634, normalized size = 8.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1))*cosh(d*x + c)^3 + 2*((a^3*b
+ a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1))*sinh(d*x + c)^3 - (2*(a^4 + a^2*b^2)*
d*f*x + 2*(a^4 + a^2*b^2)*d*cosh(1) + 2*(a^4 + a^2*b^2)*d*sinh(1) + (a^4 + a^2*b^2)*f)*cosh(d*x + c)^2 - (2*(a
^4 + a^2*b^2)*d*f*x + 2*(a^4 + a^2*b^2)*d*cosh(1) + 2*(a^4 + a^2*b^2)*d*sinh(1) + (a^4 + a^2*b^2)*f - 6*((a^3*
b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c)^2 + (a^
4 + a^2*b^2)*f - 2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1))*cosh(d*x +
c) - (b^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c)^4 - 2*b^4*f*cosh(d*x
+ c)^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 - b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*x + c)^3 - b^4*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^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x
+ c)^4 - 2*b^4*f*cosh(d*x + c)^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 - b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh
(d*x + c)^3 - b^4*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) - ((a^4 - b^4)*f*cosh(d*x + c)^4 + 4*(a^4 - b^4)*f*cosh(d
*x + c)*sinh(d*x + c)^3 + (a^4 - b^4)*f*sinh(d*x + c)^4 - 2*(a^4 - b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 - b^4)*f
*cosh(d*x + c)^2 - (a^4 - b^4)*f)*sinh(d*x + c)^2 + (a^4 - b^4)*f + 4*((a^4 - b^4)*f*cosh(d*x + c)^3 - (a^4 -
b^4)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + (a^4*f + I*a^3*b*f + (a^4*f + I*a^
3*b*f)*cosh(d*x + c)^4 + 4*(a^4*f + I*a^3*b*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*f + I*a^3*b*f)*sinh(d*x +
c)^4 - 2*(a^4*f + I*a^3*b*f)*cosh(d*x + c)^2 - 2*(a^4*f + I*a^3*b*f - 3*(a^4*f + I*a^3*b*f)*cosh(d*x + c)^2)*s
inh(d*x + c)^2 + 4*((a^4*f + I*a^3*b*f)*cosh(d*x + c)^3 - (a^4*f + I*a^3*b*f)*cosh(d*x + c))*sinh(d*x + c))*di
log(I*cosh(d*x + c) + I*sinh(d*x + c)) + (a^4*f - I*a^3*b*f + (a^4*f - I*a^3*b*f)*cosh(d*x + c)^4 + 4*(a^4*f -
I*a^3*b*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*f - I*a^3*b*f)*sinh(d*x + c)^4 - 2*(a^4*f - I*a^3*b*f)*cosh(d
*x + c)^2 - 2*(a^4*f - I*a^3*b*f - 3*(a^4*f - I*a^3*b*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^4*f - I*a^3*
b*f)*cosh(d*x + c)^3 - (a^4*f - I*a^3*b*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x +
c)) - ((a^4 - b^4)*f*cosh(d*x + c)^4 + 4*(a^4 - b^4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 - b^4)*f*sinh(d*x
+ c)^4 - 2*(a^4 - b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 - b^4)*f*cosh(d*x + c)^2 - (a^4 - b^4)*f)*sinh(d*x + c)^
2 + (a^4 - b^4)*f + 4*((a^4 - b^4)*f*cosh(d*x + c)^3 - (a^4 - b^4)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh
(d*x + c) - sinh(d*x + c)) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(
1))*cosh(d*x + c)^4 + 4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*c*f - b
^4*d*cosh(1) - b^4*d*sinh(1))*sinh(d*x + c)^4 - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2 -
2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) - 3*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2)*sinh
(d*x + c)^2 + 4*((b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^3 - (b^4*c*f - b^4*d*cosh(1) - b^4*d*
sinh(1))*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^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^4 +
4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*
sinh(1))*sinh(d*x + c)^4 - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2 - 2*(b^4*c*f - b^4*d*co
sh(1) - b^4*d*sinh(1) - 3*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4
*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^3 - (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*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^4*d*f*x +
b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4
*d*f*x + b^4*c*f)*sinh(d*x + c)^4 - 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2 - 2*(b^4*d*f*x + b^4*c*f - 3*(b^4*
d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^3 - (b^4*d*f*x + b^
4*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^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^4 + 4*(b^4*d*f
*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^4 - 2*(b^4*d*f*x + b^4*c*f)*
cosh(d*x + c)^2 - 2*(b^4*d*f*x + b^4*c*f - 3*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*
d*f*x + b^4*c*f)*cosh(d*x + c)^3 - (b^4*d*f*x +...

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)**3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6439 deep

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {e+f\,x}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)

________________________________________________________________________________________