3.5.97 \(\int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [497]

Optimal. Leaf size=699 \[ \frac {f \text {ArcTan}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \text {ArcTan}(\sinh (c+d x))}{a^3 d^2}+\frac {b^4 f \text {ArcTan}(\sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b^5 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {3 f \text {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {3 f \text {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \text {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b^5 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac {b^5 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d} \]

[Out]

3/2*f*polylog(2,-exp(d*x+c))/a/d^2-3/2*f*polylog(2,exp(d*x+c))/a/d^2-1/2*f*csch(d*x+c)/a/d^2-3/2*f*x*arctanh(c
osh(d*x+c))/a/d+2*b*(f*x+e)*coth(2*d*x+2*c)/a^2/d-1/2*(f*x+e)*csch(d*x+c)^2*sech(d*x+c)/a/d+3/2*(f*x+e)*arctan
h(cosh(d*x+c))/a/d-3/2*(f*x+e)*sech(d*x+c)/a/d+f*arctan(sinh(d*x+c))/a/d^2-b^2*f*arctan(sinh(d*x+c))/a^3/d^2-b
^2*(f*x+e)*arctanh(cosh(d*x+c))/a^3/d-b*f*ln(sinh(2*d*x+2*c))/a^2/d^2+b^2*(f*x+e)*sech(d*x+c)/a^3/d+b^4*f*arct
an(sinh(d*x+c))/a^3/(a^2+b^2)/d^2+b^2*f*x*arctanh(cosh(d*x+c))/a^3/d+b^3*f*ln(cosh(d*x+c))/a^2/(a^2+b^2)/d^2-b
^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d+b^5*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+
b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d-2*b^2*f*x*arctanh(exp(d*x+c))/a^3/d-b^2*f*polylog(2,-exp(d*x+c))/a^3/d^2+b^
2*f*polylog(2,exp(d*x+c))/a^3/d^2-b^5*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2+b
^5*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)^(3/2)/d^2-b^4*(f*x+e)*sech(d*x+c)/a^3/(a^2+b^2
)/d-b^3*(f*x+e)*tanh(d*x+c)/a^2/(a^2+b^2)/d+3*f*x*arctanh(exp(d*x+c))/a/d

________________________________________________________________________________________

Rubi [A]
time = 1.02, antiderivative size = 699, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 22, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5708, 2702, 294, 327, 213, 5570, 6406, 12, 4267, 2317, 2438, 3855, 2701, 5569, 4269, 3556, 5692, 3403, 2296, 2221, 6874, 5559} \begin {gather*} -\frac {b^2 f \text {ArcTan}(\sinh (c+d x))}{a^3 d^2}-\frac {b^2 f \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {b^2 f \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 d^2 \left (a^2+b^2\right )}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 d \left (a^2+b^2\right )}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}+\frac {b^4 f \text {ArcTan}(\sinh (c+d x))}{a^3 d^2 \left (a^2+b^2\right )}-\frac {b^5 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 )^{3/2}}+\frac {b^5 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 )^{3/2}}-\frac {b^5 (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 )^{3/2}}+\frac {b^5 (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 )^{3/2}}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 d \left (a^2+b^2\right )}+\frac {f \text {ArcTan}(\sinh (c+d x))}{a d^2}+\frac {3 f \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(f*ArcTan[Sinh[c + d*x]])/(a*d^2) - (b^2*f*ArcTan[Sinh[c + d*x]])/(a^3*d^2) + (b^4*f*ArcTan[Sinh[c + d*x]])/(a
^3*(a^2 + b^2)*d^2) + (3*f*x*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*f*x*ArcTanh[E^(c + d*x)])/(a^3*d) - (3*f*x*A
rcTanh[Cosh[c + d*x]])/(2*a*d) + (b^2*f*x*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (3*(e + f*x)*ArcTanh[Cosh[c + d*x]
])/(2*a*d) - (b^2*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (2*b*(e + f*x)*Coth[2*c + 2*d*x])/(a^2*d) - (f*C
sch[c + d*x])/(2*a*d^2) - (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2
)*d) + (b^5*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)^(3/2)*d) + (b^3*f*Log[C
osh[c + d*x]])/(a^2*(a^2 + b^2)*d^2) - (b*f*Log[Sinh[2*c + 2*d*x]])/(a^2*d^2) + (3*f*PolyLog[2, -E^(c + d*x)])
/(2*a*d^2) - (b^2*f*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) - (3*f*PolyLog[2, E^(c + d*x)])/(2*a*d^2) + (b^2*f*Pol
yLog[2, E^(c + d*x)])/(a^3*d^2) - (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2
)^(3/2)*d^2) + (b^5*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)^(3/2)*d^2) - (3*(
e + f*x)*Sech[c + d*x])/(2*a*d) + (b^2*(e + f*x)*Sech[c + d*x])/(a^3*d) - (b^4*(e + f*x)*Sech[c + d*x])/(a^3*(
a^2 + b^2)*d) - ((e + f*x)*Csch[c + d*x]^2*Sech[c + d*x])/(2*a*d) - (b^3*(e + f*x)*Tanh[c + d*x])/(a^2*(a^2 +
b^2)*d)

Rule 12

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

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 294

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*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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 2296

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

Rule 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 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 2702

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

Rule 3403

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

Rule 3556

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

Rule 3855

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

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 4269

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

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

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 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 6406

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 - u^2)), x], x] /; I
nverseFunctionFreeQ[u, x]

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}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b \int (e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {f \int \left (\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 \text {sech}(c+d x)}{2 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {(4 b) \int (e+f x) \text {csch}^2(2 c+2 d x) \, dx}{a^2}+\frac {b^2 \int (e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {f \int \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{2 a d}-\frac {(3 f) \int \tanh ^{-1}(\cosh (c+d x)) \, dx}{2 a d}+\frac {(3 f) \int \text {sech}(c+d x) \, dx}{2 a d}\\ &=\frac {3 f \tan ^{-1}(\sinh (c+d x))}{2 a d^2}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \int (e+f x) \text {sech}^2(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}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (b^2 f\right ) \int \left (-\frac {\tanh ^{-1}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a^3}-\frac {(i f) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,-i \text {csch}(c+d x)\right )}{2 a d^2}-\frac {(3 f) \int d x \text {csch}(c+d x) \, dx}{2 a d}-\frac {(2 b f) \int \coth (2 c+2 d x) \, dx}{a^2 d}\\ &=\frac {3 f \tan ^{-1}(\sinh (c+d x))}{2 a d^2}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \int \left (a (e+f x) \text {sech}^2(c+d x)-b (e+f x) \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (2 b^5\right ) \int \frac {e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {(3 f) \int x \text {csch}(c+d x) \, dx}{2 a}-\frac {(i f) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,-i \text {csch}(c+d x)\right )}{2 a d^2}+\frac {\left (b^2 f\right ) \int \tanh ^{-1}(\cosh (c+d x)) \, dx}{a^3 d}-\frac {\left (b^2 f\right ) \int \text {sech}(c+d x) \, dx}{a^3 d}\\ &=\frac {f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \tan ^{-1}(\sinh (c+d x))}{a^3 d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {\left (2 b^6\right ) \int \frac {e^{c+d x} (e+f x)}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 b^6\right ) \int \frac {e^{c+d x} (e+f x)}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \int (e+f x) \text {sech}^2(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x) \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(3 f) \int \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac {(3 f) \int \log \left (1+e^{c+d x}\right ) \, dx}{2 a d}+\frac {\left (b^2 f\right ) \int d x \text {csch}(c+d x) \, dx}{a^3 d}\\ &=\frac {f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \tan ^{-1}(\sinh (c+d x))}{a^3 d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^2 f\right ) \int x \text {csch}(c+d x) \, dx}{a^3}+\frac {(3 f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac {(3 f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{2 a d^2}+\frac {\left (b^5 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac {\left (b^5 f\right ) \int \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {\left (b^3 f\right ) \int \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^4 f\right ) \int \text {sech}(c+d x) \, dx}{a^3 \left (a^2+b^2\right ) d}\\ &=\frac {f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \tan ^{-1}(\sinh (c+d x))}{a^3 d^2}+\frac {b^4 f \tan ^{-1}(\sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {3 f \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^5 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a-2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (b^5 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 b x}{2 a+2 \sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (b^2 f\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}\\ &=\frac {f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \tan ^{-1}(\sinh (c+d x))}{a^3 d^2}+\frac {b^4 f \tan ^{-1}(\sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {3 f \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {3 f \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac {b^5 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 )^{3/2} d^2}+\frac {b^5 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 )^{3/2} d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}\\ &=\frac {f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {b^2 f \tan ^{-1}(\sinh (c+d x))}{a^3 d^2}+\frac {b^4 f \tan ^{-1}(\sinh (c+d x))}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {3 f x \tanh ^{-1}(\cosh (c+d x))}{2 a d}+\frac {b^2 f x \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {3 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x) \coth (2 c+2 d x)}{a^2 d}-\frac {f \text {csch}(c+d x)}{2 a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {b^3 f \log (\cosh (c+d x))}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (2 c+2 d x))}{a^2 d^2}+\frac {3 f \text {Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac {b^2 f \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {3 f \text {Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac {b^2 f \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {b^5 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 )^{3/2} d^2}+\frac {b^5 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 )^{3/2} d^2}-\frac {3 (e+f x) \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x) \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x) \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x) \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 9.43, size = 863, normalized size = 1.23 \begin {gather*} \frac {2 a f \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2 d+b^2 d\right )}+\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \cosh \left (\frac {1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac {1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}-\frac {b f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}-\frac {b f \log (\sinh (c+d x))}{a^2 d^2}-\frac {3 e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d}+\frac {b^2 e \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d}+\frac {3 c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}-\frac {b^2 c f \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d^2}+\frac {3 i f \left (i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{-c-d x}\right )-\text {PolyLog}\left (2,e^{-c-d x}\right )\right )\right )}{2 a d^2}-\frac {i b^2 f \left (i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (1+e^{-c-d x}\right )\right )+i \left (\text {PolyLog}\left (2,-e^{-c-d x}\right )-\text {PolyLog}\left (2,e^{-c-d x}\right )\right )\right )}{a^3 d^2}+\frac {b^5 \left (2 d e \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 c f \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-f \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 )^{3/2} 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}+\frac {\text {sech}(c+d x) (-a d e+a c f-a f (c+d x)+b d e \sinh (c+d x)-b c f \sinh (c+d x)+b f (c+d x) \sinh (c+d x))}{\left (a^2+b^2\right ) d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)*Csch[c + d*x]^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(2*a*f*ArcTan[Tanh[(c + d*x)/2]])/(d*(a^2*d + b^2*d)) + ((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] -
2*b*c*f*Cosh[(c + d*x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f
- f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) - (b*f*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^2) - (b*f*Log[Sinh[c
+ d*x]])/(a^2*d^2) - (3*e*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (b^2*e*Log[Tanh[(c + d*x)/2]])/(a^3*d) + (3*c*f*Lo
g[Tanh[(c + d*x)/2]])/(2*a*d^2) - (b^2*c*f*Log[Tanh[(c + d*x)/2]])/(a^3*d^2) + (((3*I)/2)*f*(I*(c + d*x)*(Log[
1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - PolyLog[2, E^(-c - d*x)])))/(a*d^2
) - (I*b^2*f*(I*(c + d*x)*(Log[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - Pol
yLog[2, E^(-c - d*x)])))/(a^3*d^2) + (b^5*(2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*c*f*ArcTanh[
(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x
)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*Po
lyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^3*(a^2 + b^2)^(3/2)*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) + (Sech[c + d*x]*(-(a*d*e) + a*c*f -
a*f*(c + d*x) + b*d*e*Sinh[c + d*x] - b*c*f*Sinh[c + d*x] + b*f*(c + d*x)*Sinh[c + d*x]))/((a^2 + b^2)*d^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2766\) vs. \(2(659)=1318\).
time = 11.23, size = 2767, normalized size = 3.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(2767\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*csch(d*x+c)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/2/d/(a^2+b^2)*b^2*f/a*ln(exp(d*x+c)+1)*x+1/2/d^2/(a^2+b^2)*b^2*f*c/a*ln(exp(d*x+c)-1)-3/2/d*a*b*e/(a^2+b^2)^
(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/d/a*b^3*e/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x
+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2*a*b*f/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/2/d^
2/a*b^3*f*c/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(3/2)/d^2/a^3*b^5*f*
c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d^2/a*b^5*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+b^2)^(5/2)/d^2/a*b^5*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+b^2)/d/a^3*b^4*e*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d/a^3*b^4*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d^2/
a^3*b^4*f*dilog(exp(d*x+c))-1/(a^2+b^2)/d^2/a^3*b^4*f*dilog(exp(d*x+c)+1)+2/(a^2+b^2)/d^2/a^2*b^3*f*ln(exp(d*x
+c))-1/(a^2+b^2)/d^2/a^2*b^3*f*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d^2/a^2*b^3*f*ln(exp(d*x+c)+1)-4/(a^2+b^2)/d^2*b^3
*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-1/d^2/a*b^3*f/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)
^(1/2))-2/(a^2+b^2)^(5/2)/d^2*a*f*b^3*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-3/2/(a^2+b^2)^(5/2)/
d^2/a*f*b^5*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-(3*a^3*d*f*x*exp(5*d*x+5*c)+a*b^2*d*f*x*exp(5*
d*x+5*c)+3*a^3*d*e*exp(5*d*x+5*c)+a*b^2*d*e*exp(5*d*x+5*c)-2*b^3*d*f*x*exp(4*d*x+4*c)-2*a^3*d*f*x*exp(3*d*x+3*
c)+a^3*f*exp(5*d*x+5*c)+2*a*b^2*d*f*x*exp(3*d*x+3*c)+a*b^2*f*exp(5*d*x+5*c)-2*b^3*d*e*exp(4*d*x+4*c)-2*a^3*d*e
*exp(3*d*x+3*c)-4*a^2*b*d*f*x*exp(2*d*x+2*c)+2*a*b^2*d*e*exp(3*d*x+3*c)+3*a^3*d*f*x*exp(d*x+c)-4*a^2*b*d*e*exp
(2*d*x+2*c)+a*b^2*d*f*x*exp(d*x+c)+3*a^3*d*e*exp(d*x+c)+4*a^2*b*d*f*x+a*b^2*d*e*exp(d*x+c)+2*b^3*d*f*x-a^3*f*e
xp(d*x+c)+4*a^2*b*d*e-a*b^2*f*exp(d*x+c)+2*b^3*d*e)/d^2/a^2/(exp(2*d*x+2*c)-1)^2/(a^2+b^2)/(1+exp(2*d*x+2*c))+
8/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/2/d^2/(a^2+b^2)*b^2*f/a*dilog(exp(d*x+c)+1)+1/2/d^2/(
a^2+b^2)*b^2*f*dilog(exp(d*x+c))/a-1/(a^2+b^2)^(5/2)/d^2/a^3*b^7*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+b^2)^(5/2)/d^2/a^3*b^7*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+b^2)^(5/2)/d^2/a^3*b^7*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d/a^3*b^7
*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d/a^3*b^7*f*ln((b*exp(d*x+c)
+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(5/2)/d/a*b^5*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-
a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(5/2)/d/a*b^5*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-3
/2/(a^2+b^2)^(5/2)/d^2*a^3*f*c*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3/2/d^2/(a^2+b^2)*a*f*c*ln(
exp(d*x+c)-1)+1/2/d/(a^2+b^2)*b^2*e/a*ln(exp(d*x+c)+1)-1/2/d/(a^2+b^2)*b^2*e/a*ln(exp(d*x+c)-1)+3/2/d/(a^2+b^2
)*ln(exp(d*x+c)+1)*a*f*x+3/2/d^2*a*b*f*c/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(
a^2+b^2)^(5/2)/d*a*b^3*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3/2/(a^2+b^2)^(5/2)/d*a^3*e*b*arcta
nh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3/2/(a^2+b^2)^(5/2)/d/a*b^5*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a
^2+b^2)^(1/2))+1/(a^2+b^2)^(5/2)/d^2*a^3*f*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+2/(a^2+b^2)^(5/
2)/d^2*a*f*b^3*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(5/2)/d^2/a*f*b^5*arctanh(1/2*(2*
b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*b*ln(1+exp(2*d*x+2*c))-1/(a^2+b^2)/d^2/
a^3*b^4*f*c*ln(exp(d*x+c)-1)+1/(a^2+b^2)^(3/2)/d/a^3*b^5*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1
/(a^2+b^2)^(5/2)/d/a^3*b^7*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)/d/a^3*b^4*f*ln(exp(
d*x+c)+1)*x+8/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/(a^2+b^2)^(5/2)/d^2/a^3*b^7*f*dilog((b*
exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)^(5/2)/d^2/a^3*b^7*f*dilog((-b*exp(d*x+c)+(a^2+b
^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/(a^2+b^2)^(5/2)/d^2/a*b^5*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^
2+b^2)^(1/2)))-1/(a^2+b^2)^(5/2)/d^2/a*b^5*f*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+3/2
/d/(a^2+b^2)*a*e*ln(exp(d*x+c)+1)-3/2/d/(a^2+b^2)*a*e*ln(exp(d*x+c)-1)+3/2/d^2/(a^2+b^2)*a*f*dilog(exp(d*x+c)+
1)+3/2/d^2/(a^2+b^2)*a*f*dilog(exp(d*x+c))+4/(a^2+b^2)/d^2*b*f*ln(exp(d*x+c))-1/(a^2+b^2)/d^2*b*f*ln(exp(d*x+c
)-1)-1/(a^2+b^2)/d^2*b*f*ln(exp(d*x+c)+1)

________________________________________________________________________________________

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

[Out]

-(32*b^5*integrate(-1/16*x*e^(d*x + c)/(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) + 96*a^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 64*b^2*d*integr
ate(1/64*x/(a^3*d*e^(d*x + c) + a^3*d), x) + 96*a^2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) - a^3*d), x) - 64*b^
2*d*integrate(1/64*x/(a^3*d*e^(d*x + c) - a^3*d), x) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) + 1)/(a^3*d^
2)) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) - (2*b^3*d*x*e^(4*d*x + 4*c) + 4*a^2*b*d*x*e^
(2*d*x + 2*c) + 2*(a^3*d*e^(3*c) - a*b^2*d*e^(3*c))*x*e^(3*d*x) - 2*(2*a^2*b*d + b^3*d)*x - (a^3*e^(5*c) + a*b
^2*e^(5*c) + (3*a^3*d*e^(5*c) + a*b^2*d*e^(5*c))*x)*e^(5*d*x) + (a^3*e^c + a*b^2*e^c - (3*a^3*d*e^c + a*b^2*d*
e^c)*x)*e^(d*x))/(a^4*d^2 + a^2*b^2*d^2 + (a^4*d^2*e^(6*c) + a^2*b^2*d^2*e^(6*c))*e^(6*d*x) - (a^4*d^2*e^(4*c)
+ a^2*b^2*d^2*e^(4*c))*e^(4*d*x) - (a^4*d^2*e^(2*c) + a^2*b^2*d^2*e^(2*c))*e^(2*d*x)) - 2*b*x/((a^2 + b^2)*d)
- 2*a*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + b*log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2))*f - 1/2*(2*b^5*lo
g((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^5 + a^3*b^2)*sqrt(a^2 + b
^2)*d) + 2*(4*a^2*b*e^(-2*d*x - 2*c) + 2*b^3*e^(-4*d*x - 4*c) - 4*a^2*b - 2*b^3 + (3*a^3 + a*b^2)*e^(-d*x - c)
- 2*(a^3 - a*b^2)*e^(-3*d*x - 3*c) + (3*a^3 + a*b^2)*e^(-5*d*x - 5*c))/((a^4 + a^2*b^2 - (a^4 + a^2*b^2)*e^(-
2*d*x - 2*c) - (a^4 + a^2*b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-6*d*x - 6*c))*d) - (3*a^2 - 2*b^2)*log(e
^(-d*x - c) + 1)/(a^3*d) + (3*a^2 - 2*b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 12895 vs. \(2 (663) = 1326\).
time = 0.49, size = 12895, normalized size = 18.45 \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)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(4*((2*a^5*b + 3*a^3*b^3 + a*b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*cosh(d*x + c)^6 + 4*((2*a^5*b +
3*a^3*b^3 + a*b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*sinh(d*x + c)^6 - 2*((3*a^6 + 4*a^4*b^2 + a^2*b^4
)*d*f*x + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*cosh(1) + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*sinh(1) + (a^6 + 2*a^4*b^2
+ a^2*b^4)*f)*cosh(d*x + c)^5 - 2*((3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*f*x + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*cosh
(1) + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*sinh(1) + (a^6 + 2*a^4*b^2 + a^2*b^4)*f - 12*((2*a^5*b + 3*a^3*b^3 + a*b
^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*cosh(d*x + c))*sinh(d*x + c)^5 - 4*(2*(a^5*b + a^3*b^3)*d*f*x + (
a^5*b + 2*a^3*b^3 + a*b^5)*c*f - (a^3*b^3 + a*b^5)*d*cosh(1) - (a^3*b^3 + a*b^5)*d*sinh(1))*cosh(d*x + c)^4 -
2*(4*(a^5*b + a^3*b^3)*d*f*x + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c*f - 2*(a^3*b^3 + a*b^5)*d*cosh(1) - 30*((2*a^5*
b + 3*a^3*b^3 + a*b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*cosh(d*x + c)^2 - 2*(a^3*b^3 + a*b^5)*d*sinh(1
) + 5*((3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*f*x + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*cosh(1) + (3*a^6 + 4*a^4*b^2 + a^
2*b^4)*d*sinh(1) + (a^6 + 2*a^4*b^2 + a^2*b^4)*f)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*((a^6 - a^2*b^4)*d*f*x +
(a^6 - a^2*b^4)*d*cosh(1) + (a^6 - a^2*b^4)*d*sinh(1))*cosh(d*x + c)^3 + 4*((a^6 - a^2*b^4)*d*f*x + 20*((2*a^5
*b + 3*a^3*b^3 + a*b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*cosh(d*x + c)^3 + (a^6 - a^2*b^4)*d*cosh(1) -
5*((3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*f*x + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*cosh(1) + (3*a^6 + 4*a^4*b^2 + a^2*b
^4)*d*sinh(1) + (a^6 + 2*a^4*b^2 + a^2*b^4)*f)*cosh(d*x + c)^2 + (a^6 - a^2*b^4)*d*sinh(1) - 4*(2*(a^5*b + a^3
*b^3)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f - (a^3*b^3 + a*b^5)*d*cosh(1) - (a^3*b^3 + a*b^5)*d*sinh(1))*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(a^5*b + 2*a^3*b^3 + a*b^5)*c*f - 4*(2*a^5*b + 3*a^3*b^3 + a*b^5)*d*cosh(1) -
4*((a^3*b^3 + a*b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f - 2*(a^5*b + a^3*b^3)*d*cosh(1) - 2*(a^5*b + a^3*
b^3)*d*sinh(1))*cosh(d*x + c)^2 - 4*(2*a^5*b + 3*a^3*b^3 + a*b^5)*d*sinh(1) + 4*(15*((2*a^5*b + 3*a^3*b^3 + a*
b^5)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f)*cosh(d*x + c)^4 - (a^3*b^3 + a*b^5)*d*f*x - 5*((3*a^6 + 4*a^4*b^
2 + a^2*b^4)*d*f*x + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*cosh(1) + (3*a^6 + 4*a^4*b^2 + a^2*b^4)*d*sinh(1) + (a^6
+ 2*a^4*b^2 + a^2*b^4)*f)*cosh(d*x + c)^3 - (a^5*b + 2*a^3*b^3 + a*b^5)*c*f + 2*(a^5*b + a^3*b^3)*d*cosh(1) -
6*(2*(a^5*b + a^3*b^3)*d*f*x + (a^5*b + 2*a^3*b^3 + a*b^5)*c*f - (a^3*b^3 + a*b^5)*d*cosh(1) - (a^3*b^3 + a*b^
5)*d*sinh(1))*cosh(d*x + c)^2 + 2*(a^5*b + a^3*b^3)*d*sinh(1) + 3*((a^6 - a^2*b^4)*d*f*x + (a^6 - a^2*b^4)*d*c
osh(1) + (a^6 - a^2*b^4)*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(b^6*f*cosh(d*x + c)^6 + 6*b^6*f*cosh(d
*x + c)*sinh(d*x + c)^5 + b^6*f*sinh(d*x + c)^6 - b^6*f*cosh(d*x + c)^4 - b^6*f*cosh(d*x + c)^2 + b^6*f + (15*
b^6*f*cosh(d*x + c)^2 - b^6*f)*sinh(d*x + c)^4 + 4*(5*b^6*f*cosh(d*x + c)^3 - b^6*f*cosh(d*x + c))*sinh(d*x +
c)^3 + (15*b^6*f*cosh(d*x + c)^4 - 6*b^6*f*cosh(d*x + c)^2 - b^6*f)*sinh(d*x + c)^2 + 2*(3*b^6*f*cosh(d*x + c)
^5 - 2*b^6*f*cosh(d*x + c)^3 - b^6*f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^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) + 2*(b^6*f*cosh(d*
x + c)^6 + 6*b^6*f*cosh(d*x + c)*sinh(d*x + c)^5 + b^6*f*sinh(d*x + c)^6 - b^6*f*cosh(d*x + c)^4 - b^6*f*cosh(
d*x + c)^2 + b^6*f + (15*b^6*f*cosh(d*x + c)^2 - b^6*f)*sinh(d*x + c)^4 + 4*(5*b^6*f*cosh(d*x + c)^3 - b^6*f*c
osh(d*x + c))*sinh(d*x + c)^3 + (15*b^6*f*cosh(d*x + c)^4 - 6*b^6*f*cosh(d*x + c)^2 - b^6*f)*sinh(d*x + c)^2 +
2*(3*b^6*f*cosh(d*x + c)^5 - 2*b^6*f*cosh(d*x + c)^3 - b^6*f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b
^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) - 2*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1) + (b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^
6 + 6*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^5 + (b^6*c*f - b^6*d*cosh(1) - b^6
*d*sinh(1))*sinh(d*x + c)^6 - (b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^4 - (b^6*c*f - b^6*d*cos
h(1) - b^6*d*sinh(1) - 15*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(b
^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^3 - (b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x +
c))*sinh(d*x + c)^3 - (b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^2 - (b^6*c*f - b^6*d*cosh(1) -
b^6*d*sinh(1) - 15*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^4 + 6*(b^6*c*f - b^6*d*cosh(1) - b^
6*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^5
- 2*(b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cosh(d*x + c)^3 - (b^6*c*f - b^6*d*cosh(1) - b^6*d*sinh(1))*cos
h(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b
^2)/b^2) + 2*a) + 2*(b^6*c*f - b^6*d*cosh(1) - ...

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Sympy [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)**2/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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

[Out]

Timed out

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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 )}^2\,{\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)^2*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)/(cosh(c + d*x)^2*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)

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