3.470 \(\int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=499 \[ -\frac {b^2 f \log (\cosh (c+d x))}{a d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x) \tanh (c+d x)}{a d \left (a^2+b^2\right )}+\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^3 f \tan ^{-1}(\sinh (c+d x))}{a^2 d^2 \left (a^2+b^2\right )}+\frac {b^3 (e+f x) \text {sech}(c+d x)}{a^2 d \left (a^2+b^2\right )}+\frac {b f \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac {b f \tan ^{-1}(\sinh (c+d x))}{a^2 d^2}-\frac {b (e+f x) \text {sech}(c+d x)}{a^2 d}+\frac {b (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {2 b f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {b f x \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {f \log (\sinh (2 c+2 d x))}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x)}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.98, antiderivative size = 499, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 20, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {5589, 5461, 4184, 3475, 2622, 321, 207, 5462, 6271, 12, 4182, 2279, 2391, 3770, 5573, 3322, 2264, 2190, 6742, 5451} \[ \frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b f \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {b f \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac {b^3 f \tan ^{-1}(\sinh (c+d x))}{a^2 d^2 \left (a^2+b^2\right )}-\frac {b^2 f \log (\cosh (c+d x))}{a d^2 \left (a^2+b^2\right )}+\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {b^2 (e+f x) \tanh (c+d x)}{a d \left (a^2+b^2\right )}+\frac {b^3 (e+f x) \text {sech}(c+d x)}{a^2 d \left (a^2+b^2\right )}+\frac {b f \tan ^{-1}(\sinh (c+d x))}{a^2 d^2}-\frac {b (e+f x) \text {sech}(c+d x)}{a^2 d}+\frac {b (e+f x) \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {2 b f x \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {b f x \tanh ^{-1}(\cosh (c+d x))}{a^2 d}+\frac {f \log (\sinh (2 c+2 d x))}{a d^2}-\frac {2 (e+f x) \coth (2 c+2 d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*f*ArcTan[Sinh[c + d*x]])/(a^2*d^2) - (b^3*f*ArcTan[Sinh[c + d*x]])/(a^2*(a^2 + b^2)*d^2) + (2*b*f*x*ArcTanh
[E^(c + d*x)])/(a^2*d) - (b*f*x*ArcTanh[Cosh[c + d*x]])/(a^2*d) + (b*(e + f*x)*ArcTanh[Cosh[c + d*x]])/(a^2*d)
- (2*(e + f*x)*Coth[2*c + 2*d*x])/(a*d) + (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2
*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*
d) - (b^2*f*Log[Cosh[c + d*x]])/(a*(a^2 + b^2)*d^2) + (f*Log[Sinh[2*c + 2*d*x]])/(a*d^2) + (b*f*PolyLog[2, -E^
(c + d*x)])/(a^2*d^2) - (b*f*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqr
t[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a
^2*(a^2 + b^2)^(3/2)*d^2) - (b*(e + f*x)*Sech[c + d*x])/(a^2*d) + (b^3*(e + f*x)*Sech[c + d*x])/(a^2*(a^2 + b^
2)*d) + (b^2*(e + f*x)*Tanh[c + d*x])/(a*(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 207

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

Rule 321

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2264

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 2279

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 2391

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 2622

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 3322

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 3475

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

Rule 3770

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

Rule 4182

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 4184

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 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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 5461

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 5462

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 5573

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 5589

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 6271

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 6742

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

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Mathematica [C]  time = 8.99, size = 1862, normalized size = 3.73 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

4*(-1/8*(f*(c + d*x))/((a + I*b)*d^2) + ((I/8)*((2 - I)*a^3*d*f + (3*I)*a^2*b*d*f - I*a*b^2*d*f + I*b^3*d*f +
a^2*b*c*d*f + I*a*b^2*c*d*f)*(c + d*x))/(a*(a + I*b)*(a^2 + b^2)*d^3) - ((I/16)*b*f*(c + d*x)^2)/((a^2 + b^2)*
d^2) + ((I/4)*f*ArcTan[(a*Cosh[(c + d*x)/2] - b*Cosh[(c + d*x)/2] + a*Sinh[(c + d*x)/2] + b*Sinh[(c + d*x)/2])
/(a*Cosh[(c + d*x)/2] + b*Cosh[(c + d*x)/2] - a*Sinh[(c + d*x)/2] + b*Sinh[(c + d*x)/2])])/((a + I*b)*d^2) - (
a*f*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]])/(2*(a^2 + b^2)*d^2) - (b^2*f*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]])/(
2*a*(a^2 + b^2)*d^2) - (b*c*f*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]])/(2*(a^2 + b^2)*d^2) + ((-(d*e*Cosh[(c + d*
x)/2]) + c*f*Cosh[(c + d*x)/2] - f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(8*a*d^2) + (a*f*Log[Cosh[(
c + d*x)/2]])/(4*(a^2 + b^2)*d^2) + (b^2*f*Log[Cosh[(c + d*x)/2]])/(4*a*(a^2 + b^2)*d^2) - (b*c*f*Log[Cosh[(c
+ d*x)/2]])/(4*(a^2 + b^2)*d^2) + (f*Log[Cosh[c + d*x]])/(8*(a + I*b)*d^2) + (a*f*((-I)*(c + d*x) + 2*ArcTanh[
1 - (2*I)*Tanh[(c + d*x)/2]] + Log[-1 + Cosh[c + d*x] + I*Sinh[c + d*x]]))/(4*(a^2 + b^2)*d^2) + ((I/8)*b*f*((
-I)*(c + d*x) + 2*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]] + Log[-1 + Cosh[c + d*x] + I*Sinh[c + d*x]]))/((a^2 + b
^2)*d^2) + (b^2*f*((-I)*(c + d*x) + 2*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]] + Log[-1 + Cosh[c + d*x] + I*Sinh[c
+ d*x]]))/(8*a*(a^2 + b^2)*d^2) + (b*c*f*((-I)*(c + d*x) + 2*ArcTanh[1 - (2*I)*Tanh[(c + d*x)/2]] + Log[-1 +
Cosh[c + d*x] + I*Sinh[c + d*x]]))/(8*(a^2 + b^2)*d^2) - (b*e*Log[Tanh[(c + d*x)/2]])/(4*(a^2 + b^2)*d) - (b^3
*e*Log[Tanh[(c + d*x)/2]])/(4*a^2*(a^2 + b^2)*d) + (b^3*c*f*Log[Tanh[(c + d*x)/2]])/(4*a^2*(a^2 + b^2)*d^2) +
((I/2)*b*f*((-1/8*I)*(c + d*x)^2 - (I/2)*(c + d*x)*Log[1 + E^(-c - d*x)] + (I/2)*PolyLog[2, -E^(-c - d*x)]))/(
(a^2 + b^2)*d^2) - (b*f*((-1/2*I)*(c + d*x)^2 + (I/4)*(3*Pi*(c + d*x) + (1 - I)*(c + d*x)^2 + 2*(Pi - (2*I)*(c
+ d*x))*Log[1 + I*E^(-c - d*x)] - 4*Pi*Log[1 + E^(c + d*x)] - 2*Pi*Log[-Cos[(Pi + (2*I)*(c + d*x))/4]] + 4*Pi
*Log[Cosh[(c + d*x)/2]] + (4*I)*PolyLog[2, (-I)*E^(-c - d*x)])))/(4*(a^2 + b^2)*d^2) + ((I/4)*b^3*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^2*(a^2 + b^2)*d^2) - ((I/4)*b*f*((c + d*x)^2/4 + (-3*Pi*(c + d*x) - (1 - I)*(c + d*x)^2 - 2*(Pi - (2*I)*(
c + d*x))*Log[1 + I*E^(-c - d*x)] + 4*Pi*Log[1 + E^(c + d*x)] + 2*Pi*Log[-Cos[(Pi + (2*I)*(c + d*x))/4]] - 4*P
i*Log[Cosh[(c + d*x)/2]] - (4*I)*PolyLog[2, (-I)*E^(-c - d*x)])/4 - (I/2)*(-1/2*(c + d*x)^2 + 2*(c + d*x)*Log[
1 - E^(c + d*x)] + 2*PolyLog[2, E^(c + d*x)])))/((a^2 + b^2)*d^2) + (b^4*(-2*d*e*ArcTanh[(a + b*Cosh[c + d*x]
+ b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + 2*c*f*ArcTanh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] +
f*(c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a - Sqrt[a^2 + b^2])] - f*(c + d*x)*Log[1 + (b*(Cosh
[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] + f*PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(-a + Sq
rt[a^2 + b^2])] - f*PolyLog[2, -((b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2]))]))/(4*a^2*(a^2 + b
^2)^(3/2)*d^2) + (Sech[(c + d*x)/2]*(-(d*e*Sinh[(c + d*x)/2]) + c*f*Sinh[(c + d*x)/2] - f*(c + d*x)*Sinh[(c +
d*x)/2]))/(8*a*d^2) + (Sech[c + d*x]*(-(b*d*e) + b*c*f - b*f*(c + d*x) - a*d*e*Sinh[c + d*x] + a*c*f*Sinh[c +
d*x] - a*f*(c + d*x)*Sinh[c + d*x]))/(4*(a^2 + b^2)*d^2))

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fricas [B]  time = 0.60, size = 4086, normalized size = 8.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*((2*a^5 + 3*a^3*b^2 + a*b^4)*d*f*x + (a^5 + 2*a^3*b^2 + a*b^4)*c*f)*cosh(d*x + c)^4 + 2*((2*a^5 + 3*a^3*b^
2 + a*b^4)*d*f*x + (a^5 + 2*a^3*b^2 + a*b^4)*c*f)*sinh(d*x + c)^4 + 2*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*
b^3)*d*e)*cosh(d*x + c)^3 + 2*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*b^3)*d*e + 4*((2*a^5 + 3*a^3*b^2 + a*b^4
)*d*f*x + (a^5 + 2*a^3*b^2 + a*b^4)*c*f)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*d*e -
2*(a^5 + 2*a^3*b^2 + a*b^4)*c*f + 2*((a^3*b^2 + a*b^4)*d*f*x + (a^3*b^2 + a*b^4)*d*e)*cosh(d*x + c)^2 + 2*((a^
3*b^2 + a*b^4)*d*f*x + (a^3*b^2 + a*b^4)*d*e + 6*((2*a^5 + 3*a^3*b^2 + a*b^4)*d*f*x + (a^5 + 2*a^3*b^2 + a*b^4
)*c*f)*cosh(d*x + c)^2 + 3*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*b^3)*d*e)*cosh(d*x + c))*sinh(d*x + c)^2 -
(b^5*f*cosh(d*x + c)^4 + 4*b^5*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b
^5*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*f*sinh(d*x + c)^4 - b^5*f)*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) + (b^5*f*cosh(d*x
+ c)^4 + 4*b^5*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*b^5*f*cosh(d*x +
c)*sinh(d*x + c)^3 + b^5*f*sinh(d*x + c)^4 - b^5*f)*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) - (b^5*d*e - b^5*c*f - (b^5*d*e
- b^5*c*f)*cosh(d*x + c)^4 - 4*(b^5*d*e - b^5*c*f)*cosh(d*x + c)^3*sinh(d*x + c) - 6*(b^5*d*e - b^5*c*f)*cosh(
d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*e - b^5*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 - (b^5*d*e - b^5*c*f)*sinh(d*
x + c)^4)*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) +
(b^5*d*e - b^5*c*f - (b^5*d*e - b^5*c*f)*cosh(d*x + c)^4 - 4*(b^5*d*e - b^5*c*f)*cosh(d*x + c)^3*sinh(d*x + c
) - 6*(b^5*d*e - b^5*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*e - b^5*c*f)*cosh(d*x + c)*sinh(d*x + c)^
3 - (b^5*d*e - b^5*c*f)*sinh(d*x + c)^4)*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) + (b^5*d*f*x + b^5*c*f - (b^5*d*f*x + b^5*c*f)*cosh(d*x + c)^4 - 4*(b^5*d*f*x +
b^5*c*f)*cosh(d*x + c)^3*sinh(d*x + c) - 6*(b^5*d*f*x + b^5*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*f*
x + b^5*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 - (b^5*d*f*x + b^5*c*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^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) - (b^5
*d*f*x + b^5*c*f - (b^5*d*f*x + b^5*c*f)*cosh(d*x + c)^4 - 4*(b^5*d*f*x + b^5*c*f)*cosh(d*x + c)^3*sinh(d*x +
c) - 6*(b^5*d*f*x + b^5*c*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*f*x + b^5*c*f)*cosh(d*x + c)*sinh(d*x
+ c)^3 - (b^5*d*f*x + b^5*c*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c)
- (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*((a^4*b + a^2*b^3)*f*cosh(d*x + c)^4 +
4*(a^4*b + a^2*b^3)*f*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^4*b + a^2*b^3)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 +
4*(a^4*b + a^2*b^3)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*b + a^2*b^3)*f*sinh(d*x + c)^4 - (a^4*b + a^2*b^3)
*f)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*b^3)*d*e)*cosh(d*x + c)
+ ((a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^4 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^3*sinh(d*x + c) +
6*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)*s
inh(d*x + c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*f*sinh(d*x + c)^4 - (a^4*b + 2*a^2*b^3 + b^5)*f)*dilog(cosh(d*x + c
) + sinh(d*x + c)) - ((a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^4 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c
)^3*sinh(d*x + c) + 6*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4*b + 2*a^2*b^3 + b^5
)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*f*sinh(d*x + c)^4 - (a^4*b + 2*a^2*b^3 + b^5)*f)
*dilog(-cosh(d*x + c) - sinh(d*x + c)) - ((a^5 + a^3*b^2)*f*cosh(d*x + c)^4 + 4*(a^5 + a^3*b^2)*f*cosh(d*x + c
)^3*sinh(d*x + c) + 6*(a^5 + a^3*b^2)*f*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^5 + a^3*b^2)*f*cosh(d*x + c)*si
nh(d*x + c)^3 + (a^5 + a^3*b^2)*f*sinh(d*x + c)^4 - (a^5 + a^3*b^2)*f)*log(2*cosh(d*x + c)/(cosh(d*x + c) - si
nh(d*x + c))) - (((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e + (a^5 + 2*a^3*b^2 + a*b^4)*
f)*cosh(d*x + c)^4 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e + (a^5 + 2*a^3*b^2 + a
*b^4)*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e +
(a^5 + 2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2
*a^2*b^3 + b^5)*d*e + (a^5 + 2*a^3*b^2 + a*b^4)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4*b + 2*a^2*b^3 + b^5)*
d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e + (a^5 + 2*a^3*b^2 + a*b^4)*f)*sinh(d*x + c)^4 - (a^4*b + 2*a^2*b^3 + b^
5)*d*f*x - (a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^4)*f)*log(cosh(d*x + c) + sinh(d*x + c) + 1)
+ (((a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*f)*cosh(d*x + c)^
4 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*f)*cosh(d*x + c
)^3*sinh(d*x + c) + 6*((a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)
*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2
*a^2*b^3 + b^5)*c)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4*b + 2*a^2*b^3 + b^5)*d*e - (a^5 + 2*a^3*b^2 + a*b^
4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*f)*sinh(d*x + c)^4 - (a^4*b + 2*a^2*b^3 + b^5)*d*e + (a^5 + 2*a^3*b^2 + a*b^4
+ (a^4*b + 2*a^2*b^3 + b^5)*c)*f)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (((a^4*b + 2*a^2*b^3 + b^5)*d*f*x
+ (a^4*b + 2*a^2*b^3 + b^5)*c*f)*cosh(d*x + c)^4 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b
^5)*c*f)*cosh(d*x + c)^3*sinh(d*x + c) + 6*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*c*f)*c
osh(d*x + c)^2*sinh(d*x + c)^2 + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*c*f)*cosh(d*x
+ c)*sinh(d*x + c)^3 + ((a^4*b + 2*a^2*b^3 + b^5)*d*f*x + (a^4*b + 2*a^2*b^3 + b^5)*c*f)*sinh(d*x + c)^4 - (a^
4*b + 2*a^2*b^3 + b^5)*d*f*x - (a^4*b + 2*a^2*b^3 + b^5)*c*f)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*((a^
4*b + a^2*b^3)*d*f*x - 4*((2*a^5 + 3*a^3*b^2 + a*b^4)*d*f*x + (a^5 + 2*a^3*b^2 + a*b^4)*c*f)*cosh(d*x + c)^3 +
(a^4*b + a^2*b^3)*d*e - 3*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*b^3)*d*e)*cosh(d*x + c)^2 - 2*((a^3*b^2 + a
*b^4)*d*f*x + (a^3*b^2 + a*b^4)*d*e)*cosh(d*x + c))*sinh(d*x + c))/((a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*cosh(d*x +
c)^4 + 4*(a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*co
sh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^6 + 2*a^4*b^2 + a^2*b^4)*d^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^6 + 2*a^4
*b^2 + a^2*b^4)*d^2*sinh(d*x + c)^4 - (a^6 + 2*a^4*b^2 + a^2*b^4)*d^2)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.36, size = 1771, normalized size = 3.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(a^2+b^2)/d^2/a^2*b^3*f*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d^2/a^2*b^3*f*dilog(exp(d*x+c))+4/(a^2+b^2)/d^2*a^3*
f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+1/(a^2+b^2)/d^2/a*b^2*f*ln(exp(d*x+c)-1)-2/(a^2+b^2)/d^2/a*b^2*f*ln(exp(d
*x+c))+1/(a^2+b^2)/d^2/a*b^2*f*ln(exp(d*x+c)+1)+1/(a^2+b^2)/d/a^2*b^3*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d/a^2*b^3
*e*ln(exp(d*x+c)-1)+8/(a^2+b^2)/d^2*b^3*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/(a^2+b^2)^(5/2)/d^2*b^4*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*b^4*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)/d^2*b*f*c*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d^2/a^2*b^3*f*c*ln(exp(d*x
+c)-1)+8/(a^2+b^2)/d^2*a^2*b*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))-1/d^2/(a^2+b^2)^(5/2)*a^2*b^2*f*c*arctanh(1/2*
(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d/(a^2+b^2)^(3/2)*b^2*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/
2))+1/(a^2+b^2)/d*ln(exp(d*x+c)+1)*b*f*x-4/(a^2+b^2)/d^2*a*f*ln(exp(d*x+c))-2*(f*x+e)*(a*b*exp(3*d*x+3*c)+b^2*
exp(2*d*x+2*c)-a*b*exp(d*x+c)+2*a^2+b^2)/d/(a^2+b^2)/(1+exp(2*d*x+2*c))/a/(exp(2*d*x+2*c)-1)+1/d^2/(a^2+b^2)^(
3/2)*b^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d/(a^2+b^2)^(5/2)*a^2*b^2*e*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/a^2*b^4*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^2*b^6*e*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*b^2*f/(4*
a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+1/(a^2+b^2)^(5/2)/d*b^4*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*b^4*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^2*b^4*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*b^4*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^2*b^6*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^2*b^6*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)/d/a^2*b^3*f*ln(exp(d*x+c)+1)*x+1/(a^2+b^2)^(5/2)/d^2/a^2*b^6*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^2*b^6*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/a^2*b^6*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^2*b^6*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^2/a^2*b^6*f*c*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^2*b^4*f*c*arctanh(1
/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)/d^2*b*f*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d^2*b*f*dilog(exp
(d*x+c))+1/(a^2+b^2)/d*b*e*ln(exp(d*x+c)+1)-1/(a^2+b^2)/d*b*e*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d^2*a*f*ln(exp(d*x+
c)+1)+1/(a^2+b^2)/d^2*a*f*ln(exp(d*x+c)-1)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (\frac {b^{4} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (a b e^{\left (-d x - c\right )} + b^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a b e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, a^{2} + b^{2}\right )}}{{\left (a^{3} + a b^{2} - {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d}\right )} e + {\left (16 \, b^{4} \int -\frac {x e^{\left (d x + c\right )}}{8 \, {\left (a^{4} b + a^{2} b^{3} - {\left (a^{4} b e^{\left (2 \, c\right )} + a^{2} b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{5} e^{c} + a^{3} b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 16 \, b d \int \frac {x}{16 \, {\left (a^{2} d e^{\left (d x + c\right )} + a^{2} d\right )}}\,{d x} - 16 \, b d \int \frac {x}{16 \, {\left (a^{2} d e^{\left (d x + c\right )} - a^{2} d\right )}}\,{d x} - a {\left (\frac {d x + c}{a^{2} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2} d^{2}}\right )} - a {\left (\frac {d x + c}{a^{2} d^{2}} - \frac {\log \left (e^{\left (d x + c\right )} - 1\right )}{a^{2} d^{2}}\right )} + \frac {2 \, {\left (a b x e^{\left (3 \, d x + 3 \, c\right )} + b^{2} x e^{\left (2 \, d x + 2 \, c\right )} - a b x e^{\left (d x + c\right )} + {\left (2 \, a^{2} + b^{2}\right )} x\right )}}{a^{3} d + a b^{2} d - {\left (a^{3} d e^{\left (4 \, c\right )} + a b^{2} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )}} - \frac {2 \, a x}{{\left (a^{2} + b^{2}\right )} d} + \frac {2 \, b \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} + \frac {a \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}}\right )} f \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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