\(\int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [473]

Optimal result

Integrand size = 34, antiderivative size = 978 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b f x}{2 a^2 d}-\frac {3 f x \arctan \left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \arctan (\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 a d}+\frac {2 b f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {b^5 (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 )^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^2 \left (a^2+b^2\right )^2 d}-\frac {b^5 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right )^2 d}+\frac {b f x \log (\tanh (c+d x))}{a^2 d}-\frac {b (e+f x) \log (\tanh (c+d x))}{a^2 d}+\frac {3 i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b^5 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {b^5 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2}+\frac {b^2 f \text {sech}(c+d x)}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^3 (e+f x) \text {sech}^2(c+d x)}{2 a^2 \left (a^2+b^2\right ) d}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}+\frac {b f \tanh (c+d x)}{2 a^2 d^2}-\frac {b^3 f \tanh (c+d x)}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac {b (e+f x) \tanh ^2(c+d x)}{2 a^2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 978, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.794, Rules used = {5708, 2701, 294, 327, 213, 5570, 5311, 12, 4265, 2317, 2438, 3855, 2702, 2700, 14, 2628, 4267, 3554, 8, 5692, 5680, 2221, 6874, 3799, 4270, 5559, 3852} \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {(e+f x) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}-\frac {(e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}+\frac {f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^5}{a^2 \left (a^2+b^2\right )^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^5}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {(e+f x) \text {sech}^2(c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x) b^3}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \arctan \left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {f \text {sech}(c+d x) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {(e+f x) \text {sech}(c+d x) \tanh (c+d x) b^2}{2 a \left (a^2+b^2\right ) d}+\frac {(e+f x) \tanh ^2(c+d x) b}{2 a^2 d}-\frac {f x b}{2 a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right ) b}{a^2 d}+\frac {f x \log (\tanh (c+d x)) b}{a^2 d}-\frac {(e+f x) \log (\tanh (c+d x)) b}{a^2 d}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right ) b}{2 a^2 d^2}+\frac {f \tanh (c+d x) b}{2 a^2 d^2}+\frac {(e+f x) \text {csch}(c+d x) \text {sech}^2(c+d x)}{2 a d}-\frac {3 f x \arctan \left (e^{c+d x}\right )}{a d}+\frac {3 f x \arctan (\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \arctan (\sinh (c+d x))}{2 a d}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {3 i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2} \]

[In]

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

[Out]

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

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 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 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 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 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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 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 4270

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

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

Mathematica [A] (warning: unable to verify)

Time = 11.64 (sec) , antiderivative size = 1437, normalized size of antiderivative = 1.47 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=8 \left (\frac {\left (-d e \cosh \left (\frac {1}{2} (c+d x)\right )+c f \cosh \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right ) \text {csch}(c+d x) (a+b \sinh (c+d x))}{16 a d^2 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left (-\frac {b (d e-c f+f (c+d x))^2}{2 f}+(-b d e+a f+b c f-b f (c+d x)) \log \left (1-e^{-c-d x}\right )+(-b d e-a f+b c f-b f (c+d x)) \log \left (1+e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )\right ) (a+b \sinh (c+d x))}{8 a^2 d^2 (b+a \text {csch}(c+d x))}+\frac {b^5 \text {csch}(c+d x) \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) (a+b \sinh (c+d x))}{16 a^2 \left (a^2+b^2\right )^2 d^2 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left (-2 a^2 b d e (c+d x)-4 b^3 d e (c+d x)+2 a^2 b c f (c+d x)+4 b^3 c f (c+d x)-a^2 b f (c+d x)^2-2 b^3 f (c+d x)^2-6 a^3 d e \arctan \left (e^{c+d x}\right )-10 a b^2 d e \arctan \left (e^{c+d x}\right )+6 a^3 c f \arctan \left (e^{c+d x}\right )+10 a b^2 c f \arctan \left (e^{c+d x}\right )-3 i a^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-5 i a b^2 f (c+d x) \log \left (1-i e^{c+d x}\right )+3 i a^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+5 i a b^2 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^2 b d e \log \left (1+e^{2 (c+d x)}\right )+4 b^3 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^2 b c f \log \left (1+e^{2 (c+d x)}\right )-4 b^3 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^2 b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+4 b^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+i a \left (3 a^2+5 b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-i a \left (3 a^2+5 b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a^2 b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )+2 b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right ) (a+b \sinh (c+d x))}{16 \left (a^2+b^2\right )^2 d^2 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (d e \sinh \left (\frac {1}{2} (c+d x)\right )-c f \sinh \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sinh (c+d x))}{16 a d^2 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c+d x) (a+b \sinh (c+d x)) (-a f+b f \sinh (c+d x))}{16 \left (a^2+b^2\right ) d^2 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}^2(c+d x) (a+b \sinh (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))}{16 \left (a^2+b^2\right ) d^2 (b+a \text {csch}(c+d x))}\right ) \]

[In]

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

[Out]

8*(((-(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]*Csch[
c + d*x]*(a + b*Sinh[c + d*x]))/(16*a*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*(-1/2*(b*(d*e - c*f + f*(c +
d*x))^2)/f + (-(b*d*e) + a*f + b*c*f - b*f*(c + d*x))*Log[1 - E^(-c - d*x)] + (-(b*d*e) - a*f + b*c*f - b*f*(
c + d*x))*Log[1 + E^(-c - d*x)] + b*f*PolyLog[2, -E^(-c - d*x)] + b*f*PolyLog[2, E^(-c - d*x)])*(a + b*Sinh[c
+ d*x]))/(8*a^2*d^2*(b + a*Csch[c + d*x])) + (b^5*Csch[c + d*x]*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d
*x)^2 + (4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqr
t[-(a^2 + b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 +
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f
*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*Po
lyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])*(
a + b*Sinh[c + d*x]))/(16*a^2*(a^2 + b^2)^2*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*(-2*a^2*b*d*e*(c + d*x
) - 4*b^3*d*e*(c + d*x) + 2*a^2*b*c*f*(c + d*x) + 4*b^3*c*f*(c + d*x) - a^2*b*f*(c + d*x)^2 - 2*b^3*f*(c + d*x
)^2 - 6*a^3*d*e*ArcTan[E^(c + d*x)] - 10*a*b^2*d*e*ArcTan[E^(c + d*x)] + 6*a^3*c*f*ArcTan[E^(c + d*x)] + 10*a*
b^2*c*f*ArcTan[E^(c + d*x)] - (3*I)*a^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - (5*I)*a*b^2*f*(c + d*x)*Log[1 - I
*E^(c + d*x)] + (3*I)*a^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + (5*I)*a*b^2*f*(c + d*x)*Log[1 + I*E^(c + d*x)]
+ 2*a^2*b*d*e*Log[1 + E^(2*(c + d*x))] + 4*b^3*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^2*b*c*f*Log[1 + E^(2*(c + d*
x))] - 4*b^3*c*f*Log[1 + E^(2*(c + d*x))] + 2*a^2*b*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + 4*b^3*f*(c + d*x)*L
og[1 + E^(2*(c + d*x))] + I*a*(3*a^2 + 5*b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] - I*a*(3*a^2 + 5*b^2)*f*PolyLog[2
, I*E^(c + d*x)] + a^2*b*f*PolyLog[2, -E^(2*(c + d*x))] + 2*b^3*f*PolyLog[2, -E^(2*(c + d*x))])*(a + b*Sinh[c
+ d*x]))/(16*(a^2 + b^2)^2*d^2*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*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])*(a + b*Sinh[c + d*x]))/(16*a*d^2*(b + a*Csch[c + d*
x])) + (Csch[c + d*x]*Sech[c + d*x]*(a + b*Sinh[c + d*x])*(-(a*f) + b*f*Sinh[c + d*x]))/(16*(a^2 + b^2)*d^2*(b
+ a*Csch[c + d*x])) + (Csch[c + d*x]*Sech[c + d*x]^2*(a + b*Sinh[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]))/(16*(a^2 + b^2)*d^2*(b + a*Csch[c
+ d*x])))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3279 vs. \(2 (904 ) = 1808\).

Time = 60.27 (sec) , antiderivative size = 3280, normalized size of antiderivative = 3.35

method	result	size
risch	\(\text {Expression too large to display}\)	\(3280\)

[In]

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

[Out]

4/d*a^2/(a^2+b^2)*b*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+4/d*a^2/(a^2+b^2)*b*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c)
)*x+20/d^2*a/(a^2+b^2)*c*b^2*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+2/d^2/a/(a^2+b^2)^(5/2)*c*b^5*f*arctanh(1/2*(2
*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+7/2/d^2*a/(a^2+b^2)^(5/2)*c*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^
2)^(1/2))-4/d^2*a^2/(a^2+b^2)*c*b*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+3/2/d^2*a^3/(a^2+b^2)^(5/2)*c*b*f*arcta
nh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+4/d^2*a^2/(a^2+b^2)*b*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c+4/d^2*
a^2/(a^2+b^2)*b*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*
x+c))-10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+6*I/d*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1+
I*exp(d*x+c))*x-6*I/d*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-6*I/d^2*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)
*ln(1-I*exp(d*x+c))*c+6*I/d^2*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-1/d^2/(a^2+b^2)^(5/2)*b*f*arc
tanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/d^2/(a^2+b^2)^(5/2)*b^3*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a
)/(a^2+b^2)^(1/2))*a-2/d/a/(a^2+b^2)^(5/2)*e*b^5*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+8/d/(a^2+b^
2)*f*b^3/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-8/d^2/(a^2+b^2)*c*b^3*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+8/d^2/(
a^2+b^2)*f*b^3/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c+8/d^2/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+8/d
/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+1/d^2/a^2/(a^2+b^2)^2*f*b^5*ln((-b*exp(d*x+c)+(a^2+b^2)^(1
/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+2/a/d*e*b^3/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+3
/2*a/d*e*b/(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/d^2*f*b^3/(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/a^2/(a^2+b^2)*b^3*e*ln(exp(d*x+c)-1)-1/d/a^2/(a^2+b^2)*b^3*
e*ln(exp(d*x+c)+1)-12/d*a^3/(a^2+b^2)*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/d^2/a/(a^2+b^2)*b^2*f*ln(exp(d*x+c)
-1)-1/d^2/a/(a^2+b^2)*b^2*f*ln(exp(d*x+c)+1)+1/d^2/a^2/(a^2+b^2)*f*b^3*dilog(exp(d*x+c))-1/d^2/a^2/(a^2+b^2)*f
*b^3*dilog(exp(d*x+c)+1)+1/d/a^2/(a^2+b^2)^2*b^5*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-2/a/d^2*b^3*c*f/(a^2+
b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-3/2*a/d^2*b*c*f/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*
exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+a/d^2*f*b/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1
0*I/d*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+10*I/d*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d
*x+c))*x-10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/d^2*a/(a^2+b^2)*b^2*f/(4*a^2+4*b^2
)*ln(1+I*exp(d*x+c))*c-(3*exp(5*d*x+5*c)*a^2*d*f*x+2*b^2*d*f*x*exp(5*d*x+5*c)+3*a^2*d*e*exp(5*d*x+5*c)+2*exp(4
*d*x+4*c)*a*b*d*f*x+2*b^2*d*e*exp(5*d*x+5*c)+2*exp(3*d*x+3*c)*a^2*d*f*x+a^2*f*exp(5*d*x+5*c)+2*a*b*d*e*exp(4*d
*x+4*c)+4*b^2*d*f*x*exp(3*d*x+3*c)+2*a^2*d*e*exp(3*d*x+3*c)-2*exp(2*d*x+2*c)*a*b*d*f*x+a*b*f*exp(4*d*x+4*c)+4*
b^2*d*e*exp(3*d*x+3*c)+3*a^2*d*f*x*exp(d*x+c)-2*a*b*d*e*exp(2*d*x+2*c)+2*b^2*d*f*x*exp(d*x+c)+3*a^2*d*e*exp(d*
x+c)+2*b^2*d*e*exp(d*x+c)-a^2*f*exp(d*x+c)-a*b*f)/d^2/a/(exp(2*d*x+2*c)-1)/(a^2+b^2)/(1+exp(2*d*x+2*c))^2+1/d^
2/a^2/(a^2+b^2)^2*f*b^5*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+4/d*a^2/(a^2+b^2)*b*e/(4*a^
2+4*b^2)*ln(1+exp(2*d*x+2*c))-20/d*a/(a^2+b^2)*b^2*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))-7/2/d*a/(a^2+b^2)^(5/2)*
b^3*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-3/2/d*a^3/(a^2+b^2)^(5/2)*b*e*arctanh(1/2*(2*b*exp(d*x
+c)+2*a)/(a^2+b^2)^(1/2))+12/d^2*a^3/(a^2+b^2)*c*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/d^2/a^2/(a^2+b^2)*c*f*b^
3*ln(exp(d*x+c)-1)-1/d^2/a/(a^2+b^2)^(5/2)*f*b^5*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d/a^2/(a^
2+b^2)*f*b^3*ln(exp(d*x+c)+1)*x+6*I/d^2*a^3/(a^2+b^2)*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-6*I/d^2*a^3/(a^2+b
^2)*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+4/d^2*a^2/(a^2+b^2)*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b+4/d^2*a^
2/(a^2+b^2)*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b+8/d^2/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+
8/d^2/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+1/d^2/(a^2+b^2)*c*b*f*ln(exp(d*x+c)-1)+8/d/(a^2+b^2)
*b^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-1/d/(a^2+b^2)*b*f*ln(exp(d*x+c)+1)*x+1/d^2/a^2/(a^2+b^2)^2*f*b^5*dil
og((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/d^2/a^2/(a^2+b^2)^2*f*b^5*dilog((-b*exp(d*x+c)+(a^2
+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2/(a^2+b^2)*b*f*dilog(exp(d*x+c))-1/d^2/(a^2+b^2)*b*f*dilog(exp(d*x+c
)+1)-1/d/(a^2+b^2)*b*e*ln(exp(d*x+c)+1)-1/d/(a^2+b^2)*b*e*ln(exp(d*x+c)-1)-1/d^2*a/(a^2+b^2)*f*ln(exp(d*x+c)+1
)+1/d^2*a/(a^2+b^2)*f*ln(exp(d*x+c)-1)+1/d/a^2/(a^2+b^2)^2*f*b^5*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2
+b^2)^(1/2)))*x+1/d/a^2/(a^2+b^2)^2*f*b^5*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^2
/(a^2+b^2)^2*c*f*b^5*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 15223 vs. \(2 (881) = 1762\).

Time = 0.62 (sec) , antiderivative size = 15223, normalized size of antiderivative = 15.57 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

integrate((f*x+e)*csch(d*x+c)**2*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(b^5*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^6 + 2*a^4*b^2 + a^2*b^4)*d) + (3*a^3 + 5*a*b^2)*arcta
n(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a^2*b + 2*b^3)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^
4)*d) - (2*a*b*e^(-2*d*x - 2*c) - 2*a*b*e^(-4*d*x - 4*c) + (3*a^2 + 2*b^2)*e^(-d*x - c) + 2*(a^2 + 2*b^2)*e^(-
3*d*x - 3*c) + (3*a^2 + 2*b^2)*e^(-5*d*x - 5*c))/((a^3 + a*b^2 + (a^3 + a*b^2)*e^(-2*d*x - 2*c) - (a^3 + a*b^2
)*e^(-4*d*x - 4*c) - (a^3 + a*b^2)*e^(-6*d*x - 6*c))*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c)
- 1)/(a^2*d))*e + (32*b*d*integrate(1/32*x/(a^2*d*e^(d*x + c) + a^2*d), x) - 32*b*d*integrate(1/32*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*d*x*e^(2*d*x + 2*c) - 2*(a^2*d*e^(3*c) + 2*b^2*d*e^(3*c))*x*e^(3*d*x)
+ a*b - (a^2*e^(5*c) + (3*a^2*d*e^(5*c) + 2*b^2*d*e^(5*c))*x)*e^(5*d*x) - (2*a*b*d*x*e^(4*c) + a*b*e^(4*c))*e
^(4*d*x) + (a^2*e^c - (3*a^2*d*e^c + 2*b^2*d*e^c)*x)*e^(d*x))/(a^3*d^2 + a*b^2*d^2 - (a^3*d^2*e^(6*c) + a*b^2*
d^2*e^(6*c))*e^(6*d*x) - (a^3*d^2*e^(4*c) + a*b^2*d^2*e^(4*c))*e^(4*d*x) + (a^3*d^2*e^(2*c) + a*b^2*d^2*e^(2*c
))*e^(2*d*x)) - 32*integrate(-1/16*(a*b^5*x*e^(d*x + c) - b^6*x)/(a^6*b + 2*a^4*b^3 + a^2*b^5 - (a^6*b*e^(2*c)
+ 2*a^4*b^3*e^(2*c) + a^2*b^5*e^(2*c))*e^(2*d*x) - 2*(a^7*e^c + 2*a^5*b^2*e^c + a^3*b^4*e^c)*e^(d*x)), x) - 3
2*integrate(1/32*((3*a^3*e^c + 5*a*b^2*e^c)*x*e^(d*x) + 2*(a^2*b + 2*b^3)*x)/(a^4 + 2*a^2*b^2 + b^4 + (a^4*e^(
2*c) + 2*a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x)), x))*f

Giac [F(-1)]

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

[In]

integrate((f*x+e)*csch(d*x+c)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {e+f\,x}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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