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

Optimal. Leaf size=978 \[ -\frac {b f x}{2 a^2 d}-\frac {3 f x \text {ArcTan}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \text {ArcTan}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {ArcTan}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\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 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 f \text {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 \text {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 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \text {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]

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

________________________________________________________________________________________

Rubi [A]
time = 1.15, antiderivative size = 978, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 27, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 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} \begin {gather*} \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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-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) \text {ArcTan}\left (e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac {i f \text {Li}_2\left (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) \text {ArcTan}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d}-\frac {i f \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{2 a \left (a^2+b^2\right ) d^2}+\frac {i f \text {Li}_2\left (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 \tanh ^{-1}\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 \text {Li}_2\left (-e^{2 c+2 d x}\right ) b}{2 a^2 d^2}-\frac {f \text {Li}_2\left (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 \text {ArcTan}\left (e^{c+d x}\right )}{a d}+\frac {3 f x \text {ArcTan}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \text {ArcTan}(\sinh (c+d x))}{2 a d}-\frac {f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac {3 (e+f x) \text {csch}(c+d x)}{2 a d}+\frac {3 i f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}-\frac {f \text {sech}(c+d x)}{2 a d^2} \end {gather*}

Antiderivative was successfully verified.

[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*} \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 {\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) \tan ^{-1}(\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 \tan ^{-1}(\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) \tan ^{-1}(\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 \tan ^{-1}(\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 \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 f \tanh ^{-1}(\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 \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 f \tanh ^{-1}(\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 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {f \tanh ^{-1}(\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 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\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 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\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 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f \text {Li}_2\left (i 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^2 \left (a^2+b^2\right )^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^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 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\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 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 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 )^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^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \text {Li}_2\left (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 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^4 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {3 f x \tan ^{-1}(\sinh (c+d x))}{2 a d}-\frac {3 (e+f x) \tan ^{-1}(\sinh (c+d x))}{2 a d}+\frac {2 b f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \tanh ^{-1}(\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 \text {Li}_2\left (-i e^{c+d x}\right )}{2 a d^2}-\frac {i b^4 f \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {i b^2 f \text {Li}_2\left (-i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}-\frac {3 i f \text {Li}_2\left (i e^{c+d x}\right )}{2 a d^2}+\frac {i b^4 f \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {i b^2 f \text {Li}_2\left (i e^{c+d x}\right )}{2 a \left (a^2+b^2\right ) d^2}+\frac {b^5 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 )^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^2 \left (a^2+b^2\right )^2 d^2}-\frac {b^5 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right )^2 d^2}+\frac {b f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \text {Li}_2\left (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*}

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Mathematica [A]
time = 9.22, size = 1337, normalized size = 1.37 \begin {gather*} 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 {b e \text {csch}(c+d x) \log (\sinh (c+d x)) (a+b \sinh (c+d x))}{8 a^2 d (b+a \text {csch}(c+d x))}+\frac {b c f \text {csch}(c+d x) \log (\sinh (c+d x)) (a+b \sinh (c+d x))}{8 a^2 d^2 (b+a \text {csch}(c+d x))}+\frac {f \text {csch}(c+d x) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \sinh (c+d x))}{8 a d^2 (b+a \text {csch}(c+d x))}+\frac {i b f \text {csch}(c+d x) \left (i (c+d x) \log \left (1-e^{-2 (c+d x)}\right )-\frac {1}{2} i \left (-(c+d x)^2+\text {PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )\right ) (a+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 (-\frac {1}{2} f (c+d x)^2+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) (a+b \sinh (c+d x))}{8 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 \text {ArcTan}\left (e^{c+d x}\right )-10 a b^2 d e \text {ArcTan}\left (e^{c+d x}\right )+6 a^3 c f \text {ArcTan}\left (e^{c+d x}\right )+10 a b^2 c f \text {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 \text {PolyLog}\left (2,-i e^{c+d x}\right )-i a \left (3 a^2+5 b^2\right ) f \text {PolyLog}\left (2,i e^{c+d x}\right )+a^2 b f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+2 b^3 f \text {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 ) \end {gather*}

Antiderivative was successfully verified.

[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])) - (b*e*Csch[c + d*x]*Log[Sinh[c + d*x]]*(a +
b*Sinh[c + d*x]))/(8*a^2*d*(b + a*Csch[c + d*x])) + (b*c*f*Csch[c + d*x]*Log[Sinh[c + d*x]]*(a + b*Sinh[c + d*
x]))/(8*a^2*d^2*(b + a*Csch[c + d*x])) + (f*Csch[c + d*x]*Log[Tanh[(c + d*x)/2]]*(a + b*Sinh[c + d*x]))/(8*a*d
^2*(b + a*Csch[c + d*x])) + ((I/8)*b*f*Csch[c + d*x]*(I*(c + d*x)*Log[1 - E^(-2*(c + d*x))] - (I/2)*(-(c + d*x
)^2 + PolyLog[2, E^(-2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/(a^2*d^2*(b + a*Csch[c + d*x])) + (b^5*Csch[c + d*
x]*(-1/2*(f*(c + d*x)^2) + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b
*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e*Log[a + b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + f*PolyLog
[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])*(a + b*S
inh[c + d*x]))/(8*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)*Log[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*Csc
h[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]))
)

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Maple [B] 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 = 5.92, size = 3280, normalized size = 3.35

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

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))+2/d/a*b^3*e/(a^2+b^2)^(3/2)*arct
anh(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))+10*I/(a^2+b^2)/d*a*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+10*I/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*
b^2)*ln(1+I*exp(d*x+c))*c-10*I/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c-10*I/(a^2+b^2)/d*a*b^2
*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-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))+4/(a^2+b^2)/d*a^2*e/(4*a^2+4*b^2)*b*ln(1+exp(2*d*x+2*c))-6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*dilog
(1-I*exp(d*x+c))+6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*dilog(1+I*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^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)-1/(a^2+b^2)/d^2/a^2*f*b^3*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d^2/a^2*f*b^3*dilog(exp(d*x+c))
-1/(a^2+b^2)/d*ln(exp(d*x+c)+1)*b*f*x+1/(a^2+b^2)/d^2*b*f*c*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d/a^2*f*b^3*ln(exp(d*
x+c)+1)*x+1/(a^2+b^2)/d^2/a^2*b^3*f*c*ln(exp(d*x+c)-1)+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))+4/(a^2+b^2)/d*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x+4/(a^2+b^2)/d*a^2*f/(4*a^2+4
*b^2)*ln(1-I*exp(d*x+c))*b*x+4/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c+4/(a^2+b^2)/d^2*a^2*f/
(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c-4/(a^2+b^2)/d^2*a^2*f*c/(4*a^2+4*b^2)*b*ln(1+exp(2*d*x+2*c))+7/2/(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))+2/(a^2+b^2)^(5/2)/d^2/a*f*b^5*c*arctan
h(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+20/(a^2+b^2)/d^2*a*b^2*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))+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))+6*I/(a^2+b^2)/d*a^3*f/(4*a^2+4*b
^2)*ln(1+I*exp(d*x+c))*x-6*I/(a^2+b^2)/d*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+6*I/(a^2+b^2)/d^2*a^3*f/(4*a
^2+4*b^2)*ln(1+I*exp(d*x+c))*c-6*I/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+10*I/(a^2+b^2)/d^2*a
*b^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-10*I/(a^2+b^2)/d^2*a*b^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-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))-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*b*f*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d^2*b*f*dilog(exp(d*
x+c))-12/(a^2+b^2)/d*a^3*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/(a^2+b^2)^2/d^2/a^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/(a^2+b^2)^2/d^2/a^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/(a^2+b^2)^2/d/a^2*b^5*e*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+8/(a^2+b^2)/d*b^3*e/(4*a^2+4
*b^2)*ln(1+exp(2*d*x+2*c))+8/(a^2+b^2)/d^2*f*b^3/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+8/(a^2+b^2)/d^2*f*b^3/(4*
a^2+4*b^2)*dilog(1-I*exp(d*x+c))-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)+1/(
a^2+b^2)^2/d/a^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-7/2/(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))-20/(a^2+b^2)/d*a*b^2*e/(4*a^2+4*b^2)*arctan(exp(d*x+c))-3/
2/(a^2+b^2)^(5/2)/d*a^3*e*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^2/d^2/a^2*f*b^5*c*ln
(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/(a^2+b^2)^2/d^2/a^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^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))+12/(a^2+b^2)/d^2*
a^3*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))+4/(a^2+b^2)/d^2*a^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b+4/(a^2+b^
2)/d^2*a^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b-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))+8/(a^2+b^2)/d^2*f*b^3/(4*a^2+4*b^2)*ln(
1+I*exp(d*x+c))*c+8/(a^2+b^2)/d^2*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+8/(a^2+b^2)/d*f*b^3/(4*a^2+4*b^2)*l
n(1+I*exp(d*x+c))*x+8/(a^2+b^2)/d*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-8/(a^2+b^2)/d^2*f*b^3*c/(4*a^2+4*b^
2)*ln(1+exp(2*d*x+2*c))+1/(a^2+b^2)^2/d^2/a^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
+1/(a^2+b^2)^2/d/a^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-(3*a^2*d*f*x*exp(5*d*x
+5*c)+2*b^2*d*f*x*exp(5*d*x+5*c)+3*a^2*d*e*exp(5*d*x+5*c)+2*a*b*d*f*x*exp(4*d*x+4*c)+2*b^2*d*e*exp(5*d*x+5*c)+
2*a^2*d*f*x*exp(3*d*x+3*c)+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*a*b*d*f*x*exp(2*d*x+2*c)+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^2+b^2)/(1+exp(2*d*x+2*c))^2/a/(exp(2*d*x+2*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)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(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) - 32*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 + (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)*arctan(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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 17940 vs. \(2 (893) = 1786\).
time = 0.69, size = 17940, normalized size = 18.34 \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)^2*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*cosh(1) + (3*a^5 + 5*a^3*b^2 +
2*a*b^4)*d*sinh(1) + (a^5 + a^3*b^2)*f)*cosh(d*x + c)^5 + 2*((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*
a^3*b^2 + 2*a*b^4)*d*cosh(1) + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*sinh(1) + (a^5 + a^3*b^2)*f)*sinh(d*x + c)^5 +
2*(2*(a^4*b + a^2*b^3)*d*f*x + 2*(a^4*b + a^2*b^3)*d*cosh(1) + 2*(a^4*b + a^2*b^3)*d*sinh(1) + (a^4*b + a^2*b^
3)*f)*cosh(d*x + c)^4 + 2*(2*(a^4*b + a^2*b^3)*d*f*x + 2*(a^4*b + a^2*b^3)*d*cosh(1) + 2*(a^4*b + a^2*b^3)*d*s
inh(1) + (a^4*b + a^2*b^3)*f + 5*((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*cosh(1
) + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*sinh(1) + (a^5 + a^3*b^2)*f)*cosh(d*x + c))*sinh(d*x + c)^4 + 4*((a^5 + 3*
a^3*b^2 + 2*a*b^4)*d*f*x + (a^5 + 3*a^3*b^2 + 2*a*b^4)*d*cosh(1) + (a^5 + 3*a^3*b^2 + 2*a*b^4)*d*sinh(1))*cosh
(d*x + c)^3 + 4*((a^5 + 3*a^3*b^2 + 2*a*b^4)*d*f*x + (a^5 + 3*a^3*b^2 + 2*a*b^4)*d*cosh(1) + 5*((3*a^5 + 5*a^3
*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*cosh(1) + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*sinh(1) + (a
^5 + a^3*b^2)*f)*cosh(d*x + c)^2 + (a^5 + 3*a^3*b^2 + 2*a*b^4)*d*sinh(1) + 2*(2*(a^4*b + a^2*b^3)*d*f*x + 2*(a
^4*b + a^2*b^3)*d*cosh(1) + 2*(a^4*b + a^2*b^3)*d*sinh(1) + (a^4*b + a^2*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^
3 - 4*((a^4*b + a^2*b^3)*d*f*x + (a^4*b + a^2*b^3)*d*cosh(1) + (a^4*b + a^2*b^3)*d*sinh(1))*cosh(d*x + c)^2 -
4*((a^4*b + a^2*b^3)*d*f*x - 5*((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*cosh(1)
+ (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*sinh(1) + (a^5 + a^3*b^2)*f)*cosh(d*x + c)^3 + (a^4*b + a^2*b^3)*d*cosh(1) -
3*(2*(a^4*b + a^2*b^3)*d*f*x + 2*(a^4*b + a^2*b^3)*d*cosh(1) + 2*(a^4*b + a^2*b^3)*d*sinh(1) + (a^4*b + a^2*b
^3)*f)*cosh(d*x + c)^2 + (a^4*b + a^2*b^3)*d*sinh(1) - 3*((a^5 + 3*a^3*b^2 + 2*a*b^4)*d*f*x + (a^5 + 3*a^3*b^2
+ 2*a*b^4)*d*cosh(1) + (a^5 + 3*a^3*b^2 + 2*a*b^4)*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(a^4*b + a^2
*b^3)*f + 2*((3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*f*x + (3*a^5 + 5*a^3*b^2 + 2*a*b^4)*d*cosh(1) + (3*a^5 + 5*a^3*b^
2 + 2*a*b^4)*d*sinh(1) - (a^5 + a^3*b^2)*f)*cosh(d*x + c) - 2*(b^5*f*cosh(d*x + c)^6 + 6*b^5*f*cosh(d*x + c)*s
inh(d*x + c)^5 + b^5*f*sinh(d*x + c)^6 + b^5*f*cosh(d*x + c)^4 - b^5*f*cosh(d*x + c)^2 - b^5*f + (15*b^5*f*cos
h(d*x + c)^2 + b^5*f)*sinh(d*x + c)^4 + 4*(5*b^5*f*cosh(d*x + c)^3 + b^5*f*cosh(d*x + c))*sinh(d*x + c)^3 + (1
5*b^5*f*cosh(d*x + c)^4 + 6*b^5*f*cosh(d*x + c)^2 - b^5*f)*sinh(d*x + c)^2 + 2*(3*b^5*f*cosh(d*x + c)^5 + 2*b^
5*f*cosh(d*x + c)^3 - b^5*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d
*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(b^5*f*cosh(d*x + c)^6 + 6*b^5*f*cosh(d*x + c
)*sinh(d*x + c)^5 + b^5*f*sinh(d*x + c)^6 + b^5*f*cosh(d*x + c)^4 - b^5*f*cosh(d*x + c)^2 - b^5*f + (15*b^5*f*
cosh(d*x + c)^2 + b^5*f)*sinh(d*x + c)^4 + 4*(5*b^5*f*cosh(d*x + c)^3 + b^5*f*cosh(d*x + c))*sinh(d*x + c)^3 +
(15*b^5*f*cosh(d*x + c)^4 + 6*b^5*f*cosh(d*x + c)^2 - b^5*f)*sinh(d*x + c)^2 + 2*(3*b^5*f*cosh(d*x + c)^5 + 2
*b^5*f*cosh(d*x + c)^3 - b^5*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cos
h(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*((a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)
^6 + 6*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)*sinh(d*x + c)^5 + (a^4*b + 2*a^2*b^3 + b^5)*f*sinh(d*x + c)^6
+ (a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^4 + (15*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^2 + (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*cosh(d*x + c)^2 + 4*(5*(a^4*b + 2*a^2*b^3 + b^
5)*f*cosh(d*x + c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^4*b + 2*a^2*b^3 + b
^5)*f*cosh(d*x + c)^4 + 6*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^2 - (a^4*b + 2*a^2*b^3 + b^5)*f)*sinh(d*x
+ c)^2 - (a^4*b + 2*a^2*b^3 + b^5)*f + 2*(3*(a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c)^5 + 2*(a^4*b + 2*a^2*b^3
+ b^5)*f*cosh(d*x + c)^3 - (a^4*b + 2*a^2*b^3 + b^5)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + si
nh(d*x + c)) - ((-I*(3*a^5 + 5*a^3*b^2)*f + 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^6 - 6*(I*(3*a^5 + 5*a^3*b^2
)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^5 + (-I*(3*a^5 + 5*a^3*b^2)*f + 2*(a^4*b + 2*a^2*b^
3)*f)*sinh(d*x + c)^6 + (-I*(3*a^5 + 5*a^3*b^2)*f + 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^4 - (15*(I*(3*a^5 +
5*a^3*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^2 + I*(3*a^5 + 5*a^3*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*
sinh(d*x + c)^4 - 4*(5*(I*(3*a^5 + 5*a^3*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^3 + (I*(3*a^5 + 5*a^3
*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^3 + (I*(3*a^5 + 5*a^3*b^2)*f - 2*(a^4*b + 2*a^
2*b^3)*f)*cosh(d*x + c)^2 - (15*(I*(3*a^5 + 5*a^3*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^4 + 6*(I*(3*
a^5 + 5*a^3*b^2)*f - 2*(a^4*b + 2*a^2*b^3)*f)*cosh(d*x + c)^2 - I*(3*a^5 + 5*a^3*b^2)*f + 2*(a^4*b + 2*a^2*b^3
)*f)*sinh(d*x + c)^2 + I*(3*a^5 + 5*a^3*b^2)*f ...

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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)**2*sech(d*x+c)**3/(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)^2*sech(d*x+c)^3/(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 )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\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)^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)

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