3.5.46 \(\int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [446]

Optimal. Leaf size=746 \[ \frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {i b^3 f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f \text {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {i b^3 f \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f \text {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^2}-\frac {f \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d} \]

[Out]

1/2*f*x/a/d-2*b^3*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d-b*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)/d-2*f*x*arct
anh(exp(2*d*x+2*c))/a/d+b^4*(f*x+e)*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)^2/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/a/(a^2+b^2)^2/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d-f*x*ln(tanh(d
*x+c))/a/d+(f*x+e)*ln(tanh(d*x+c))/a/d-1/2*I*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+1/2*I*b*f*polylog(2,-I*
exp(d*x+c))/(a^2+b^2)/d^2-I*b^3*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2+I*b^3*f*polylog(2,-I*exp(d*x+c))/(a^
2+b^2)^2/d^2+1/2*b^4*f*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)^2/d^2-1/2*f*polylog(2,-exp(2*d*x+2*c))/a/d^2+1/2
*f*polylog(2,exp(2*d*x+2*c))/a/d^2-b^4*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-b^4*f*
polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-1/2*b*f*sech(d*x+c)/(a^2+b^2)/d^2-1/2*b^2*(f*x+
e)*sech(d*x+c)^2/a/(a^2+b^2)/d-1/2*f*tanh(d*x+c)/a/d^2+1/2*b^2*f*tanh(d*x+c)/a/(a^2+b^2)/d^2-1/2*b*(f*x+e)*sec
h(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d-1/2*(f*x+e)*tanh(d*x+c)^2/a/d

________________________________________________________________________________________

Rubi [A]
time = 0.88, antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 20, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5708, 2700, 14, 5570, 2628, 12, 4267, 2317, 2438, 3554, 8, 5692, 5680, 2221, 6874, 4265, 3799, 4270, 5559, 3852} \begin {gather*} -\frac {b (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {2 b^3 (e+f x) \text {ArcTan}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {b^2 f \tanh (c+d x)}{2 a d^2 \left (a^2+b^2\right )}-\frac {b f \text {sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a d \left (a^2+b^2\right )}-\frac {b (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d \left (a^2+b^2\right )}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}+\frac {b^4 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a d^2 \left (a^2+b^2\right )^2}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac {b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \tanh (c+d x)}{2 a d^2}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {f x}{2 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*x)/(2*a*d) - (2*b^3*(e + f*x)*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (b*(e + f*x)*ArcTan[E^(c + d*x)])/((
a^2 + b^2)*d) - (2*f*x*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2])])/(a*(a^2 + b^2)^2*d) - (b^4*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^
2*d) + (b^4*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d) - (f*x*Log[Tanh[c + d*x]])/(a*d) + ((e + f
*x)*Log[Tanh[c + d*x]])/(a*d) + (I*b^3*f*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) + ((I/2)*b*f*PolyLo
g[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - (I*b^3*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - ((I/2)*b
*f*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])
/(a*(a^2 + b^2)^2*d^2) - (b^4*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) +
(b^4*f*PolyLog[2, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)^2*d^2) - (f*PolyLog[2, -E^(2*c + 2*d*x)])/(2*a*d^2) + (f
*PolyLog[2, E^(2*c + 2*d*x)])/(2*a*d^2) - (b*f*Sech[c + d*x])/(2*(a^2 + b^2)*d^2) - (b^2*(e + f*x)*Sech[c + d*
x]^2)/(2*a*(a^2 + b^2)*d) - (f*Tanh[c + d*x])/(2*a*d^2) + (b^2*f*Tanh[c + d*x])/(2*a*(a^2 + b^2)*d^2) - (b*(e
+ f*x)*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d) - ((e + f*x)*Tanh[c + d*x]^2)/(2*a*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 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 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 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 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}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {b \int (e+f x) \text {sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {f \int \left (\frac {\log (\tanh (c+d x))}{d}-\frac {\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \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 \left (a^2+b^2\right )}+\frac {f \int \tanh ^2(c+d x) \, dx}{2 a d}-\frac {f \int \log (\tanh (c+d x)) \, dx}{a d}\\ &=\frac {b^4 (e+f x)^2}{2 a \left (a^2+b^2\right )^2 f}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {f \tanh (c+d x)}{2 a d^2}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x) \text {sech}^3(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x) \text {sech}^2(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac {f \int 1 \, dx}{2 a d}+\frac {f \int 2 d x \text {csch}(2 c+2 d x) \, dx}{a d}\\ &=\frac {f x}{2 a d}+\frac {b^4 (e+f x)^2}{2 a \left (a^2+b^2\right )^2 f}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {b^4 \int (e+f x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac {b \int (e+f x) \text {sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}+\frac {(2 f) \int x \text {csch}(2 c+2 d x) \, dx}{a}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac {\left (b^2 f\right ) \int \text {sech}^2(c+d x) \, dx}{2 a \left (a^2+b^2\right ) d}\\ &=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )^2}+\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {\left (i b^2 f\right ) \text {Subst}(\int 1 \, dx,x,-i \tanh (c+d x))}{2 a \left (a^2+b^2\right ) d^2}-\frac {f \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {f \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {\left (i b^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac {\left (i b^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac {(i b f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{2 \left (a^2+b^2\right ) d}-\frac {(i b f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{2 \left (a^2+b^2\right ) d}\\ &=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {\left (b^4 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}\\ &=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}-\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^2}\\ &=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {i b^3 f \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f \text {Li}_2\left (-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {i b^3 f \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f \text {Li}_2\left (i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^2}-\frac {f \text {Li}_2\left (-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \text {Li}_2\left (e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}\\ \end {align*}

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Mathematica [A]
time = 9.06, size = 886, normalized size = 1.19 \begin {gather*} \frac {e \log (\sinh (c+d x))}{a d}-\frac {c f \log (\sinh (c+d x))}{a d^2}-\frac {i f \left (i (c+d x) \log \left (1-e^{-2 (c+d x)}\right )-\frac {1}{2} i \left (-(c+d x)^2+\text {PolyLog}\left (2,e^{-2 (c+d x)}\right )\right )\right )}{a d^2}-\frac {b^4 \left (-\frac {1}{2} f (c+d x)^2+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))+f \text {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+f \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {-2 a^3 d e (c+d x)-4 a b^2 d e (c+d x)+2 a^3 c f (c+d x)+4 a b^2 c f (c+d x)-a^3 f (c+d x)^2-2 a b^2 f (c+d x)^2+2 a^2 b d e \text {ArcTan}\left (e^{c+d x}\right )+6 b^3 d e \text {ArcTan}\left (e^{c+d x}\right )-2 a^2 b c f \text {ArcTan}\left (e^{c+d x}\right )-6 b^3 c f \text {ArcTan}\left (e^{c+d x}\right )+i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )+3 i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-3 i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )+4 a b^2 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )-4 a b^2 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+4 a b^2 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b \left (a^2+3 b^2\right ) f \text {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (a^2+3 b^2\right ) f \text {PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+2 a b^2 f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (-b f-a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (a d e-a c f+a f (c+d x)-b d e \sinh (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Log[Sinh[c + d*x]])/(a*d) - (c*f*Log[Sinh[c + d*x]])/(a*d^2) - (I*f*(I*(c + d*x)*Log[1 - E^(-2*(c + d*x))]
- (I/2)*(-(c + d*x)^2 + PolyLog[2, E^(-2*(c + d*x))])))/(a*d^2) - (b^4*(-1/2*(f*(c + d*x)^2) + f*(c + d*x)*Log
[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d*e
*Log[a + b*Sinh[c + d*x]] - c*f*Log[a + b*Sinh[c + d*x]] + f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])
] + f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a*(a^2 + b^2)^2*d^2) - (-2*a^3*d*e*(c + d*x) - 4
*a*b^2*d*e*(c + d*x) + 2*a^3*c*f*(c + d*x) + 4*a*b^2*c*f*(c + d*x) - a^3*f*(c + d*x)^2 - 2*a*b^2*f*(c + d*x)^2
+ 2*a^2*b*d*e*ArcTan[E^(c + d*x)] + 6*b^3*d*e*ArcTan[E^(c + d*x)] - 2*a^2*b*c*f*ArcTan[E^(c + d*x)] - 6*b^3*c
*f*ArcTan[E^(c + d*x)] + I*a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*f*(c + d*x)*Log[1 - I*E^(c + d
*x)] - I*a^2*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)] - (3*I)*b^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^3*d*e*L
og[1 + E^(2*(c + d*x))] + 4*a*b^2*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^3*c*f*Log[1 + E^(2*(c + d*x))] - 4*a*b^2*
c*f*Log[1 + E^(2*(c + d*x))] + 2*a^3*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + 4*a*b^2*f*(c + d*x)*Log[1 + E^(2*(
c + d*x))] - I*b*(a^2 + 3*b^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*(a^2 + 3*b^2)*f*PolyLog[2, I*E^(c + d*x)]
+ a^3*f*PolyLog[2, -E^(2*(c + d*x))] + 2*a*b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c
+ d*x]*(-(b*f) - a*f*Sinh[c + d*x]))/(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(a*d*e - a*c*f + a*f*(c + d*x) -
b*d*e*Sinh[c + d*x] + b*c*f*Sinh[c + d*x] - b*f*(c + d*x)*Sinh[c + d*x]))/(2*(a^2 + b^2)*d^2)

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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. 2579 vs. \(2 (693 ) = 1386\).
time = 8.08, size = 2580, normalized size = 3.46

method	result	size
risch	\(\text {Expression too large to display}\)	\(2580\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d/(a^2+b^2)*b^2*f/a*ln(exp(d*x+c)+1)*x+4/d^2/(a^2+b^2)*a^2*f*c/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-8/d/(a^2+b
^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*x-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*c-8/d/
(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*x-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*
c-1/2/d^2/(a^2+b^2)^(5/2)*a^2*b^2*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+8/d^2/(a^2+b^2)*b^2*f*
c/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))+2*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b-2*I/d^2/(
a^2+b^2)*a^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b+6*I/d/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+
6*I/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-6*I/d/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+
c))*x-6*I/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c-1/d^2/(a^2+b^2)*b^2*f*c/a*ln(exp(d*x+c)-1)+12
/d^2/(a^2+b^2)*b^3*f*c/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/2/d^2/(a^2+b^2)^(3/2)*b^2*f*c*arctanh(1/2*(2*b*exp(d
*x+c)+2*a)/(a^2+b^2)^(1/2))+1/d^2/(a^2+b^2)^2*b^4*f*c/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/d/(a^2+b^2)*a^
2*e/(4*a^2+4*b^2)*b*arctan(exp(d*x+c))-1/d/(a^2+b^2)^2*b^4*f/a*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+b^2)^2*b^4*f/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d/(a^2+
b^2)^2*b^4*f/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/2/d/(a^2+b^2)^(5/2)*e*a^2*b^2*arct
anh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-8/d/(a^2+b^2)*b^2*e/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))-6*I/d^2
/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+6*I/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c
))-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-1/2/d/(a^2+b^2)^(3/2)*b^2*e*arctanh(1/2*(2*b*exp(
d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/d/(a^2+b^2)*a^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-1/d^2/(a^2+b^2)^2*b^4*f/a*
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)^2*b^4*f/a*dilog((b*exp(d*x+c)+(a
^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d/(a^2+b^2)^2*b^4*e/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/2/d/(a^2
+b^2)^(5/2)*b^4*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-12/d/(a^2+b^2)*b^3*e/(4*a^2+4*b^2)*arctan(
exp(d*x+c))-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+1/d^2/(a^2+b^2)*b^2*f/a*dilog(exp(d*x+c)
+1)-1/d^2/(a^2+b^2)*b^2*f*dilog(exp(d*x+c))/a+2*I/d/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x+2*I/d
^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c-2*I/d/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))
*b*x-2*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c-1/d^2/(a^2+b^2)^2*b^4*f/a*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+b^2)*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-4/d^2/(a^2+b^2
)*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-4/d/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x-4/d^2/(a^2+b
^2)*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*a-8/d^2
/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*a+4/d^2/(a^2+b^2)*a^3*f*c/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*
c))-1/2/d^2/(a^2+b^2)^(5/2)*b^4*f*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2/(a^2+b^2)*a*f*c*ln
(exp(d*x+c)-1)+1/d/(a^2+b^2)*b^2*e/a*ln(exp(d*x+c)+1)+1/d/(a^2+b^2)*b^2*e/a*ln(exp(d*x+c)-1)+1/d/(a^2+b^2)*ln(
exp(d*x+c)+1)*a*f*x+(-b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2
*c)+b*d*f*x*exp(d*x+c)-b*f*exp(3*d*x+3*c)+a*f*exp(2*d*x+2*c)+b*d*e*exp(d*x+c)-f*b*exp(d*x+c)+f*a)/d^2/(a^2+b^2
)/(1+exp(2*d*x+2*c))^2+1/d/(a^2+b^2)*a*e*ln(exp(d*x+c)+1)+1/d/(a^2+b^2)*a*e*ln(exp(d*x+c)-1)+1/d^2/(a^2+b^2)*a
*f*dilog(exp(d*x+c)+1)-1/d^2/(a^2+b^2)*a*f*dilog(exp(d*x+c))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8973 vs. \(2 (685) = 1370\).
time = 0.62, size = 8973, normalized size = 12.03 \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)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1) + (a^3*b + a*b^3)*f)*co
sh(d*x + c)^3 + 2*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1) + (a^3*b + a*
b^3)*f)*sinh(d*x + c)^3 - 2*(2*(a^4 + a^2*b^2)*d*f*x + 2*(a^4 + a^2*b^2)*d*cosh(1) + 2*(a^4 + a^2*b^2)*d*sinh(
1) + (a^4 + a^2*b^2)*f)*cosh(d*x + c)^2 - 2*(2*(a^4 + a^2*b^2)*d*f*x + 2*(a^4 + a^2*b^2)*d*cosh(1) + 2*(a^4 +
a^2*b^2)*d*sinh(1) + (a^4 + a^2*b^2)*f - 3*((a^3*b + a*b^3)*d*f*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3
)*d*sinh(1) + (a^3*b + a*b^3)*f)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(a^4 + a^2*b^2)*f - 2*((a^3*b + a*b^3)*d*f
*x + (a^3*b + a*b^3)*d*cosh(1) + (a^3*b + a*b^3)*d*sinh(1) - (a^3*b + a*b^3)*f)*cosh(d*x + c) + 2*(b^4*f*cosh(
d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c)^4 + 2*b^4*f*cosh(d*x + c)^2 + b^4*f +
2*(3*b^4*f*cosh(d*x + c)^2 + b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*x + c)^3 + b^4*f*cosh(d*x + c))*sinh(d*
x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) -
b)/b + 1) + 2*(b^4*f*cosh(d*x + c)^4 + 4*b^4*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f*sinh(d*x + c)^4 + 2*b^4*
f*cosh(d*x + c)^2 + b^4*f + 2*(3*b^4*f*cosh(d*x + c)^2 + b^4*f)*sinh(d*x + c)^2 + 4*(b^4*f*cosh(d*x + c)^3 + b
^4*f*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x +
c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^
4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*f*c
osh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f)*sinh(d*x + c)^2 +
(a^4 + 2*a^2*b^2 + b^4)*f + 4*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x
+ c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + ((2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*
cosh(d*x + c)^4 + 4*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*(a^4 +
2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^4 + 2*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh
(d*x + c)^2 + 2*(3*(2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f + I
*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f + 4*((2*(a^4 + 2*a^2*b^2
)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*f + I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c))*
sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + ((2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cos
h(d*x + c)^4 + 4*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*(a^4 + 2*a
^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*sinh(d*x + c)^4 + 2*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*
x + c)^2 + 2*(3*(2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f - I*(a
^3*b + 3*a*b^3)*f)*sinh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f + 4*((2*(a^4 + 2*a^2*b^2)*f
- I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c)^3 + (2*(a^4 + 2*a^2*b^2)*f - I*(a^3*b + 3*a*b^3)*f)*cosh(d*x + c))*sin
h(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^4 + 4*(a^4
+ 2*a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^4 + 2*(a^4 + 2*a^
2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f)*s
inh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f + 4*((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 +
b^4)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 2*(b^4*c*f - b^4*d*cosh(1) - b^4
*d*sinh(1) + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^4 + 4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*si
nh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*sinh(d*x + c)^4 + 2*(b^4*c*f
- b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2 + 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) + 3*(b^4*c*f - b
^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*
cosh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c
) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) + (b^4*c
*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^4 + 4*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c
)*sinh(d*x + c)^3 + (b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*sinh(d*x + c)^4 + 2*(b^4*c*f - b^4*d*cosh(1) - b
^4*d*sinh(1))*cosh(d*x + c)^2 + 2*(b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1) + 3*(b^4*c*f - b^4*d*cosh(1) - b^4*
d*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c)^3 + (
b^4*c*f - b^4*d*cosh(1) - b^4*d*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(b^4*...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*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 )\,\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)*(a + b*sinh(c + d*x))),x)

[Out]

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

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