3.492 \(\int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\)
Optimal. Leaf size=1219 \[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}+\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {(e+f x)^2 \text {csch}(c+d x) b}{a^2 d}+\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b}{a^2 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b}{a^2 d^2}-\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right ) b}{a^2 d^3}+\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b}{a^2 d^3}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {e f x}{a d}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3} \]
[Out]
2*b*(f*x+e)^2*arctan(exp(d*x+c))/a^2/d-2*b^2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^3/d-f*(f*x+e)*coth(d*x+c)/a/d
^2+b*(f*x+e)^2*csch(d*x+c)/a^2/d+1/2*b^2*f^2*polylog(3,-exp(2*d*x+2*c))/a^3/d^3-2*b^3*(f*x+e)^2*arctan(exp(d*x
+c))/a^2/(a^2+b^2)/d-b^2*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^3/d^2+b^2*f*(f*x+e)*polylog(2,exp(2*d*x+2*c))/
a^3/d^2-1/2*b^4*f^2*polylog(3,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^3-2*I*b*f^2*polylog(3,I*exp(d*x+c))/a^2/d^3+4*b
*f*(f*x+e)*arctanh(exp(d*x+c))/a^2/d^2+b^4*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^2+2*I*b*f^2*po
lylog(3,-I*exp(d*x+c))/a^2/d^3-2*I*b*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a^2/d^2-2*I*b^3*f^2*polylog(3,-I*exp(d
*x+c))/a^2/(a^2+b^2)/d^3+2*b*f^2*polylog(2,-exp(d*x+c))/a^2/d^3-2*b*f^2*polylog(2,exp(d*x+c))/a^2/d^3-1/2*b^2*
f^2*polylog(3,exp(2*d*x+2*c))/a^3/d^3+f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a/d^2-f*(f*x+e)*polylog(2,exp(2*d*x
+2*c))/a/d^2+2*I*b*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^2/d^2+2*I*b^3*f^2*polylog(3,I*exp(d*x+c))/a^2/(a^2+b^2)
/d^3-2*I*b^3*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2-1/2*(f*x+e)^2*coth(d*x+c)^2/a/d-1/2*f^2*polyl
og(3,-exp(2*d*x+2*c))/a/d^3+1/2*f^2*polylog(3,exp(2*d*x+2*c))/a/d^3+2*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a/d+e*
f*x/a/d+1/2*f^2*x^2/a/d+2*I*b^3*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+b^4*(f*x+e)^2*ln(1+exp(2*
d*x+2*c))/a^3/(a^2+b^2)/d-b^4*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*(f*x+e)^2*l
n(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a
^3/(a^2+b^2)/d^3+2*b^4*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^3-2*b^4*f*(f*x+e)*poly
log(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2-2*b^4*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2
)^(1/2)))/a^3/(a^2+b^2)/d^2+f^2*ln(sinh(d*x+c))/a/d^3
________________________________________________________________________________________
Rubi [A] time = 2.22, antiderivative size = 1219, normalized size of antiderivative =
1.00, number of steps used = 71, number of rules used = 26, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) =
0.765, Rules used = {5589, 2620, 14, 5462, 6741, 12, 6742, 3720, 3475, 2551, 4182,
2531, 2282, 6589, 2621, 321, 207, 5205, 4180, 2279, 2391, 5461, 5573, 5561, 2190, 3718}
\[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^4}{a^3 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^4}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right ) b^2}{a^3 d}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right ) b^2}{a^3 d^2}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}-\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right ) b^2}{2 a^3 d^3}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}+\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac {(e+f x)^2 \text {csch}(c+d x) b}{a^2 d}+\frac {2 f^2 \text {PolyLog}\left (2,-e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right ) b}{a^2 d^2}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right ) b}{a^2 d^2}-\frac {2 f^2 \text {PolyLog}\left (2,e^{c+d x}\right ) b}{a^2 d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) b}{a^2 d^3}-\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right ) b}{a^2 d^3}+\frac {f^2 x^2}{2 a d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {e f x}{a d}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}-\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3} \]
Antiderivative was successfully verified.
[In]
Int[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]
[Out]
(e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) + (2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a^2*d) - (2*b^3*(e + f*x)^2*ArcTan[
E^(c + d*x)])/(a^2*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) + (2*(e + f*x)^2*ArcTanh[
E^(2*c + 2*d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^3*d) - (f*(e + f*x)*Coth[c + d*x])/(
a*d^2) - ((e + f*x)^2*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^2*Csch[c + d*x])/(a^2*d) - (b^4*(e + f*x)^2*Log[
1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^3*(a^2 + b^2)*d) + (b^4*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d) + (f
^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^3) - ((2*I)*b*f*(e + f*x)*PolyLog[2
, (-I)*E^(c + d*x)])/(a^2*d^2) + ((2*I)*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) +
((2*I)*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a^2*d^2) - ((2*I)*b^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/
(a^2*(a^2 + b^2)*d^2) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a^3*(a^2 + b^2)*d^
2) + (f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a*d^2) - (b^2*f*(e + f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^3*
d^2) - (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^3*
d^2) + ((2*I)*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^2*d^3) - ((2*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a^
2*(a^2 + b^2)*d^3) - ((2*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/(a^2*d^3) + ((2*I)*b^3*f^2*PolyLog[3, I*E^(c + d*
x)])/(a^2*(a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)
*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*(a^2 + b^2)*d^3) - (b^4*f^2*Poly
Log[3, -E^(2*(c + d*x))])/(2*a^3*(a^2 + b^2)*d^3) - (f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) + (b^2*f^2*Po
lyLog[3, -E^(2*c + 2*d*x)])/(2*a^3*d^3) + (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3) - (b^2*f^2*PolyLog[3, E^
(2*c + 2*d*x)])/(2*a^3*d^3)
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 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rule 321
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]
Rule 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2391
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2551
Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]
Rule 2620
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 2621
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 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 3718
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 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rule 4180
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 4182
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]
Rule 5205
Int[((a_.) + ArcTan[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 + u^2), x], x]
, x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m +
1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]
Rule 5461
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Rule 5462
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
Rule 5561
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 5573
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]
Rule 5589
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Rule 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rule 6741
Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(2 f) \int (e+f x) \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a}\\ &=\frac {b (e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(2 f) \int \frac {(e+f x) \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right )}{2 d} \, dx}{a}+\frac {(2 b f) \int (e+f x) \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2}\\ &=\frac {b (e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(2 b f) \int \frac {(e+f x) \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right )}{d} \, dx}{a^2}-\frac {f \int (e+f x) \left (-\coth ^2(c+d x)-2 \log (\tanh (c+d x))\right ) \, dx}{a d}\\ &=\frac {b^4 (e+f x)^3}{3 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b^3 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {f \int \left (-(e+f x) \coth ^2(c+d x)-2 (e+f x) \log (\tanh (c+d x))\right ) \, dx}{a d}+\frac {(2 b f) \int (e+f x) \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right ) \, dx}{a^2 d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d}\\ &=\frac {b^4 (e+f x)^3}{3 a^3 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {b^3 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x)^2 \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \int (e+f x) \coth ^2(c+d x) \, dx}{a d}+\frac {(2 f) \int (e+f x) \log (\tanh (c+d x)) \, dx}{a d}+\frac {(2 b f) \int \left (-(e+f x) \tan ^{-1}(\sinh (c+d x))-(e+f x) \text {csch}(c+d x)\right ) \, dx}{a^2 d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (b^2 f^2\right ) \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}-\frac {\left (b^2 f^2\right ) \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a^3 d^2}\\ &=-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b (e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\int 2 d (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a d}+\frac {f \int (e+f x) \, dx}{a d}-\frac {(2 b f) \int (e+f x) \tan ^{-1}(\sinh (c+d x)) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \text {csch}(c+d x) \, dx}{a^2 d}+\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (2 i b^3 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {\left (b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {f^2 \int \coth (c+d x) \, dx}{a d^2}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (2 b^4 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {2 \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a}+\frac {b \int d (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (2 b^4 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 b f^2\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (2 b f^2\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i b^3 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {\left (2 i b^3 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {b \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a^2}+\frac {(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 i b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {\left (2 i b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (b^4 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {(2 i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {(2 i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d}-\frac {\left (b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}+\frac {f^2 \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}-\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}+\frac {f^2 \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}+\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a^2 d^2}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (2 i b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}\\ &=\frac {e f x}{a d}+\frac {f^2 x^2}{2 a d}+\frac {2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 d^2}+\frac {2 i b^3 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 d^2}-\frac {2 i b^3 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f^2 \text {Li}_2\left (e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 b^4 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}-\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^3 d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^3 d^2}+\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 d^3}-\frac {2 i b^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a^2 d^3}+\frac {2 i b^3 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 b^4 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {b^4 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}+\frac {b^2 f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^3 d^3}+\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac {b^2 f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^3 d^3}\\ \end {align*}
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Mathematica [B] time = 39.34, size = 3044, normalized size = 2.50 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((e + f*x)^2*Csch[c + d*x]^3*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]
[Out]
((-e^2 - 2*e*f*x - f^2*x^2)*Csch[c/2 + (d*x)/2]^2)/(8*a*d) + (-12*a*d^3*e^2*E^(2*c)*x - 12*a*d^3*e*E^(2*c)*f*x
^2 - 4*a*d^3*E^(2*c)*f^2*x^3 + 12*b*d^2*e^2*ArcTan[E^(c + d*x)] + 12*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + (
12*I)*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (12*I)*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b*d^2*f^2
*x^2*Log[1 - I*E^(c + d*x)] + (6*I)*b*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (12*I)*b*d^2*e*f*x*Log[1 +
I*E^(c + d*x)] - (12*I)*b*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*b*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)
] - (6*I)*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*a*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6*a*d^2*e^2*E^
(2*c)*Log[1 + E^(2*(c + d*x))] + 12*a*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*d^2*e*E^(2*c)*f*x*Log[1 + E^(2
*(c + d*x))] + 6*a*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (
12*I)*b*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (12*I)*b*d*(1 + E^(2*c))*f*(e + f*x)*PolyLo
g[2, I*E^(c + d*x)] + 6*a*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 6*a*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))]
+ 6*a*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (12*I)*b*f^2*P
olyLog[3, (-I)*E^(c + d*x)] + (12*I)*b*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (12*I)*b*f^2*PolyLog[3, I*E^
(c + d*x)] - (12*I)*b*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a*f^2*PolyLog[3, -E^(2*(c + d*x))] - 3*a*E^(2*
c)*f^2*PolyLog[3, -E^(2*(c + d*x))])/(6*(a^2 + b^2)*d^3*(1 + E^(2*c))) - (-12*a^2*d^3*e^2*E^(2*c)*x + 12*b^2*d
^3*e^2*E^(2*c)*x + 12*a^2*d*E^(2*c)*f^2*x - 12*a^2*d^3*e*E^(2*c)*f*x^2 + 12*b^2*d^3*e*E^(2*c)*f*x^2 - 4*a^2*d^
3*E^(2*c)*f^2*x^3 + 4*b^2*d^3*E^(2*c)*f^2*x^3 + 24*a*b*d*e*f*ArcTanh[E^(c + d*x)] - 24*a*b*d*e*E^(2*c)*f*ArcTa
nh[E^(c + d*x)] - 12*a*b*d*f^2*x*Log[1 - E^(c + d*x)] + 12*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] + 12*a*b*d
*f^2*x*Log[1 + E^(c + d*x)] - 12*a*b*d*E^(2*c)*f^2*x*Log[1 + E^(c + d*x)] - 6*a^2*d^2*e^2*Log[1 - E^(2*(c + d*
x))] + 6*b^2*d^2*e^2*Log[1 - E^(2*(c + d*x))] + 6*a^2*d^2*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] - 6*b^2*d^2*e^2
*E^(2*c)*Log[1 - E^(2*(c + d*x))] + 6*a^2*f^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*E^(2*c)*f^2*Log[1 - E^(2*(c + d
*x))] - 12*a^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] + 12*a^2*d^2*e*E
^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*f^2*x^2*Lo
g[1 - E^(2*(c + d*x))] + 6*b^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] + 6*a^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(
c + d*x))] - 6*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, -E^(c +
d*x)] + 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, E^(c + d*x)] - 6*a^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d
*e*f*PolyLog[2, E^(2*(c + d*x))] + 6*a^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*e*E^(2*c)*f*PolyL
og[2, E^(2*(c + d*x))] - 6*a^2*d*f^2*x*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d*f^2*x*PolyLog[2, E^(2*(c + d*x))]
+ 6*a^2*d*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] + 3*a
^2*f^2*PolyLog[3, E^(2*(c + d*x))] - 3*b^2*f^2*PolyLog[3, E^(2*(c + d*x))] - 3*a^2*E^(2*c)*f^2*PolyLog[3, E^(2
*(c + d*x))] + 3*b^2*E^(2*c)*f^2*PolyLog[3, E^(2*(c + d*x))])/(6*a^3*d^3*(-1 + E^(2*c))) + (b^4*(6*d^3*e^2*E^(
2*c)*x + 6*d^3*e*E^(2*c)*f*x^2 + 2*d^3*E^(2*c)*f^2*x^3 + 3*d^2*e^2*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))
] - 3*d^2*e^2*E^(2*c)*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 6*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*
E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d^2*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^
(2*c)])] + 3*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*d^2*E^(2*c)*f^2*x^
2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 6*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^
c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d^2*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2
*c)])] + 3*d^2*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*d^2*E^(2*c)*f^2*x^2*
Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((
b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2
)*E^(2*c)]))] + 6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*f^2*Pol
yLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x
))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(3*a^3*(a^2 + b^2)*d^3*(-1 + E^(2*c))) + ((-3*a^3*d*e^2*x - 3*a^3*d
*e*f*x^2 - a^3*d*f^2*x^3 + 3*a^2*b*e^2*Cosh[c] + 3*b^3*e^2*Cosh[c] + 6*a^2*b*e*f*x*Cosh[c] + 6*b^3*e*f*x*Cosh[
c] + 3*a^2*b*f^2*x^2*Cosh[c] + 3*b^3*f^2*x^2*Cosh[c])*Csch[c/2]*Sech[c/2]*Sech[c])/(6*a^2*(a^2 + b^2)*d) + ((e
^2 + 2*e*f*x + f^2*x^2)*Sech[c/2 + (d*x)/2]^2)/(8*a*d) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(b*d*e^2*Sinh[(d*x)/
2]) - a*e*f*Sinh[(d*x)/2] - 2*b*d*e*f*x*Sinh[(d*x)/2] - a*f^2*x*Sinh[(d*x)/2] - b*d*f^2*x^2*Sinh[(d*x)/2]))/(2
*a^2*d^2) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(-(b*d*e^2*Sinh[(d*x)/2]) + a*e*f*Sinh[(d*x)/2] - 2*b*d*e*f*x*Sinh[
(d*x)/2] + a*f^2*x*Sinh[(d*x)/2] - b*d*f^2*x^2*Sinh[(d*x)/2]))/(2*a^2*d^2)
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fricas [C] time = 0.93, size = 13381, normalized size = 10.98 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-(2*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(d*x + c)^4 + 2*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^
2*b^2)*c*f^2)*sinh(d*x + c)^4 - 2*(a^4 + a^2*b^2)*d*e*f + 2*(a^4 + a^2*b^2)*c*f^2 - 2*((a^3*b + a*b^3)*d^2*f^2
*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2)*cosh(d*x + c)^3 - 2*((a^3*b + a*b^3)*d^2*f^2*x^2
+ 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2 - 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)
*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((a^4 + a^2*b^2)*d^2*f^2*x^2 + (a^4 + a^2*b^2)*d^2*e^2 + (a^4 + a^2*b^2)*d
*e*f - 2*(a^4 + a^2*b^2)*c*f^2 + (2*(a^4 + a^2*b^2)*d^2*e*f - (a^4 + a^2*b^2)*d*f^2)*x)*cosh(d*x + c)^2 + 2*((
a^4 + a^2*b^2)*d^2*f^2*x^2 + (a^4 + a^2*b^2)*d^2*e^2 + (a^4 + a^2*b^2)*d*e*f - 2*(a^4 + a^2*b^2)*c*f^2 + 6*((a
^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(d*x + c)^2 + (2*(a^4 + a^2*b^2)*d^2*e*f - (a^4 + a^2*b^2)*
d*f^2)*x - 3*((a^3*b + a*b^3)*d^2*f^2*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2)*cosh(d*x +
c))*sinh(d*x + c)^2 + 2*((a^3*b + a*b^3)*d^2*f^2*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2)*
cosh(d*x + c) + 2*(b^4*d*f^2*x + b^4*d*e*f + (b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^4 + 4*(b^4*d*f^2*x + b^4*
d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d*f^2*x + b^4*d*e*f)*sinh(d*x + c)^4 - 2*(b^4*d*f^2*x + b^4*d*e*f)
*cosh(d*x + c)^2 - 2*(b^4*d*f^2*x + b^4*d*e*f - 3*(b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 +
4*((b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^3 - (b^4*d*f^2*x + b^4*d*e*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*d*f^2*x + b^4*d*e*f + (b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^4 + 4*(b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x +
c)*sinh(d*x + c)^3 + (b^4*d*f^2*x + b^4*d*e*f)*sinh(d*x + c)^4 - 2*(b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^2
- 2*(b^4*d*f^2*x + b^4*d*e*f - 3*(b^4*d*f^2*x + b^4*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d*f^2*x
+ b^4*d*e*f)*cosh(d*x + c)^3 - (b^4*d*f^2*x + b^4*d*e*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 - b^4)*d*f
^2*x + ((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c)^4 + 4*((a^4 - b^4)*d*f^2*
x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 - b^4)*d*f^2*x + (a^4 - b^4
)*d*e*f + (a^3*b + a*b^3)*f^2)*sinh(d*x + c)^4 + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2 - 2*((a^4 - b^4)*d*f^
2*x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c)^2 - 2*((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f +
(a^3*b + a*b^3)*f^2 - 3*((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c)^2)*sinh(
d*x + c)^2 + 4*(((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c)^3 - ((a^4 - b^4)
*d*f^2*x + (a^4 - b^4)*d*e*f + (a^3*b + a*b^3)*f^2)*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d
*x + c)) - (2*a^4*d*f^2*x + 2*I*a^3*b*d*f^2*x + 2*a^4*d*e*f + 2*I*a^3*b*d*e*f + (2*a^4*d*f^2*x + 2*I*a^3*b*d*f
^2*x + 2*a^4*d*e*f + 2*I*a^3*b*d*e*f)*cosh(d*x + c)^4 + (8*a^4*d*f^2*x + 8*I*a^3*b*d*f^2*x + 8*a^4*d*e*f + 8*I
*a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^4*d*f^2*x + 2*I*a^3*b*d*f^2*x + 2*a^4*d*e*f + 2*I*a^3*b*d*e
*f)*sinh(d*x + c)^4 - (4*a^4*d*f^2*x + 4*I*a^3*b*d*f^2*x + 4*a^4*d*e*f + 4*I*a^3*b*d*e*f)*cosh(d*x + c)^2 - (4
*a^4*d*f^2*x + 4*I*a^3*b*d*f^2*x + 4*a^4*d*e*f + 4*I*a^3*b*d*e*f - (12*a^4*d*f^2*x + 12*I*a^3*b*d*f^2*x + 12*a
^4*d*e*f + 12*I*a^3*b*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^4*d*f^2*x + 8*I*a^3*b*d*f^2*x + 8*a^4*d*
e*f + 8*I*a^3*b*d*e*f)*cosh(d*x + c)^3 - (8*a^4*d*f^2*x + 8*I*a^3*b*d*f^2*x + 8*a^4*d*e*f + 8*I*a^3*b*d*e*f)*c
osh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (2*a^4*d*f^2*x - 2*I*a^3*b*d*f^2*x + 2
*a^4*d*e*f - 2*I*a^3*b*d*e*f + (2*a^4*d*f^2*x - 2*I*a^3*b*d*f^2*x + 2*a^4*d*e*f - 2*I*a^3*b*d*e*f)*cosh(d*x +
c)^4 + (8*a^4*d*f^2*x - 8*I*a^3*b*d*f^2*x + 8*a^4*d*e*f - 8*I*a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*
a^4*d*f^2*x - 2*I*a^3*b*d*f^2*x + 2*a^4*d*e*f - 2*I*a^3*b*d*e*f)*sinh(d*x + c)^4 - (4*a^4*d*f^2*x - 4*I*a^3*b*
d*f^2*x + 4*a^4*d*e*f - 4*I*a^3*b*d*e*f)*cosh(d*x + c)^2 - (4*a^4*d*f^2*x - 4*I*a^3*b*d*f^2*x + 4*a^4*d*e*f -
4*I*a^3*b*d*e*f - (12*a^4*d*f^2*x - 12*I*a^3*b*d*f^2*x + 12*a^4*d*e*f - 12*I*a^3*b*d*e*f)*cosh(d*x + c)^2)*sin
h(d*x + c)^2 + ((8*a^4*d*f^2*x - 8*I*a^3*b*d*f^2*x + 8*a^4*d*e*f - 8*I*a^3*b*d*e*f)*cosh(d*x + c)^3 - (8*a^4*d
*f^2*x - 8*I*a^3*b*d*f^2*x + 8*a^4*d*e*f - 8*I*a^3*b*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x +
c) - I*sinh(d*x + c)) + 2*((a^4 - b^4)*d*f^2*x + ((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^
2)*cosh(d*x + c)^4 + 4*((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 + ((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^2)*sinh(d*x + c)^4 + (a^4 - b^4)*d*e*f -
(a^3*b + a*b^3)*f^2 - 2*((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^2)*cosh(d*x + c)^2 - 2*(
(a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^2 - 3*((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (
a^3*b + a*b^3)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b +
a*b^3)*f^2)*cosh(d*x + c)^3 - ((a^4 - b^4)*d*f^2*x + (a^4 - b^4)*d*e*f - (a^3*b + a*b^3)*f^2)*cosh(d*x + c))*s
inh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2 + (b^4*d^2*e^
2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c)
*sinh(d*x + c)^3 + (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^4*d^2*e^2 - 2*b^4*c*d*e*
f + b^4*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2 - 3*(b^4*d^2*e^2 - 2*b^4*c*d*e
*f + b^4*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x +
c)^3 - (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*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) + (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2 + (b^4*d^2*e^2 -
2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c)^3 + (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^4*d^2*e^2 - 2*b^4*c*d*e*f +
b^4*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2 - 3*(b^4*d^2*e^2 - 2*b^4*c*d*e*f
+ b^4*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c)
^3 - (b^4*d^2*e^2 - 2*b^4*c*d*e*f + b^4*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sin
h(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f
^2 + (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^4*d^2*f^2*x^2 +
2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*
f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^
4*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2 - 3*(b^4*d^2*f
^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d^2*f^2*x^2
+ 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^3 - (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4
*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4
*c^2*f^2 + (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^4*d^2*f^2*
x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^4*d^2*f^2*x^2 + 2*b^4*
d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*
f - b^4*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2 - 3*(b^4
*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^4*d^2*f
^2*x^2 + 2*b^4*d^2*e*f*x + 2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c)^3 - (b^4*d^2*f^2*x^2 + 2*b^4*d^2*e*f*x +
2*b^4*c*d*e*f - b^4*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(
d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + ((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 + (
(a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 + 2*((a^4 - b^4)
*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^4 + 4*((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2*(a
^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)*s
inh(d*x + c)^3 + ((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^
2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*sinh(d*x + c)^4 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*
b^2)*f^2 - 2*((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 +
2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*
d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 - 3*((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2
*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c
)^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a
*b^3)*d*f^2)*x + 4*(((a^4 - b^4)*d^2*f^2*x^2 + (a^4 - b^4)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)
*f^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^3 - ((a^4 - b^4)*d^2*f^2*x^2 + (a^4 -
b^4)*d^2*e^2 - 2*(a^3*b + a*b^3)*d*e*f - (a^4 + a^2*b^2)*f^2 + 2*((a^4 - b^4)*d^2*e*f - (a^3*b + a*b^3)*d*f^2)
*x)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (a^4*d^2*e^2 + I*a^3*b*d^2*e^2 - 2*
a^4*c*d*e*f - 2*I*a^3*b*c*d*e*f + a^4*c^2*f^2 + I*a^3*b*c^2*f^2 + (a^4*d^2*e^2 + I*a^3*b*d^2*e^2 - 2*a^4*c*d*e
*f - 2*I*a^3*b*c*d*e*f + a^4*c^2*f^2 + I*a^3*b*c^2*f^2)*cosh(d*x + c)^4 + (4*a^4*d^2*e^2 + 4*I*a^3*b*d^2*e^2 -
8*a^4*c*d*e*f - 8*I*a^3*b*c*d*e*f + 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*d
^2*e^2 + I*a^3*b*d^2*e^2 - 2*a^4*c*d*e*f - 2*I*a^3*b*c*d*e*f + a^4*c^2*f^2 + I*a^3*b*c^2*f^2)*sinh(d*x + c)^4
- (2*a^4*d^2*e^2 + 2*I*a^3*b*d^2*e^2 - 4*a^4*c*d*e*f - 4*I*a^3*b*c*d*e*f + 2*a^4*c^2*f^2 + 2*I*a^3*b*c^2*f^2)*
cosh(d*x + c)^2 - (2*a^4*d^2*e^2 + 2*I*a^3*b*d^2*e^2 - 4*a^4*c*d*e*f - 4*I*a^3*b*c*d*e*f + 2*a^4*c^2*f^2 + 2*I
*a^3*b*c^2*f^2 - (6*a^4*d^2*e^2 + 6*I*a^3*b*d^2*e^2 - 12*a^4*c*d*e*f - 12*I*a^3*b*c*d*e*f + 6*a^4*c^2*f^2 + 6*
I*a^3*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a^4*d^2*e^2 + 4*I*a^3*b*d^2*e^2 - 8*a^4*c*d*e*f - 8*I*
a^3*b*c*d*e*f + 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f^2)*cosh(d*x + c)^3 - (4*a^4*d^2*e^2 + 4*I*a^3*b*d^2*e^2 - 8*a^
4*c*d*e*f - 8*I*a^3*b*c*d*e*f + 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x
+ c) + sinh(d*x + c) + I) - (a^4*d^2*e^2 - I*a^3*b*d^2*e^2 - 2*a^4*c*d*e*f + 2*I*a^3*b*c*d*e*f + a^4*c^2*f^2 -
I*a^3*b*c^2*f^2 + (a^4*d^2*e^2 - I*a^3*b*d^2*e^2 - 2*a^4*c*d*e*f + 2*I*a^3*b*c*d*e*f + a^4*c^2*f^2 - I*a^3*b*
c^2*f^2)*cosh(d*x + c)^4 + (4*a^4*d^2*e^2 - 4*I*a^3*b*d^2*e^2 - 8*a^4*c*d*e*f + 8*I*a^3*b*c*d*e*f + 4*a^4*c^2*
f^2 - 4*I*a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*d^2*e^2 - I*a^3*b*d^2*e^2 - 2*a^4*c*d*e*f + 2*I*
a^3*b*c*d*e*f + a^4*c^2*f^2 - I*a^3*b*c^2*f^2)*sinh(d*x + c)^4 - (2*a^4*d^2*e^2 - 2*I*a^3*b*d^2*e^2 - 4*a^4*c*
d*e*f + 4*I*a^3*b*c*d*e*f + 2*a^4*c^2*f^2 - 2*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2 - (2*a^4*d^2*e^2 - 2*I*a^3*b*d^
2*e^2 - 4*a^4*c*d*e*f + 4*I*a^3*b*c*d*e*f + 2*a^4*c^2*f^2 - 2*I*a^3*b*c^2*f^2 - (6*a^4*d^2*e^2 - 6*I*a^3*b*d^2
*e^2 - 12*a^4*c*d*e*f + 12*I*a^3*b*c*d*e*f + 6*a^4*c^2*f^2 - 6*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)
^2 + ((4*a^4*d^2*e^2 - 4*I*a^3*b*d^2*e^2 - 8*a^4*c*d*e*f + 8*I*a^3*b*c*d*e*f + 4*a^4*c^2*f^2 - 4*I*a^3*b*c^2*f
^2)*cosh(d*x + c)^3 - (4*a^4*d^2*e^2 - 4*I*a^3*b*d^2*e^2 - 8*a^4*c*d*e*f + 8*I*a^3*b*c*d*e*f + 4*a^4*c^2*f^2 -
4*I*a^3*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + ((a^4 - b^4)*d^2*e^
2 + ((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3
*b + a*b^3)*c)*f^2)*cosh(d*x + c)^4 + 4*((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4
+ a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 - b^4)*d^2*e^2 +
2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*f^2)*sinh(d
*x + c)^4 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*
f^2 - 2*((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*
(a^3*b + a*b^3)*c)*f^2)*cosh(d*x + c)^2 - 2*((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (
a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*f^2 - 3*((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4
- b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 + 4*(((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4 + a^2*b^2 - (a^4 - b^4)*c^2 + 2*(
a^3*b + a*b^3)*c)*f^2)*cosh(d*x + c)^3 - ((a^4 - b^4)*d^2*e^2 + 2*(a^3*b + a*b^3 - (a^4 - b^4)*c)*d*e*f - (a^4
+ a^2*b^2 - (a^4 - b^4)*c^2 + 2*(a^3*b + a*b^3)*c)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sin
h(d*x + c) - 1) - (a^4*d^2*f^2*x^2 - I*a^3*b*d^2*f^2*x^2 + 2*a^4*d^2*e*f*x - 2*I*a^3*b*d^2*e*f*x + 2*a^4*c*d*e
*f - 2*I*a^3*b*c*d*e*f - a^4*c^2*f^2 + I*a^3*b*c^2*f^2 + (a^4*d^2*f^2*x^2 - I*a^3*b*d^2*f^2*x^2 + 2*a^4*d^2*e*
f*x - 2*I*a^3*b*d^2*e*f*x + 2*a^4*c*d*e*f - 2*I*a^3*b*c*d*e*f - a^4*c^2*f^2 + I*a^3*b*c^2*f^2)*cosh(d*x + c)^4
+ (4*a^4*d^2*f^2*x^2 - 4*I*a^3*b*d^2*f^2*x^2 + 8*a^4*d^2*e*f*x - 8*I*a^3*b*d^2*e*f*x + 8*a^4*c*d*e*f - 8*I*a^
3*b*c*d*e*f - 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*d^2*f^2*x^2 - I*a^3*b*d^
2*f^2*x^2 + 2*a^4*d^2*e*f*x - 2*I*a^3*b*d^2*e*f*x + 2*a^4*c*d*e*f - 2*I*a^3*b*c*d*e*f - a^4*c^2*f^2 + I*a^3*b*
c^2*f^2)*sinh(d*x + c)^4 - (2*a^4*d^2*f^2*x^2 - 2*I*a^3*b*d^2*f^2*x^2 + 4*a^4*d^2*e*f*x - 4*I*a^3*b*d^2*e*f*x
+ 4*a^4*c*d*e*f - 4*I*a^3*b*c*d*e*f - 2*a^4*c^2*f^2 + 2*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2 - (2*a^4*d^2*f^2*x^2
- 2*I*a^3*b*d^2*f^2*x^2 + 4*a^4*d^2*e*f*x - 4*I*a^3*b*d^2*e*f*x + 4*a^4*c*d*e*f - 4*I*a^3*b*c*d*e*f - 2*a^4*c^
2*f^2 + 2*I*a^3*b*c^2*f^2 - (6*a^4*d^2*f^2*x^2 - 6*I*a^3*b*d^2*f^2*x^2 + 12*a^4*d^2*e*f*x - 12*I*a^3*b*d^2*e*f
*x + 12*a^4*c*d*e*f - 12*I*a^3*b*c*d*e*f - 6*a^4*c^2*f^2 + 6*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ ((4*a^4*d^2*f^2*x^2 - 4*I*a^3*b*d^2*f^2*x^2 + 8*a^4*d^2*e*f*x - 8*I*a^3*b*d^2*e*f*x + 8*a^4*c*d*e*f - 8*I*a
^3*b*c*d*e*f - 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f^2)*cosh(d*x + c)^3 - (4*a^4*d^2*f^2*x^2 - 4*I*a^3*b*d^2*f^2*x^2
+ 8*a^4*d^2*e*f*x - 8*I*a^3*b*d^2*e*f*x + 8*a^4*c*d*e*f - 8*I*a^3*b*c*d*e*f - 4*a^4*c^2*f^2 + 4*I*a^3*b*c^2*f
^2)*cosh(d*x + c))*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) - (a^4*d^2*f^2*x^2 + I*a^3*b*d^2*
f^2*x^2 + 2*a^4*d^2*e*f*x + 2*I*a^3*b*d^2*e*f*x + 2*a^4*c*d*e*f + 2*I*a^3*b*c*d*e*f - a^4*c^2*f^2 - I*a^3*b*c^
2*f^2 + (a^4*d^2*f^2*x^2 + I*a^3*b*d^2*f^2*x^2 + 2*a^4*d^2*e*f*x + 2*I*a^3*b*d^2*e*f*x + 2*a^4*c*d*e*f + 2*I*a
^3*b*c*d*e*f - a^4*c^2*f^2 - I*a^3*b*c^2*f^2)*cosh(d*x + c)^4 + (4*a^4*d^2*f^2*x^2 + 4*I*a^3*b*d^2*f^2*x^2 + 8
*a^4*d^2*e*f*x + 8*I*a^3*b*d^2*e*f*x + 8*a^4*c*d*e*f + 8*I*a^3*b*c*d*e*f - 4*a^4*c^2*f^2 - 4*I*a^3*b*c^2*f^2)*
cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*d^2*f^2*x^2 + I*a^3*b*d^2*f^2*x^2 + 2*a^4*d^2*e*f*x + 2*I*a^3*b*d^2*e*f*x
+ 2*a^4*c*d*e*f + 2*I*a^3*b*c*d*e*f - a^4*c^2*f^2 - I*a^3*b*c^2*f^2)*sinh(d*x + c)^4 - (2*a^4*d^2*f^2*x^2 + 2
*I*a^3*b*d^2*f^2*x^2 + 4*a^4*d^2*e*f*x + 4*I*a^3*b*d^2*e*f*x + 4*a^4*c*d*e*f + 4*I*a^3*b*c*d*e*f - 2*a^4*c^2*f
^2 - 2*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2 - (2*a^4*d^2*f^2*x^2 + 2*I*a^3*b*d^2*f^2*x^2 + 4*a^4*d^2*e*f*x + 4*I*a
^3*b*d^2*e*f*x + 4*a^4*c*d*e*f + 4*I*a^3*b*c*d*e*f - 2*a^4*c^2*f^2 - 2*I*a^3*b*c^2*f^2 - (6*a^4*d^2*f^2*x^2 +
6*I*a^3*b*d^2*f^2*x^2 + 12*a^4*d^2*e*f*x + 12*I*a^3*b*d^2*e*f*x + 12*a^4*c*d*e*f + 12*I*a^3*b*c*d*e*f - 6*a^4*
c^2*f^2 - 6*I*a^3*b*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((4*a^4*d^2*f^2*x^2 + 4*I*a^3*b*d^2*f^2*x^2 +
8*a^4*d^2*e*f*x + 8*I*a^3*b*d^2*e*f*x + 8*a^4*c*d*e*f + 8*I*a^3*b*c*d*e*f - 4*a^4*c^2*f^2 - 4*I*a^3*b*c^2*f^2)
*cosh(d*x + c)^3 - (4*a^4*d^2*f^2*x^2 + 4*I*a^3*b*d^2*f^2*x^2 + 8*a^4*d^2*e*f*x + 8*I*a^3*b*d^2*e*f*x + 8*a^4*
c*d*e*f + 8*I*a^3*b*c*d*e*f - 4*a^4*c^2*f^2 - 4*I*a^3*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-I*cosh(d*x
+ c) - I*sinh(d*x + c) + 1) + ((a^4 - b^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f + ((a^4 - b^4)*d^2*f^2*x^2 + 2
*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3*b + a*b^3)*
d*f^2)*x)*cosh(d*x + c)^4 + 4*((a^4 - b^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b +
a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^4 - b^
4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f
+ (a^3*b + a*b^3)*d*f^2)*x)*sinh(d*x + c)^4 - ((a^4 - b^4)*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 - 2*((a^4 - b^4)*d^2
*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3
*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^4 - b^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2
- 2*(a^3*b + a*b^3)*c)*f^2 - 3*((a^4 - b^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b
+ a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^2 + 2*((a^4 - b^4)*d^2*e*f
+ (a^3*b + a*b^3)*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x + 4*(((a^4 - b
^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f
+ (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c)^3 - ((a^4 - b^4)*d^2*f^2*x^2 + 2*(a^4 - b^4)*c*d*e*f - ((a^4 - b^4)
*c^2 - 2*(a^3*b + a*b^3)*c)*f^2 + 2*((a^4 - b^4)*d^2*e*f + (a^3*b + a*b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x +
c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*(b^4*f^2*cosh(d*x + c)^4 + 4*b^4*f^2*cosh(d*x + c)*sinh(d*x +
c)^3 + b^4*f^2*sinh(d*x + c)^4 - 2*b^4*f^2*cosh(d*x + c)^2 + b^4*f^2 + 2*(3*b^4*f^2*cosh(d*x + c)^2 - b^4*f^2
)*sinh(d*x + c)^2 + 4*(b^4*f^2*cosh(d*x + c)^3 - b^4*f^2*cosh(d*x + c))*sinh(d*x + c))*polylog(3, (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) - 2*(b^4*f^2*cosh(d*x +
c)^4 + 4*b^4*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*f^2*sinh(d*x + c)^4 - 2*b^4*f^2*cosh(d*x + c)^2 + b^4*f^
2 + 2*(3*b^4*f^2*cosh(d*x + c)^2 - b^4*f^2)*sinh(d*x + c)^2 + 4*(b^4*f^2*cosh(d*x + c)^3 - b^4*f^2*cosh(d*x +
c))*sinh(d*x + c))*polylog(3, (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) - 2*((a^4 - b^4)*f^2*cosh(d*x + c)^4 + 4*(a^4 - b^4)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a
^4 - b^4)*f^2*sinh(d*x + c)^4 - 2*(a^4 - b^4)*f^2*cosh(d*x + c)^2 + (a^4 - b^4)*f^2 + 2*(3*(a^4 - b^4)*f^2*cos
h(d*x + c)^2 - (a^4 - b^4)*f^2)*sinh(d*x + c)^2 + 4*((a^4 - b^4)*f^2*cosh(d*x + c)^3 - (a^4 - b^4)*f^2*cosh(d*
x + c))*sinh(d*x + c))*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + (2*a^4*f^2 + 2*I*a^3*b*f^2 + (2*a^4*f^2 + 2
*I*a^3*b*f^2)*cosh(d*x + c)^4 + (8*a^4*f^2 + 8*I*a^3*b*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a^4*f^2 + 2*I*a
^3*b*f^2)*sinh(d*x + c)^4 - (4*a^4*f^2 + 4*I*a^3*b*f^2)*cosh(d*x + c)^2 - (4*a^4*f^2 + 4*I*a^3*b*f^2 - (12*a^4
*f^2 + 12*I*a^3*b*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^4*f^2 + 8*I*a^3*b*f^2)*cosh(d*x + c)^3 - (8*a^
4*f^2 + 8*I*a^3*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) + (2*a^4*f^
2 - 2*I*a^3*b*f^2 + (2*a^4*f^2 - 2*I*a^3*b*f^2)*cosh(d*x + c)^4 + (8*a^4*f^2 - 8*I*a^3*b*f^2)*cosh(d*x + c)*si
nh(d*x + c)^3 + (2*a^4*f^2 - 2*I*a^3*b*f^2)*sinh(d*x + c)^4 - (4*a^4*f^2 - 4*I*a^3*b*f^2)*cosh(d*x + c)^2 - (4
*a^4*f^2 - 4*I*a^3*b*f^2 - (12*a^4*f^2 - 12*I*a^3*b*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a^4*f^2 - 8*I*
a^3*b*f^2)*cosh(d*x + c)^3 - (8*a^4*f^2 - 8*I*a^3*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, -I*cosh(d*x
+ c) - I*sinh(d*x + c)) - 2*((a^4 - b^4)*f^2*cosh(d*x + c)^4 + 4*(a^4 - b^4)*f^2*cosh(d*x + c)*sinh(d*x + c)^3
+ (a^4 - b^4)*f^2*sinh(d*x + c)^4 - 2*(a^4 - b^4)*f^2*cosh(d*x + c)^2 + (a^4 - b^4)*f^2 + 2*(3*(a^4 - b^4)*f^
2*cosh(d*x + c)^2 - (a^4 - b^4)*f^2)*sinh(d*x + c)^2 + 4*((a^4 - b^4)*f^2*cosh(d*x + c)^3 - (a^4 - b^4)*f^2*co
sh(d*x + c))*sinh(d*x + c))*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) + 2*((a^3*b + a*b^3)*d^2*f^2*x^2 + 2*(a
^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2 + 4*((a^4 + a^2*b^2)*d*f^2*x + (a^4 + a^2*b^2)*c*f^2)*cosh(d
*x + c)^3 - 3*((a^3*b + a*b^3)*d^2*f^2*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2)*cosh(d*x +
c)^2 + 2*((a^4 + a^2*b^2)*d^2*f^2*x^2 + (a^4 + a^2*b^2)*d^2*e^2 + (a^4 + a^2*b^2)*d*e*f - 2*(a^4 + a^2*b^2)*c
*f^2 + (2*(a^4 + a^2*b^2)*d^2*e*f - (a^4 + a^2*b^2)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + a^3*b^2)*d
^3*cosh(d*x + c)^4 + 4*(a^5 + a^3*b^2)*d^3*cosh(d*x + c)*sinh(d*x + c)^3 + (a^5 + a^3*b^2)*d^3*sinh(d*x + c)^4
- 2*(a^5 + a^3*b^2)*d^3*cosh(d*x + c)^2 + (a^5 + a^3*b^2)*d^3 + 2*(3*(a^5 + a^3*b^2)*d^3*cosh(d*x + c)^2 - (a
^5 + a^3*b^2)*d^3)*sinh(d*x + c)^2 + 4*((a^5 + a^3*b^2)*d^3*cosh(d*x + c)^3 - (a^5 + a^3*b^2)*d^3*cosh(d*x + c
))*sinh(d*x + c))
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [F] time = 2.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right )^{3} \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
[Out]
int((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b
^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x -
3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) +
(a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^2 + 2*(a*f^2*x + a*e*f + (b*d*f^2*x^2*e^(3*c) + 2*b*d*e*f*x*e^(3*
c))*e^(3*d*x) - (a*d*f^2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) - (b*d*f^2*x^2*e
^c + 2*b*d*e*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) - (2*b*d*e*f +
a*f^2)*x/(a^2*d^2) + (2*b*d*e*f - a*f^2)*x/(a^2*d^2) + (2*b*d*e*f + a*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - (2
*b*d*e*f - a*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) - (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) -
2*polylog(3, -e^(d*x + c)))*(a^2*f^2 - b^2*f^2)/(a^3*d^3) - (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d
*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 - b^2*f^2)/(a^3*d^3) - 2*(a^2*d*e*f - b^2*d*e*f - a*b*f^2)*(d*x
*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) - 2*(a^2*d*e*f - b^2*d*e*f + a*b*f^2)*(d*x*log(-e^(d*x
+ c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) + 1/3*((a^2*f^2 - b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - b^2*d*e*f + a*b*
f^2)*d^2*x^2)/(a^3*d^3) + 1/3*((a^2*f^2 - b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - b^2*d*e*f - a*b*f^2)*d^2*x^2)/(a^3
*d^3) + integrate(2*(b^5*f^2*x^2 + 2*b^5*e*f*x - (a*b^4*f^2*x^2*e^c + 2*a*b^4*e*f*x*e^c)*e^(d*x))/(a^5*b + a^3
*b^3 - (a^5*b*e^(2*c) + a^3*b^3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^c)*e^(d*x)), x) + integrate(-2*(a*
f^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d
*x)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^2}{\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))),x)
[Out]
int((e + f*x)^2/(cosh(c + d*x)*sinh(c + d*x)^3*(a + b*sinh(c + d*x))), x)
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)**2*csch(d*x+c)**3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)
[Out]
Timed out
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