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

Optimal. Leaf size=1245 \[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} 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^5}{a^3 \left (a^2+b^2\right )^{3/2} 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^5}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac {f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) b^3}{a^2 \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d}-\frac {2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right ) b^2}{a^3 d^3}+\frac {2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {2 f^2 \text {Li}_3\left (-e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {2 f^2 \text {Li}_3\left (e^{c+d x}\right ) b^2}{a^3 d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) b^2}{a^3 d}+\frac {2 (e+f x)^2 b}{a^2 d}+\frac {2 (e+f x)^2 \coth (2 c+2 d x) b}{a^2 d}-\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right ) b}{a^2 d^2}-\frac {f^2 \text {Li}_2\left (e^{4 (c+d x)}\right ) b}{2 a^2 d^3}+\frac {4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 3.44, antiderivative size = 1245, normalized size of antiderivative = 1.00, number of steps used = 88, number of rules used = 33, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5589, 2622, 288, 321, 207, 5462, 6688, 12, 6742, 6273, 4182, 2531, 2282, 6589, 4133, 453, 203, 4180, 2279, 2391, 2621, 5203, 3770, 5461, 4184, 3716, 2190, 6741, 5573, 3322, 2264, 3718, 5451} \[ -\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} 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^5}{a^3 \left (a^2+b^2\right )^{3/2} 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^5}{a^3 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^5}{a^3 \left (a^2+b^2\right )^{3/2} d^3}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right ) b^4}{a^3 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) b^4}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 b^3}{a^2 \left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac {f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) b^3}{a^2 \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d^2}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right ) b^2}{a^3 d^3}+\frac {2 f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right ) b^2}{a^3 d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right ) b^2}{a^3 d^3}-\frac {2 f^2 \text {PolyLog}\left (3,e^{c+d x}\right ) b^2}{a^3 d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) b^2}{a^3 d}+\frac {2 (e+f x)^2 b}{a^2 d}+\frac {2 (e+f x)^2 \coth (2 c+2 d x) b}{a^2 d}-\frac {2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right ) b}{a^2 d^2}-\frac {f^2 \text {PolyLog}\left (2,e^{4 (c+d x)}\right ) b}{2 a^2 d^3}+\frac {4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}+\frac {3 f (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

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

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 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 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 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 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 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 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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 3770

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

Rule 4133

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[((1 - ff^2*x^2)^((m - 1)/2)*(b + a*(ff*x)^n)^p)/(ff*x)^(n*
p), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

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 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5203

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 + u^2), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5462

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5589

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6273

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan
h[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 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}^3(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b \int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(2 f) \int (e+f x) \left (\frac {3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 \text {sech}(c+d x)}{2 d}-\frac {\text {csch}^2(c+d x) \text {sech}(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac {3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {(4 b) \int (e+f x)^2 \text {csch}^2(2 c+2 d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}-\frac {(2 f) \int \frac {(e+f x) \left (3 \tanh ^{-1}(\cosh (c+d x))-\left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x)\right )}{2 d} \, dx}{a}\\ &=\frac {3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \int (e+f x)^2 \text {sech}^2(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}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (2 b^2 f\right ) \int (e+f x) \left (-\frac {\tanh ^{-1}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a^3}-\frac {f \int (e+f x) \left (3 \tanh ^{-1}(\cosh (c+d x))-\left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x)\right ) \, dx}{a d}-\frac {(4 b f) \int (e+f x) \coth (2 c+2 d x) \, dx}{a^2 d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}+\frac {3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 \int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (2 b^5\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {\left (2 b^2 f\right ) \int \frac {(e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right )}{d} \, dx}{a^3}-\frac {f \int \left (3 (e+f x) \tanh ^{-1}(\cosh (c+d x))-(e+f x) \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x)\right ) \, dx}{a d}+\frac {(8 b f) \int \frac {e^{2 (2 c+2 d x)} (e+f x)}{1-e^{2 (2 c+2 d x)}} \, dx}{a^2 d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}+\frac {3 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{2 a d}-\frac {b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {\left (2 b^6\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 b^6\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \int (e+f x) \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x) \, dx}{a d}-\frac {(3 f) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right ) \, dx}{a^3 d}+\frac {\left (2 b f^2\right ) \int \log \left (1-e^{2 (2 c+2 d x)}\right ) \, dx}{a^2 d^2}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {3 \int d (e+f x)^2 \text {csch}(c+d x) \, dx}{2 a d}+\frac {f \int \left (e \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x)+f x \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x)\right ) \, dx}{a d}-\frac {\left (2 b^2 f\right ) \int \left (-(e+f x) \tanh ^{-1}(\cosh (c+d x))+(e+f x) \text {sech}(c+d x)\right ) \, dx}{a^3 d}+\frac {\left (2 b^5 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}-\frac {\left (2 b^5 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d}+\frac {\left (2 b^3 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (2 b^4 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (b f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 (2 c+2 d x)}\right )}{2 a^2 d^3}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {3 \int (e+f x)^2 \text {csch}(c+d x) \, dx}{2 a}+\frac {\left (2 b^2 f\right ) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a^3 d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{a^3 d}+\frac {\left (4 b^3 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right ) d}+\frac {(e f) \int \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x) \, dx}{a d}+\frac {\left (2 b^5 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (2 b^5 f^2\right ) \int \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (2 i b^4 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (2 i b^4 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f^2 \int x \left (3+\text {csch}^2(c+d x)\right ) \text {sech}(c+d x) \, dx}{a d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \int d (e+f x)^2 \text {csch}(c+d x) \, dx}{a^3 d}+\frac {(3 f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d}-\frac {(3 f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {(e f) \operatorname {Subst}\left (\int \frac {-1-3 x^2}{x^2 \left (1+x^2\right )} \, dx,x,\sinh (c+d x)\right )}{a d^2}+\frac {\left (2 b^5 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 )^{3/2} d^3}-\frac {\left (2 b^5 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 )^{3/2} d^3}-\frac {\left (2 i b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 i b^4 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 i b^2 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (2 i b^2 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (2 b^3 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}+\frac {f^2 \int \left (3 x \text {sech}(c+d x)+x \text {csch}^2(c+d x) \text {sech}(c+d x)\right ) \, dx}{a d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {b^2 \int (e+f x)^2 \text {csch}(c+d x) \, dx}{a^3}+\frac {(2 e f) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{a d^2}+\frac {\left (2 i b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\left (2 i b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\left (b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {\left (3 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}+\frac {f^2 \int x \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a d}+\frac {\left (3 f^2\right ) \int x \text {sech}(c+d x) \, dx}{a d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {f^2 x \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac {\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac {\left (3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {\left (3 i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (3 i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}-\frac {f^2 \int \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}-\frac {f^2 x \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (3 i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (3 i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {f^2 \int \tan ^{-1}(\sinh (c+d x)) \, dx}{a d^2}+\frac {f^2 \int \text {csch}(c+d x) \, dx}{a d^2}+\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac {\left (2 b^2 f^2\right ) \int \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a^3 d^2}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {3 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (2 b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {\left (2 b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^3}-\frac {f^2 \int d x \text {sech}(c+d x) \, dx}{a d^2}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {6 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {3 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}-\frac {f^2 \int x \text {sech}(c+d x) \, dx}{a d}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {3 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {3 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {3 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac {\left (i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}-\frac {\left (i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac {2 b (e+f x)^2}{a^2 d}-\frac {b^3 (e+f x)^2}{a^2 \left (a^2+b^2\right ) d}+\frac {4 f^2 x \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {4 b^2 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 d^2}+\frac {4 b^4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {2 e f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac {3 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}+\frac {2 b (e+f x)^2 \coth (2 c+2 d x)}{a^2 d}-\frac {e f \text {csch}(c+d x)}{a d^2}-\frac {f^2 x \text {csch}(c+d x)}{a d^2}-\frac {b^5 (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 )^{3/2} d}+\frac {b^5 (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 )^{3/2} d}+\frac {2 b^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {2 b f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a^2 d^2}+\frac {3 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 d^3}-\frac {2 i b^4 f^2 \text {Li}_2\left (-i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 d^3}+\frac {2 i b^4 f^2 \text {Li}_2\left (i e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^3}-\frac {3 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^5 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 )^{3/2} d^2}+\frac {2 b^5 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 )^{3/2} d^2}+\frac {b^3 f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_2\left (e^{4 (c+d x)}\right )}{2 a^2 d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac {3 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \text {Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^5 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 )^{3/2} d^3}-\frac {2 b^5 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 )^{3/2} d^3}-\frac {3 (e+f x)^2 \text {sech}(c+d x)}{2 a d}+\frac {b^2 (e+f x)^2 \text {sech}(c+d x)}{a^3 d}-\frac {b^4 (e+f x)^2 \text {sech}(c+d x)}{a^3 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \tanh (c+d x)}{a^2 \left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [B]  time = 27.06, size = 2574, normalized size = 2.07 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(8*a*b*d^2*e*E^(2*c)*f*x + 4*a*b*d^2*E^(2*c)*f^2*x^2 - 6*a^2*d^2*e^2*ArcTanh[E^(c + d*x)] + 4*b^2*d^2*e^2*ArcT
anh[E^(c + d*x)] + 6*a^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] + 4
*a^2*f^2*ArcTanh[E^(c + d*x)] - 4*a^2*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] + 6*a^2*d^2*e*f*x*Log[1 - E^(c + d*x)]
- 4*b^2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 6*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 4*b^2*d^2*e*E^(2*c)*f*
x*Log[1 - E^(c + d*x)] + 3*a^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 2*b^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 3*a
^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] - 6*a^2*d^2*e*f*x
*Log[1 + E^(c + d*x)] + 4*b^2*d^2*e*f*x*Log[1 + E^(c + d*x)] + 6*a^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] -
4*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 3*a^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 2*b^2*d^2*f^2*x^2*Log[
1 + E^(c + d*x)] + 3*a^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] - 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d
*x)] + 4*a*b*d*e*f*Log[1 - E^(2*(c + d*x))] - 4*a*b*d*e*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + 4*a*b*d*f^2*x*Log
[1 - E^(2*(c + d*x))] - 4*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(2*(c + d*x))] + 2*(3*a^2 - 2*b^2)*d*(-1 + E^(2*c))*f*
(e + f*x)*PolyLog[2, -E^(c + d*x)] - 2*(3*a^2 - 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] +
2*a*b*f^2*PolyLog[2, E^(2*(c + d*x))] - 2*a*b*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] + 6*a^2*f^2*PolyLog[3, -
E^(c + d*x)] - 4*b^2*f^2*PolyLog[3, -E^(c + d*x)] - 6*a^2*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 4*b^2*E^(2*c)
*f^2*PolyLog[3, -E^(c + d*x)] - 6*a^2*f^2*PolyLog[3, E^(c + d*x)] + 4*b^2*f^2*PolyLog[3, E^(c + d*x)] + 6*a^2*
E^(2*c)*f^2*PolyLog[3, E^(c + d*x)] - 4*b^2*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)])/(2*a^3*d^3*(-1 + E^(2*c))) +
(b^5*(2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a
^2 + b^2])] - d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*d*f*(e + f*x)*PolyLog
[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
]))] + 2*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))]))/(a^3*(a^2 + b^2)^(3/2)*d^3) - (2*b*e*f*Sech[c]*(Cosh[c]*Log[Cosh[c]*Cosh[d*x] + Sinh[c]*Sinh[
d*x]] - d*x*Sinh[c]))/((a^2 + b^2)*d^2*(Cosh[c]^2 - Sinh[c]^2)) + (4*a*e*f*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x
)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]])/((a^2 + b^2)*d^2*Sqrt[Cosh[c]^2 - Sinh[c]^2]) - (b*f^2*Csch[c]*((d^2*x^2)/
E^ArcTanh[Coth[c]] - (I*Coth[c]*(-(d*x*(-Pi + (2*I)*ArcTanh[Coth[c]])) - Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*
ArcTanh[Coth[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))] + Pi*Log[Cosh[d*x]] + (2*I)*ArcTanh[Coth[c]]
*Log[I*Sinh[d*x + ArcTanh[Coth[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))]))/Sqrt[1 - Coth[c]
^2])*Sech[c])/((a^2 + b^2)*d^3*Sqrt[Csch[c]^2*(-Cosh[c]^2 + Sinh[c]^2)]) + (2*a*f^2*(((-I)*Csch[c]*(I*(d*x + A
rcTanh[Coth[c]])*(Log[1 - E^(-(d*x) - ArcTanh[Coth[c]])] - Log[1 + E^(-(d*x) - ArcTanh[Coth[c]])]) + I*(PolyLo
g[2, -E^(-(d*x) - ArcTanh[Coth[c]])] - PolyLog[2, E^(-(d*x) - ArcTanh[Coth[c]])])))/Sqrt[1 - Coth[c]^2] - (2*A
rcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]]*ArcTanh[Coth[c]])/Sqrt[Cosh[c]^2 - Sinh[c
]^2]))/((a^2 + b^2)*d^3) + (Csch[c]*Csch[c + d*x]^2*Sech[c]*Sech[c + d*x]*(2*a^3*e*f*Cosh[2*d*x] + 2*a*b^2*e*f
*Cosh[2*d*x] + 2*a^3*f^2*x*Cosh[2*d*x] + 2*a*b^2*f^2*x*Cosh[2*d*x] + 4*a^2*b*d*e^2*Cosh[c - d*x] + 8*a^2*b*d*e
*f*x*Cosh[c - d*x] + 4*a^2*b*d*f^2*x^2*Cosh[c - d*x] + 2*b^3*d*e^2*Cosh[c + d*x] + 4*b^3*d*e*f*x*Cosh[c + d*x]
+ 2*b^3*d*f^2*x^2*Cosh[c + d*x] + 2*b^3*d*e^2*Cosh[3*c + d*x] + 4*b^3*d*e*f*x*Cosh[3*c + d*x] + 2*b^3*d*f^2*x
^2*Cosh[3*c + d*x] - 2*a^3*e*f*Cosh[4*c + 2*d*x] - 2*a*b^2*e*f*Cosh[4*c + 2*d*x] - 2*a^3*f^2*x*Cosh[4*c + 2*d*
x] - 2*a*b^2*f^2*x*Cosh[4*c + 2*d*x] - 4*a^2*b*d*e^2*Cosh[c + 3*d*x] - 2*b^3*d*e^2*Cosh[c + 3*d*x] - 8*a^2*b*d
*e*f*x*Cosh[c + 3*d*x] - 4*b^3*d*e*f*x*Cosh[c + 3*d*x] - 4*a^2*b*d*f^2*x^2*Cosh[c + 3*d*x] - 2*b^3*d*f^2*x^2*C
osh[c + 3*d*x] - 2*b^3*d*e^2*Cosh[3*c + 3*d*x] - 4*b^3*d*e*f*x*Cosh[3*c + 3*d*x] - 2*b^3*d*f^2*x^2*Cosh[3*c +
3*d*x] + 2*a^3*d*e^2*Sinh[2*c] - 2*a*b^2*d*e^2*Sinh[2*c] + 4*a^3*d*e*f*x*Sinh[2*c] - 4*a*b^2*d*e*f*x*Sinh[2*c]
+ 2*a^3*d*f^2*x^2*Sinh[2*c] - 2*a*b^2*d*f^2*x^2*Sinh[2*c] + 3*a^3*d*e^2*Sinh[2*d*x] + a*b^2*d*e^2*Sinh[2*d*x]
+ 6*a^3*d*e*f*x*Sinh[2*d*x] + 2*a*b^2*d*e*f*x*Sinh[2*d*x] + 3*a^3*d*f^2*x^2*Sinh[2*d*x] + a*b^2*d*f^2*x^2*Sin
h[2*d*x] - 3*a^3*d*e^2*Sinh[4*c + 2*d*x] - a*b^2*d*e^2*Sinh[4*c + 2*d*x] - 6*a^3*d*e*f*x*Sinh[4*c + 2*d*x] - 2
*a*b^2*d*e*f*x*Sinh[4*c + 2*d*x] - 3*a^3*d*f^2*x^2*Sinh[4*c + 2*d*x] - a*b^2*d*f^2*x^2*Sinh[4*c + 2*d*x]))/(16
*a^2*(a^2 + b^2)*d^2)

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fricas [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)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 5.70, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right )^{3} \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx \]

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)^2/(a+b*sinh(d*x+c)),x)

[Out]

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

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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)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

2*b*e*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - log(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2)) + 4*a*f^2*integrate(x*e^(
d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) + 4*b*f^2*integrate(x/(a^2*d*e^(2
*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 1/2*(2*b^5*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2
))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/((a^5 + a^3*b^2)*sqrt(a^2 + b^2)*d) + 2*(4*a^2*b*e^(-2*d*x - 2*c) +
2*b^3*e^(-4*d*x - 4*c) - 4*a^2*b - 2*b^3 + (3*a^3 + a*b^2)*e^(-d*x - c) - 2*(a^3 - a*b^2)*e^(-3*d*x - 3*c) +
(3*a^3 + a*b^2)*e^(-5*d*x - 5*c))/((a^4 + a^2*b^2 - (a^4 + a^2*b^2)*e^(-2*d*x - 2*c) - (a^4 + a^2*b^2)*e^(-4*d
*x - 4*c) + (a^4 + a^2*b^2)*e^(-6*d*x - 6*c))*d) - (3*a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (3*a^2 - 2*
b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^2 + 4*a*e*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) - (2*(2*a^2*b*d*f^2 +
b^3*d*f^2)*x^2 + 4*(2*a^2*b*d*e*f + b^3*d*e*f)*x + (2*a^3*e*f*e^(5*c) + 2*a*b^2*e*f*e^(5*c) + (3*a^3*d*f^2*e^(
5*c) + a*b^2*d*f^2*e^(5*c))*x^2 + 2*((3*d*e*f + f^2)*a^3*e^(5*c) + (d*e*f + f^2)*a*b^2*e^(5*c))*x)*e^(5*d*x) -
2*(b^3*d*f^2*x^2*e^(4*c) + 2*b^3*d*e*f*x*e^(4*c))*e^(4*d*x) - 2*((a^3*d*f^2*e^(3*c) - a*b^2*d*f^2*e^(3*c))*x^
2 + 2*(a^3*d*e*f*e^(3*c) - a*b^2*d*e*f*e^(3*c))*x)*e^(3*d*x) - 4*(a^2*b*d*f^2*x^2*e^(2*c) + 2*a^2*b*d*e*f*x*e^
(2*c))*e^(2*d*x) - (2*a^3*e*f*e^c + 2*a*b^2*e*f*e^c - (3*a^3*d*f^2*e^c + a*b^2*d*f^2*e^c)*x^2 - 2*((3*d*e*f -
f^2)*a^3*e^c + (d*e*f - f^2)*a*b^2*e^c)*x)*e^(d*x))/(a^4*d^2 + a^2*b^2*d^2 + (a^4*d^2*e^(6*c) + a^2*b^2*d^2*e^
(6*c))*e^(6*d*x) - (a^4*d^2*e^(4*c) + a^2*b^2*d^2*e^(4*c))*e^(4*d*x) - (a^4*d^2*e^(2*c) + a^2*b^2*d^2*e^(2*c))
*e^(2*d*x)) + (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) + 1/2*(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)))*(3*a^2*f^2 - 2*b^2*f^2)/(a^3*d^3) - 1/2*(d^2*x^2*l
og(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*(3*a^2*f^2 - 2*b^2*f^2)/(a^3*d^3)
+ (3*a^2*d*e*f - 2*b^2*d*e*f - 2*a*b*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) - (3*a^2
*d*e*f - 2*b^2*d*e*f + 2*a*b*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) + 1/6*((3*a^2*f^2
- 2*b^2*f^2)*d^3*x^3 + 3*(3*a^2*d*e*f - 2*b^2*d*e*f + 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/6*((3*a^2*f^2 - 2*b^2
*f^2)*d^3*x^3 + 3*(3*a^2*d*e*f - 2*b^2*d*e*f - 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - integrate(-2*(b^5*f^2*x^2*e^c +
2*b^5*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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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