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

Optimal. Leaf size=982 \[ \frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \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^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^3}{a^2 \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^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^3}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \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^2}{a \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^2}{a \left (a^2+b^2\right ) d^2}+\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) b^2}{a \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}{a^2 d}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right ) b}{a^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right ) b}{a^2 d^2}-\frac {f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right ) b}{2 a^2 d^3}+\frac {f^2 \text {Li}_3\left (e^{2 c+2 d x}\right ) b}{2 a^2 d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^2}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a d^3} \]

[Out]

-b^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/a^2/(a^2+b^2)/d+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/(
a^2+b^2)/d+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d-2*b^3*f^2*polylog(3,-b*exp(d*x
+c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2
)/d^3+2*b^2*(f*x+e)^2*arctan(exp(d*x+c))/a/(a^2+b^2)/d+b*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^2/d^2-b*f*(f*x
+e)*polylog(2,exp(2*d*x+2*c))/a^2/d^2+2*I*f^2*polylog(3,I*exp(d*x+c))/a/d^3+1/2*b^3*f^2*polylog(3,-exp(2*d*x+2
*c))/a^2/(a^2+b^2)/d^3-2*I*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a/d^2+2*I*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^
2-b^3*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/a^2/(a^2+b^2)/d^2-2*I*b^2*f^2*polylog(3,I*exp(d*x+c))/a/(a^2+b^2)/d
^3+2*b*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^2/d-1/2*b*f^2*polylog(3,-exp(2*d*x+2*c))/a^2/d^3-2*I*f^2*polylog(3,
-I*exp(d*x+c))/a/d^3-4*f*(f*x+e)*arctanh(exp(d*x+c))/a/d^2+1/2*b*f^2*polylog(3,exp(2*d*x+2*c))/a^2/d^3+2*I*b^2
*f^2*polylog(3,-I*exp(d*x+c))/a/(a^2+b^2)/d^3-2*I*b^2*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/(a^2+b^2)/d^2+2*f^2
*polylog(2,exp(d*x+c))/a/d^3-(f*x+e)^2*csch(d*x+c)/a/d-2*f^2*polylog(2,-exp(d*x+c))/a/d^3-2*(f*x+e)^2*arctan(e
xp(d*x+c))/a/d+2*I*b^2*f*(f*x+e)*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+
c)/(a-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/(a^
2+b^2)/d^2

________________________________________________________________________________________

Rubi [A]  time = 1.67, antiderivative size = 982, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 21, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.618, Rules used = {5589, 2621, 321, 207, 5462, 6741, 12, 6742, 5205, 4180, 2531, 2282, 6589, 4182, 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^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^3}{a^2 \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^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^3}{a^2 \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^3}{a^2 \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^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac {f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^3}{2 a^2 \left (a^2+b^2\right ) d^3}+\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \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^2}{a \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^2}{a \left (a^2+b^2\right ) d^2}+\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right ) b^2}{a \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}{a^2 d}+\frac {f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right ) b}{a^2 d^2}-\frac {f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right ) b}{a^2 d^2}-\frac {f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right ) b}{2 a^2 d^3}+\frac {f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right ) b}{2 a^2 d^3}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}-\frac {2 f^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{a d^2}+\frac {2 f^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}-\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{a d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d) + (2*b^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*(a^2 + b^2)*d) - (4*f*
(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a^2*d) - ((e + f*x)^2*Cs
ch[c + d*x])/(a*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) + (b
^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d) - (b^3*(e + f*x)^2*Log[1 +
E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + ((2*I)*f*(e + f*x)*PolyLog[
2, (-I)*E^(c + d*x)])/(a*d^2) - ((2*I)*b^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - ((2
*I)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*d^2) + ((2*I)*b^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2
+ b^2)*d^2) + (2*f^2*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a -
Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^
2]))])/(a^2*(a^2 + b^2)*d^2) - (b^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a^2*(a^2 + b^2)*d^2) + (b*f*(e
+ f*x)*PolyLog[2, -E^(2*c + 2*d*x)])/(a^2*d^2) - (b*f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a^2*d^2) - ((2*I
)*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^3) + ((2*I)*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3)
+ ((2*I)*f^2*PolyLog[3, I*E^(c + d*x)])/(a*d^3) - ((2*I)*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^
3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) - (2*b^3*f^2*PolyL
og[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)*d^3) + (b^3*f^2*PolyLog[3, -E^(2*(c + d*x))]
)/(2*a^2*(a^2 + b^2)*d^3) - (b*f^2*PolyLog[3, -E^(2*c + 2*d*x)])/(2*a^2*d^3) + (b*f^2*PolyLog[3, E^(2*c + 2*d*
x)])/(2*a^2*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 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 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 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 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}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}-\frac {b \int (e+f x)^2 \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {(2 f) \int (e+f x) \left (-\frac {\tan ^{-1}(\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a}\\ &=-\frac {(e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}-\frac {(2 b) \int (e+f x)^2 \text {csch}(2 c+2 d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(2 f) \int \frac {(e+f x) \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right )}{d} \, dx}{a}\\ &=-\frac {b^3 (e+f x)^3}{3 a^2 \left (a^2+b^2\right ) f}-\frac {(e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^2 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(2 f) \int (e+f x) \left (-\tan ^{-1}(\sinh (c+d x))-\text {csch}(c+d x)\right ) \, dx}{a d}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^2 d}\\ &=-\frac {b^3 (e+f x)^3}{3 a^2 \left (a^2+b^2\right ) f}-\frac {(e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}+\frac {b^2 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int (e+f x)^2 \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}-\frac {(2 f) \int \left (-(e+f x) \tan ^{-1}(\sinh (c+d x))-(e+f x) \text {csch}(c+d x)\right ) \, dx}{a d}-\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (b f^2\right ) \int \text {Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a^2 d^2}+\frac {\left (b f^2\right ) \int \text {Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a^2 d^2}\\ &=\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \tan ^{-1}(\sinh (c+d x))}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {\left (2 b^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac {(2 f) \int (e+f x) \tan ^{-1}(\sinh (c+d x)) \, dx}{a d}+\frac {(2 f) \int (e+f x) \text {csch}(c+d x) \, dx}{a d}-\frac {\left (2 i b^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac {\left (2 i b^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d}-\frac {\left (b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {\left (b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\left (2 b^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (2 b^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}\\ &=\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\int d (e+f x)^2 \text {sech}(c+d x) \, dx}{a d}+\frac {\left (2 b^3 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (2 b^3 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^2 \left (a^2+b^2\right ) d^3}-\frac {\left (2 b^3 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^2 \left (a^2+b^2\right ) d^3}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 f^2\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 i b^2 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}-\frac {\left (2 i b^2 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\int (e+f x)^2 \text {sech}(c+d x) \, dx}{a}-\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 i b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {\left (2 i b^2 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {\left (b^3 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}\\ &=-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}+\frac {2 i b^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {(2 i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a d}-\frac {(2 i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a d}+\frac {\left (b^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}\\ &=-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \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 )}{a d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}+\frac {2 i b^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {b^3 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \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 )}{a d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}+\frac {2 i b^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {b^3 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}-\frac {\left (2 i f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 i f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}-\frac {4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {(e+f x)^2 \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {2 f^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^2}-\frac {2 i b^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \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 )}{a d^2}+\frac {2 i b^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {Li}_2\left (-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {Li}_2\left (e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac {b^3 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {Li}_3\left (-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {Li}_3\left (e^{2 c+2 d x}\right )}{2 a^2 d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 11.77, size = 1971, normalized size = 2.01 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(12*b*d^3*e^2*E^(2*c)*x - 12*b*d^3*e^2*(1 + E^(2*c))*x - 12*b*d^3*e*f*x^2 - 4*b*d^3*f^2*x^3 + 12*a*d^2*e^
2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 6*b*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*a*
d*e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + Po
lyLog[2, I*E^(c + d*x)]) + 6*b*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*
(c + d*x))]) + (6*I)*a*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c + d*x)] - 2*
d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLo
g[3, I*E^(c + d*x)]) + b*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog[2,
-E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))]))/((a^2 + b^2)*d^3*(1 + E^(2*c))) - (b^3*(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*Lo
g[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*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)]))]))/(3*a^2*(a^2 + b^2)*d^3*(-1 + E^(2*c))) + (b*d^3*(e + f*x)^3*(-1 + Coth[
c]) + 3*d*e*f*(b*d*e - 2*a*f)*(d*x - Log[1 - Cosh[c + d*x] - Sinh[c + d*x]]) - 6*d*f^2*(b*d*e + a*f)*x*Log[1 +
Cosh[c + d*x] - Sinh[c + d*x]] - 3*b*d^2*f^3*x^2*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]] - 6*d*f^2*(b*d*e - a*
f)*x*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]] - 3*b*d^2*f^3*x^2*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]] + 3*d*e*f
*(b*d*e + 2*a*f)*(d*x - Log[1 + Cosh[c + d*x] + Sinh[c + d*x]]) + 6*f^2*(b*d*e - a*f)*PolyLog[2, Cosh[c + d*x]
- Sinh[c + d*x]] + 6*f^2*(b*d*e + a*f)*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]] + 6*b*f^3*(d*x*PolyLog[2, C
osh[c + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x] - Sinh[c + d*x]]) + 6*b*f^3*(d*x*PolyLog[2, -Cosh[c +
d*x] + Sinh[c + d*x]] + PolyLog[3, -Cosh[c + d*x] + Sinh[c + d*x]]))/(3*a^2*d^3*f) + ((-3*a*b*d*e^2*x - 3*a*b
*d*e*f*x^2 - a*b*d*f^2*x^3 - 3*a^2*e^2*Cosh[c] - 3*b^2*e^2*Cosh[c] - 6*a^2*e*f*x*Cosh[c] - 6*b^2*e*f*x*Cosh[c]
- 3*a^2*f^2*x^2*Cosh[c] - 3*b^2*f^2*x^2*Cosh[c])*Csch[c/2]*Sech[c/2]*Sech[c])/(6*a*(a^2 + b^2)*d) + (Csch[c/2
]*Csch[c/2 + (d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d) + (Sech[c/2
]*Sech[c/2 + (d*x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]))/(2*a*d)

________________________________________________________________________________________

fricas [C]  time = 0.63, size = 5666, normalized size = 5.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 2.84, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \mathrm {csch}\left (d x +c \right )^{2} \mathrm {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ {\left (\frac {b^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + a^{2} b^{2}\right )} d} + \frac {2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d}\right )} e^{2} - \frac {2 \, {\left (f^{2} x^{2} e^{c} + 2 \, e f x e^{c}\right )} e^{\left (d x\right )}}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - \frac {2 \, e f \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {2 \, e f \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} - \frac {{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} b f^{2}}{a^{2} d^{3}} - \frac {{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} b f^{2}}{a^{2} d^{3}} - \frac {2 \, {\left (b d e f + a f^{2}\right )} {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )}}{a^{2} d^{3}} - \frac {2 \, {\left (b d e f - a f^{2}\right )} {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )}}{a^{2} d^{3}} + \frac {b d^{3} f^{2} x^{3} + 3 \, {\left (b d e f + a f^{2}\right )} d^{2} x^{2}}{3 \, a^{2} d^{3}} + \frac {b d^{3} f^{2} x^{3} + 3 \, {\left (b d e f - a f^{2}\right )} d^{2} x^{2}}{3 \, a^{2} d^{3}} - \int \frac {2 \, {\left (b^{4} f^{2} x^{2} + 2 \, b^{4} e f x - {\left (a b^{3} f^{2} x^{2} e^{c} + 2 \, a b^{3} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{4} b + a^{2} b^{3} - {\left (a^{4} b e^{\left (2 \, c\right )} + a^{2} b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{5} e^{c} + a^{3} b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int \frac {2 \, {\left (b f^{2} x^{2} + 2 \, b e f x + {\left (a f^{2} x^{2} e^{c} + 2 \, a e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} + b^{2} + {\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + a^2*b^2)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 + b^
2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-
d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e^2 - 2*(f^2*x^2*e^c + 2*e*f*x*e^c)*e^(d*x)/(a*d*e^(2
*d*x + 2*c) - a*d) - 2*e*f*log(e^(d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/(a*d^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)))*b*f^2/(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)))*b*f^2/(a^2*d^3) - 2*(b*d*e*f + a*f^2)*(d*x
*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d*e*f - a*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilo
g(e^(d*x + c)))/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 +
3*(b*d*e*f - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(2*(b^4*f^2*x^2 + 2*b^4*e*f*x - (a*b^3*f^2*x^2*e^c + 2*a*b^
3*e*f*x*e^c)*e^(d*x))/(a^4*b + a^2*b^3 - (a^4*b*e^(2*c) + a^2*b^3*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + a^3*b^2*e^
c)*e^(d*x)), x) - integrate(2*(b*f^2*x^2 + 2*b*e*f*x + (a*f^2*x^2*e^c + 2*a*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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________