3.5.65 \(\int \frac {(e+f x)^2 \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [465]

Optimal. Leaf size=982 \[ -\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x)^2 \text {ArcTan}\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 {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 b^2 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{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 )}{a d^2}+\frac {2 i b^2 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b f (e+f x) \text {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {b f (e+f x) \text {PolyLog}\left (2,e^{2 c+2 d x}\right )}{a^2 d^2}-\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}+\frac {2 i b^2 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{a d^3}-\frac {2 i b^2 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^3}-\frac {b f^2 \text {PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a^2 d^3}+\frac {b f^2 \text {PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a^2 d^3} \]

[Out]

-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*(f*x+e)*polylog(2
,I*exp(d*x+c))/a/(a^2+b^2)/d^2+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*(f*x+e)^2*arctan(exp(d*x+c))/a/d-2*f^2*polylog(2,-exp(d*x+c))/a/d
^3+2*f^2*polylog(2,exp(d*x+c))/a/d^3-(f*x+e)^2*csch(d*x+c)/a/d-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-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+2*I*f*(f*x+e)*polyl
og(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*e
xp(d*x+c))/a/(a^2+b^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*b*(f*x+e)^2*arctanh(exp(2*d*x+2*c))/a^2/d+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 = 1.26, 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 = {5708, 2701, 327, 213, 5570, 6873, 12, 6874, 5313, 4265, 2611, 2320, 6724, 4267, 2317, 2438, 5569, 5692, 5680, 2221, 3799} \begin {gather*} \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 \text {ArcTan}\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 \text {ArcTan}\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} \end {gather*}

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 213

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

Rule 327

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

Rule 2221

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

Rule 2317

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

Rule 2320

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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

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 2701

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

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5313

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 5569

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 5570

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

Rule 5680

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5692

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

Rule 5708

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

Rule 6724

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 6873

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

Rule 6874

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^3}+\frac {\left (2 f^2\right ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 9.86, size = 1467, normalized size = 1.49 \begin {gather*} -\frac {12 b d^3 e^2 e^{2 c} x+12 b d^3 e e^{2 c} f x^2+4 b d^3 e^{2 c} f^2 x^3+12 a d^2 e^2 \left (1+e^{2 c}\right ) \text {ArcTan}\left (e^{c+d x}\right )-6 b d^2 e^2 \left (1+e^{2 c}\right ) \log \left (1+e^{2 (c+d x)}\right )+12 i a d e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\text {PolyLog}\left (2,-i e^{c+d x}\right )+\text {PolyLog}\left (2,i e^{c+d x}\right )\right )-6 b d e \left (1+e^{2 c}\right ) f \left (2 d x \log \left (1+e^{2 (c+d x)}\right )+\text {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )+6 i a \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,i e^{c+d x}\right )+2 \text {PolyLog}\left (3,-i e^{c+d x}\right )-2 \text {PolyLog}\left (3,i e^{c+d x}\right )\right )-3 b \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \log \left (1+e^{2 (c+d x)}\right )+2 d x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-\text {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )}{6 \left (a^2+b^2\right ) d^3 \left (1+e^{2 c}\right )}+\frac {-12 b e^2 x+\frac {12 b e^2 e^{2 c} x}{-1+e^{2 c}}+\frac {12 b e f x^2}{-1+e^{2 c}}+\frac {4 b f^2 x^3}{-1+e^{2 c}}-\frac {24 a e f \tanh ^{-1}\left (e^{c+d x}\right )}{d^2}+\frac {6 b e^2 \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )}{d}+\frac {12 a f^2 \left (d x \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-\text {PolyLog}\left (2,-e^{c+d x}\right )+\text {PolyLog}\left (2,e^{c+d x}\right )\right )}{d^3}+\frac {6 b e f \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )}{d^2}+\frac {b f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1-e^{2 (c+d x)}\right )\right )-6 d x \text {PolyLog}\left (2,e^{2 (c+d x)}\right )+3 \text {PolyLog}\left (3,e^{2 (c+d x)}\right )\right )}{d^3}}{6 a^2}+\frac {b^3 \left (-\frac {2 e^{2 c} x \left (3 e^2+3 e f x+f^2 x^2\right )}{-1+e^{2 c}}+\frac {3 \left (d^2 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 d f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-2 f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{d^3}\right )}{3 a^2 \left (a^2+b^2\right )}+\frac {\left (-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)\right ) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right ) \text {sech}(c)}{6 a \left (a^2+b^2\right ) d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \end {gather*}

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*E^(2*c)*f*x^2 + 4*b*d^3*E^(2*c)*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))*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)] + PolyLog[2, I*E^(c + d*x)])
- 6*b*d*e*(1 + E^(2*c))*f*(2*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*PolyLog[3, I*E^(c + d*x)]) - 3*b*(1 + E
^(2*c))*f^2*(2*d^2*x^2*Log[1 + E^(2*(c + d*x))] + 2*d*x*PolyLog[2, -E^(2*(c + d*x))] - PolyLog[3, -E^(2*(c + d
*x))]))/((a^2 + b^2)*d^3*(1 + E^(2*c))) + (-12*b*e^2*x + (12*b*e^2*E^(2*c)*x)/(-1 + E^(2*c)) + (12*b*e*f*x^2)/
(-1 + E^(2*c)) + (4*b*f^2*x^3)/(-1 + E^(2*c)) - (24*a*e*f*ArcTanh[E^(c + d*x)])/d^2 + (6*b*e^2*(2*d*x - Log[1
- E^(2*(c + d*x))]))/d + (12*a*f^2*(d*x*(Log[1 - E^(c + d*x)] - Log[1 + E^(c + d*x)]) - PolyLog[2, -E^(c + d*x
)] + PolyLog[2, E^(c + d*x)]))/d^3 + (6*b*e*f*(2*d*x*(d*x - Log[1 - E^(2*(c + d*x))]) - PolyLog[2, E^(2*(c + d
*x))]))/d^2 + (b*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*P
olyLog[3, E^(2*(c + d*x))]))/d^3)/(6*a^2) + (b^3*((-2*E^(2*c)*x*(3*e^2 + 3*e*f*x + f^2*x^2))/(-1 + E^(2*c)) +
(3*(d^2*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*d^2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 + b^2)*E^(2*c)])] + 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)])] + 2*d^
2*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 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)])] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^
2)*E^(2*c)]))] + 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*
PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 2*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a
*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/d^3))/(3*a^2*(a^2 + b^2)) + ((-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*Cos
h[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)

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Maple [F]
time = 2.24, 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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8198 vs. \(2 (914) = 1828\).
time = 0.57, size = 8198, normalized size = 8.35 \begin {gather*} \text {Too large to display} \end {gather*}

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\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 \end {gather*}

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)

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