3.5.40 \(\int \frac {(e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [440]

Optimal. Leaf size=1164 \[ -\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

[Out]

6*I*b^2*f^2*(f*x+e)*polylog(2,I*exp(d*x+c))/a/(a^2+b^2)/d^3-3*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2+3*f*(f*
x+e)^2*polylog(2,exp(d*x+c))/a/d^2+6*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-6*f*(f*x+e)^2*arctan(exp(d*x+c))
/a/d^2-6*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3-b*(f*x+e)^3/(a^2+b^2)/d+(f*x+e)^3*sech(d*x+c)/a/d+6*I*b^2*f^3
*polylog(3,-I*exp(d*x+c))/a/(a^2+b^2)/d^4-6*I*b^2*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/(a^2+b^2)/d^3-2*(f*x+
e)^3*arctanh(exp(d*x+c))/a/d-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*polylog(4,exp(d*x+c))/a/d^4+3*b*f^2*(f*x
+e)*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)/d^3+6*I*f^3*polylog(3,I*exp(d*x+c))/a/d^4-b^2*(f*x+e)^3*sech(d*x+c)/a
/(a^2+b^2)/d-6*I*f^2*(f*x+e)*polylog(2,I*exp(d*x+c))/a/d^3-3*b^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b
^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^2+3*b^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(
3/2)/d^2+6*b^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^3-6*b^3*f^2*(f*x+e
)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^3-3/2*b*f^3*polylog(3,-exp(2*d*x+2*c))/(a^2
+b^2)/d^4-b*(f*x+e)^3*tanh(d*x+c)/(a^2+b^2)/d-6*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+6*b^2*f*(f*x+e)^2*arctan(
exp(d*x+c))/a/(a^2+b^2)/d^2+6*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3-6*I*b^2*f^3*polylog(3,I*exp(d*x+c))
/a/(a^2+b^2)/d^4+3*b*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/(a^2+b^2)/d^2-b^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^
2)^(1/2)))/a/(a^2+b^2)^(3/2)/d+b^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d-6*b^3*
f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^4+6*b^3*f^3*polylog(4,-b*exp(d*x+c)/(a+(a
^2+b^2)^(1/2)))/a/(a^2+b^2)^(3/2)/d^4

________________________________________________________________________________________

Rubi [A]
time = 1.76, antiderivative size = 1164, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 22, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {5708, 2702, 327, 213, 5570, 6873, 12, 6874, 6408, 4267, 2611, 6744, 2320, 6724, 4265, 5692, 3403, 2296, 2221, 4269, 3799, 5559} \begin {gather*} -\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \text {sech}(c+d x) b^2}{a \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 b}{\left (a^2+b^2\right ) d}+\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^2}+\frac {3 f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac {3 f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b}{2 \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \tanh (c+d x) b}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(e + f*x)^3)/((a^2 + b^2)*d)) - (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (6*b^2*f*(e + f*x)^2*ArcT
an[E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^3*Log[1 + (
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (3*b*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) -
(3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((6*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^
3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog
[2, I*E^(c + d*x)])/(a*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (3*f*(
e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) + (6*f^2*(e
+ f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - ((6*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*b^2*f^3*
PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) - ((6*I)*b^2
*f^3*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b
^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (6*b^3*f^2*
(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (3*b*f^3*PolyLog[3
, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^4) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*PolyLog[4, E^(c +
d*x)])/(a*d^4) - (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) +
(6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) + ((e + f*x)^3*Sec
h[c + d*x])/(a*d) - (b^2*(e + f*x)^3*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^3*Tanh[c + d*x])/((a^2 +
b^2)*d)

Rule 12

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

Rule 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 2296

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

Rule 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 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 2702

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

Rule 3403

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

Rule 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 4269

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

Rule 5559

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

Rule 5570

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

Rule 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 6408

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((a + b*ArcTan
h[u])/(d*(m + 1))), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/(1 - u^2)), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

Rule 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 6744

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

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)^3 \text {csch}(c+d x) \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {csch}(c+d x) \text {sech}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^3 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b \int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac {b^3 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int (e+f x)^2 \left (-\frac {\tanh ^{-1}(\cosh (c+d x))}{d}+\frac {\text {sech}(c+d x)}{d}\right ) \, dx}{a}\\ &=-\frac {(e+f x)^3 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b \int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int \frac {(e+f x)^2 \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right )}{d} \, dx}{a}\\ &=-\frac {(e+f x)^3 \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {\left (2 b^4\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 b^4\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}-\frac {b \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{a^2+b^2}+\frac {b^2 \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac {(3 f) \int (e+f x)^2 \left (-\tanh ^{-1}(\cosh (c+d x))+\text {sech}(c+d x)\right ) \, dx}{a d}\\ &=-\frac {(e+f x)^3 \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(3 f) \int \left (-(e+f x)^2 \tanh ^{-1}(\cosh (c+d x))+(e+f x)^2 \text {sech}(c+d x)\right ) \, dx}{a d}+\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}-\frac {\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}+\frac {(3 b f) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a \left (a^2+b^2\right ) d}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^3 \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {(3 f) \int (e+f x)^2 \tanh ^{-1}(\cosh (c+d x)) \, dx}{a d}-\frac {(3 f) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{a d}+\frac {(6 b f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (6 b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 b^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac {\left (6 i b^2 f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\int d (-e-f x)^3 \text {csch}(c+d x) \, dx}{a d}+\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 i f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (6 b^3 f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (6 b^3 f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (6 i b^2 f^3\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}-\frac {\left (6 i b^2 f^3\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a \left (a^2+b^2\right ) d^3}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\int (-e-f x)^3 \text {csch}(c+d x) \, dx}{a}-\frac {\left (6 b^3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {\left (6 b^3 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {\left (6 i b^2 f^3\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^4}-\frac {\left (6 i b^2 f^3\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^4}-\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 i f^3\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{a d^3}-\frac {\left (3 b f^3\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^3}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 i b^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 i b^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(3 f) \int (-e-f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(3 f) \int (-e-f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}-\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac {\left (3 b f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^2\right ) \int (-e-f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac {\left (6 f^2\right ) \int (-e-f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac {b (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}+\frac {6 b^2 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {b^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d}+\frac {3 b f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {6 i f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a d^3}-\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}-\frac {6 i f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a d^3}+\frac {6 i b^2 f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^3}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac {3 b f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 i f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a d^4}+\frac {6 i b^2 f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}+\frac {6 i f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a d^4}-\frac {6 i b^2 f^3 \text {Li}_3\left (i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {6 b^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac {3 b f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {6 b^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac {(e+f x)^3 \text {sech}(c+d x)}{a d}-\frac {b^2 (e+f x)^3 \text {sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [A]
time = 16.32, size = 1991, normalized size = 1.71 \begin {gather*} 4 \left (\frac {f \text {csch}(c+d x) \left (-\frac {4 b d^3 e^{2 c} x \left (3 e^2+3 e f x+f^2 x^2\right )}{1+e^{2 c}}+3 \left (-4 a d^2 e^2 \text {ArcTan}\left (e^{c+d x}\right )-4 i a d^2 e f x \log \left (1-i e^{c+d x}\right )-2 i a d^2 f^2 x^2 \log \left (1-i e^{c+d x}\right )+4 i a d^2 e f x \log \left (1+i e^{c+d x}\right )+2 i a d^2 f^2 x^2 \log \left (1+i e^{c+d x}\right )+2 b d^2 e^2 \log \left (1+e^{2 (c+d x)}\right )+4 b d^2 e f x \log \left (1+e^{2 (c+d x)}\right )+2 b d^2 f^2 x^2 \log \left (1+e^{2 (c+d x)}\right )+4 i a d f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )-4 i a d f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )+2 b d e f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+2 b d f^2 x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-4 i a f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )+4 i a f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )-b f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right ) (a+b \sinh (c+d x))}{8 \left (a^2+b^2\right ) d^4 (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \left (d^3 e^3 \log \left (1-e^{c+d x}\right )+3 d^3 e^2 f x \log \left (1-e^{c+d x}\right )+3 d^3 e f^2 x^2 \log \left (1-e^{c+d x}\right )+d^3 f^3 x^3 \log \left (1-e^{c+d x}\right )-d^3 e^3 \log \left (1+e^{c+d x}\right )-3 d^3 e^2 f x \log \left (1+e^{c+d x}\right )-3 d^3 e f^2 x^2 \log \left (1+e^{c+d x}\right )-d^3 f^3 x^3 \log \left (1+e^{c+d x}\right )-3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )+3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )+6 d e f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+6 d f^3 x \text {PolyLog}\left (3,-e^{c+d x}\right )-6 d e f^2 \text {PolyLog}\left (3,e^{c+d x}\right )-6 d f^3 x \text {PolyLog}\left (3,e^{c+d x}\right )-6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )+6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )\right ) (a+b \sinh (c+d x))}{4 a d^4 (b+a \text {csch}(c+d x))}+\frac {b^3 \text {csch}(c+d x) \left (2 d^3 e^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )+3 \sqrt {-a^2-b^2} d^3 e^2 e^c 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 )+3 \sqrt {-a^2-b^2} d^3 e e^c 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 )+\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^3 e^2 e^c 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 )-3 \sqrt {-a^2-b^2} d^3 e e^c 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 )-\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \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 )-3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \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 )-6 \sqrt {-a^2-b^2} d e e^c 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 )-6 \sqrt {-a^2-b^2} d e^c f^3 x \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 )+6 \sqrt {-a^2-b^2} d e e^c 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 )+6 \sqrt {-a^2-b^2} d e^c f^3 x \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 )+6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right ) (a+b \sinh (c+d x))}{4 a \left (-a^2-b^2\right )^{3/2} d^4 \sqrt {\left (a^2+b^2\right ) e^{2 c}} (b+a \text {csch}(c+d x))}+\frac {\text {csch}(c+d x) \text {sech}(c) \text {sech}(c+d x) \left (a e^3 \cosh (c)+3 a e^2 f x \cosh (c)+3 a e f^2 x^2 \cosh (c)+a f^3 x^3 \cosh (c)-b e^3 \sinh (d x)-3 b e^2 f x \sinh (d x)-3 b e f^2 x^2 \sinh (d x)-b f^3 x^3 \sinh (d x)\right ) (a+b \sinh (c+d x))}{4 \left (a^2+b^2\right ) d (b+a \text {csch}(c+d x))}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*((f*Csch[c + d*x]*((-4*b*d^3*E^(2*c)*x*(3*e^2 + 3*e*f*x + f^2*x^2))/(1 + E^(2*c)) + 3*(-4*a*d^2*e^2*ArcTan[E
^(c + d*x)] - (4*I)*a*d^2*e*f*x*Log[1 - I*E^(c + d*x)] - (2*I)*a*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (4*I)*a*
d^2*e*f*x*Log[1 + I*E^(c + d*x)] + (2*I)*a*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] + 2*b*d^2*e^2*Log[1 + E^(2*(c +
d*x))] + 4*b*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 2*b*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + (4*I)*a*d*f*(e +
f*x)*PolyLog[2, (-I)*E^(c + d*x)] - (4*I)*a*d*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)] + 2*b*d*e*f*PolyLog[2, -E^
(2*(c + d*x))] + 2*b*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - (4*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (4*I)*a
*f^2*PolyLog[3, I*E^(c + d*x)] - b*f^2*PolyLog[3, -E^(2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/(8*(a^2 + b^2)*d^
4*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*(d^3*e^3*Log[1 - E^(c + d*x)] + 3*d^3*e^2*f*x*Log[1 - E^(c + d*x)] +
3*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] + d^3*f^3*x^3*Log[1 - E^(c + d*x)] - d^3*e^3*Log[1 + E^(c + d*x)] - 3*d^
3*e^2*f*x*Log[1 + E^(c + d*x)] - 3*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] - d^3*f^3*x^3*Log[1 + E^(c + d*x)] - 3*d
^2*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)] + 3*d^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)] + 6*d*e*f^2*PolyLog[3,
-E^(c + d*x)] + 6*d*f^3*x*PolyLog[3, -E^(c + d*x)] - 6*d*e*f^2*PolyLog[3, E^(c + d*x)] - 6*d*f^3*x*PolyLog[3,
E^(c + d*x)] - 6*f^3*PolyLog[4, -E^(c + d*x)] + 6*f^3*PolyLog[4, E^(c + d*x)])*(a + b*Sinh[c + d*x]))/(4*a*d^
4*(b + a*Csch[c + d*x])) + (b^3*Csch[c + d*x]*(2*d^3*e^3*Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTan[(a + b*E^(c + d*x))/
Sqrt[-a^2 - b^2]] + 3*Sqrt[-a^2 - b^2]*d^3*e^2*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(
2*c)])] + 3*Sqrt[-a^2 - b^2]*d^3*e*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])]
+ Sqrt[-a^2 - b^2]*d^3*E^c*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*Sqrt[-a^
2 - b^2]*d^3*e^2*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*Sqrt[-a^2 - b^2]*d
^3*e*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - Sqrt[-a^2 - b^2]*d^3*E^c*f^3
*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 3*Sqrt[-a^2 - b^2]*d^2*E^c*f*(e + f*x)^2
*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 3*Sqrt[-a^2 - b^2]*d^2*E^c*f*(e + f*x)
^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[-a^2 - b^2]*d*e*E^c*f^2*PolyL
og[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[-a^2 - b^2]*d*E^c*f^3*x*PolyLog[3, -(
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[-a^2 - b^2]*d*e*E^c*f^2*PolyLog[3, -((b*E^(2*
c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[-a^2 - b^2]*d*E^c*f^3*x*PolyLog[3, -((b*E^(2*c + d*x)
)/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[-a^2 - b^2]*E^c*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[-a^2 - b^2]*E^c*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b
^2)*E^(2*c)]))])*(a + b*Sinh[c + d*x]))/(4*a*(-a^2 - b^2)^(3/2)*d^4*Sqrt[(a^2 + b^2)*E^(2*c)]*(b + a*Csch[c +
d*x])) + (Csch[c + d*x]*Sech[c]*Sech[c + d*x]*(a*e^3*Cosh[c] + 3*a*e^2*f*x*Cosh[c] + 3*a*e*f^2*x^2*Cosh[c] + a
*f^3*x^3*Cosh[c] - b*e^3*Sinh[d*x] - 3*b*e^2*f*x*Sinh[d*x] - 3*b*e*f^2*x^2*Sinh[d*x] - b*f^3*x^3*Sinh[d*x])*(a
+ b*Sinh[c + d*x]))/(4*(a^2 + b^2)*d*(b + a*Csch[c + d*x])))

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Maple [F]
time = 2.72, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a*f^3*integrate(x^2*e^(d*x + c)/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 6*b*f
^3*integrate(x^2/(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 12*b*f^2*e*integrate(x/
(a^2*d*e^(2*d*x + 2*c) + b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - 3*b*f*(2*(d*x + c)/((a^2 + b^2)*d^2) - l
og(e^(2*d*x + 2*c) + 1)/((a^2 + b^2)*d^2))*e^2 - 12*a*f^2*integrate(x*e^(d*x + c + 1)/(a^2*d*e^(2*d*x + 2*c) +
b^2*d*e^(2*d*x + 2*c) + a^2*d + b^2*d), x) - (b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c)
- a + sqrt(a^2 + b^2)))/((a^3 + a*b^2)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^2 + b^2)*e
^(-2*d*x - 2*c))*d) + log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^3 - 6*a*f*arctan(e^(d*x + c
))*e^2/((a^2 + b^2)*d^2) + 2*(b*f^3*x^3 + 3*b*f^2*x^2*e + 3*b*f*x*e^2 + (a*f^3*x^3*e^c + 3*a*f^2*x^2*e^(c + 1)
+ 3*a*f*x*e^(c + 2))*e^(d*x))/(a^2*d + b^2*d + (a^2*d*e^(2*c) + b^2*d*e^(2*c))*e^(2*d*x)) - 3*(d*x*log(e^(d*x
+ c) + 1) + dilog(-e^(d*x + c)))*f*e^2/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))*f*e^2/(a*
d^2) - 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)))*f^2*e/(a*d^3)
+ 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)))*f^2*e/(a*d^3) - (d
^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e
^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^
(d*x + c)) + 6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) - integrate(-2*(b^3*f^3*x^3*e^c + 3*b^3*f^2*x^2*e^(c + 1)
+ 3*b^3*f*x*e^(c + 2))*e^(d*x)/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 2*(a^4*e^c + a^2*b
^2*e^c)*e^(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. 16781 vs. \(2 (1087) = 2174\).
time = 0.72, size = 16781, normalized size = 14.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a^3*b + a*b^3)*c^3*f^3 - 6*(a^3*b + a*b^3)*c^2*d*f^2*cosh(1) + 6*(a^3*b + a*b^3)*c*d^2*f*cosh(1)^2 - 2*(a
^3*b + a*b^3)*d^3*cosh(1)^3 - 2*(a^3*b + a*b^3)*d^3*sinh(1)^3 + 2*((a^3*b + a*b^3)*d^3*f^3*x^3 + (a^3*b + a*b^
3)*c^3*f^3 + 3*((a^3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*c*d^2*f)*cosh(1)^2 + 3*((a^3*b + a*b^3)*d^3*f*x + (a
^3*b + a*b^3)*c*d^2*f)*sinh(1)^2 + 3*((a^3*b + a*b^3)*d^3*f^2*x^2 - (a^3*b + a*b^3)*c^2*d*f^2)*cosh(1) + 3*((a
^3*b + a*b^3)*d^3*f^2*x^2 - (a^3*b + a*b^3)*c^2*d*f^2 + 2*((a^3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*c*d^2*f)*
cosh(1))*sinh(1))*cosh(d*x + c)^2 + 6*((a^3*b + a*b^3)*c*d^2*f - (a^3*b + a*b^3)*d^3*cosh(1))*sinh(1)^2 + 2*((
a^3*b + a*b^3)*d^3*f^3*x^3 + (a^3*b + a*b^3)*c^3*f^3 + 3*((a^3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*c*d^2*f)*c
osh(1)^2 + 3*((a^3*b + a*b^3)*d^3*f*x + (a^3*b + a*b^3)*c*d^2*f)*sinh(1)^2 + 3*((a^3*b + a*b^3)*d^3*f^2*x^2 -
(a^3*b + a*b^3)*c^2*d*f^2)*cosh(1) + 3*((a^3*b + a*b^3)*d^3*f^2*x^2 - (a^3*b + a*b^3)*c^2*d*f^2 + 2*((a^3*b +
a*b^3)*d^3*f*x + (a^3*b + a*b^3)*c*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 3*(b^4*d^2*f^3*x^2 + 2*b^4*d^2*f
^2*x*cosh(1) + b^4*d^2*f*cosh(1)^2 + b^4*d^2*f*sinh(1)^2 + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*d^
2*f*cosh(1)^2 + b^4*d^2*f*sinh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(b^4*
d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*d^2*f*cosh(1)^2 + b^4*d^2*f*sinh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2
*f*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*d^2*f*cosh
(1)^2 + b^4*d^2*f*sinh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 2*(b^4*d^2*f^2*
x + b^4*d^2*f*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^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) - 3*(b^4*d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*
d^2*f*cosh(1)^2 + b^4*d^2*f*sinh(1)^2 + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*d^2*f*cosh(1)^2 + b^4
*d^2*f*sinh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(b^4*d^2*f^3*x^2 + 2*b^4
*d^2*f^2*x*cosh(1) + b^4*d^2*f*cosh(1)^2 + b^4*d^2*f*sinh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(1))*sinh(1)
)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*f^2*x*cosh(1) + b^4*d^2*f*cosh(1)^2 + b^4*d^2*f*s
inh(1)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 2*(b^4*d^2*f^2*x + b^4*d^2*f*cosh(
1))*sinh(1))*sqrt((a^2 + b^2)/b^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) + (b^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4
*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + (b^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^
3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + 3*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2
*f*cosh(1) + b^4*d^3*cosh(1)^2)*sinh(1))*cosh(d*x + c)^2 + 3*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 + 2*(b^
4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + 3*(b^4
*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1)^2)*sinh(1))
*cosh(d*x + c)*sinh(d*x + c) + (b^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh
(1)^3 - b^4*d^3*sinh(1)^3 + 3*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cos
h(1) + b^4*d^3*cosh(1)^2)*sinh(1))*sinh(d*x + c)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1
)^2)*sinh(1))*sqrt((a^2 + b^2)/b^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^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3
+ (b^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + 3
*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1)^2)*sin
h(1))*cosh(d*x + c)^2 + 3*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 + 2*(b^4*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1)
+ 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + 3*(b^4*c*d^2*f - b^4*d^3*cosh(1))*sinh(1)
^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1)^2)*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (b^4
*c^3*f^3 - 3*b^4*c^2*d*f^2*cosh(1) + 3*b^4*c*d^2*f*cosh(1)^2 - b^4*d^3*cosh(1)^3 - b^4*d^3*sinh(1)^3 + 3*(b^4*
c*d^2*f - b^4*d^3*cosh(1))*sinh(1)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1)^2)*sinh(1))*
sinh(d*x + c)^2 - 3*(b^4*c^2*d*f^2 - 2*b^4*c*d^2*f*cosh(1) + b^4*d^3*cosh(1)^2)*sinh(1))*sqrt((a^2 + b^2)/b^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^4*d^3*f^3*x^3 + b^4*c^3*f^3
+ 3*(b^4*d^3*f*x + b^4*c*d^2*f)*cosh(1)^2 + (b^4*d^3*f^3*x^3 + b^4*c^3*f^3 + 3*(b^4*d^3*f*x + b^4*c*d^2*f)*co
sh(1)^2 + 3*(b^4*d^3*f*x + b^4*c*d^2*f)*sinh(1)^2 + 3*(b^4*d^3*f^2*x^2 - b^4*c^2*d*f^2)*cosh(1) + 3*(b^4*d^3*f
^2*x^2 - b^4*c^2*d*f^2 + 2*(b^4*d^3*f*x + b^4*c...

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Sympy [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)**3*csch(d*x+c)*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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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)^3*csch(d*x+c)*sech(d*x+c)^2/(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 )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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