\(\int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [304]

Optimal result

Integrand size = 26, antiderivative size = 786 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 \left (a^2+b^2\right ) d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 786, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5692, 5680, 2221, 2611, 6744, 2320, 6724, 6874, 4265, 3799} \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 d^4 \left (a^2+b^2\right )}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d^3 \left (a^2+b^2\right )}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {b (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )}-\frac {b (e+f x)^3 \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )} \]

[In]

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

[Out]

(2*a*(e + f*x)^3*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 +
b^2])])/((a^2 + b^2)*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)*d) - (b
*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d) - ((3*I)*a*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/
((a^2 + b^2)*d^2) + ((3*I)*a*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*P
olyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -((b*E^
(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*(a
^2 + b^2)*d^2) + ((6*I)*a*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) - ((6*I)*a*f^2*(e + f*
x)*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^
2 + b^2]))])/((a^2 + b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^
2 + b^2)*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^3) - ((6*I)*a*f^3*PolyLog[4,
(-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) + ((6*I)*a*f^3*PolyLog[4, I*E^(c + d*x)])/((a^2 + b^2)*d^4) + (6*b*f^3*P
olyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)*d^4) - (3*b*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*(a^2 + b^2)*d^4)

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 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 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 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 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 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2} \\ & = -\frac {b (e+f x)^4}{4 \left (a^2+b^2\right ) f}+\frac {\int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2+b^2}+\frac {b^2 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^2+b^2} \\ & = -\frac {b (e+f x)^4}{4 \left (a^2+b^2\right ) f}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {a \int (e+f x)^3 \text {sech}(c+d x) \, dx}{a^2+b^2}-\frac {b \int (e+f x)^3 \tanh (c+d x) \, dx}{a^2+b^2}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {(2 b) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{a^2+b^2}-\frac {(3 i a f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {(3 i a f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d^2} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (6 i a f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (6 i a f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right ) d^3} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {\left (3 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {\left (6 i a f^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}+\frac {\left (6 i a f^3\right ) \int \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^3}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}-\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {\left (6 i a f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {\left (3 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) \, dx}{2 \left (a^2+b^2\right ) d^3} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}-\frac {\left (3 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 \left (a^2+b^2\right ) d^4} \\ & = \frac {2 a (e+f x)^3 \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {b (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 i a f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^2}+\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i a f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^4}-\frac {3 b f^3 \operatorname {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 \left (a^2+b^2\right ) d^4} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3078\) vs. \(2(786)=1572\).

Time = 11.35 (sec) , antiderivative size = 3078, normalized size of antiderivative = 3.92 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(8*b*d^4*e^3*E^(2*c)*x + 12*b*d^4*e^2*E^(2*c)*f*x^2 + 8*b*d^4*e*E^(2*c)*f^2*x^3 + 2*b*d^4*E^(2*c)*f^3*x^4 + 8*
a*d^3*e^3*ArcTan[E^(c + d*x)] + 8*a*d^3*e^3*E^(2*c)*ArcTan[E^(c + d*x)] + (12*I)*a*d^3*e^2*f*x*Log[1 - I*E^(c
+ d*x)] + (12*I)*a*d^3*e^2*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (12*I)*a*d^3*e*f^2*x^2*Log[1 - I*E^(c + d*x)]
+ (12*I)*a*d^3*e*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (4*I)*a*d^3*f^3*x^3*Log[1 - I*E^(c + d*x)] + (4*I)*a
*d^3*E^(2*c)*f^3*x^3*Log[1 - I*E^(c + d*x)] - (12*I)*a*d^3*e^2*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e^2*E
^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e*f^2*x^2*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^3*e*E^(2*c)*f^2
*x^2*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*f^3*x^3*Log[1 + I*E^(c + d*x)] - (4*I)*a*d^3*E^(2*c)*f^3*x^3*Log[1 +
I*E^(c + d*x)] - 4*b*d^3*e^3*Log[1 + E^(2*(c + d*x))] - 4*b*d^3*e^3*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 12*b*d
^3*e^2*f*x*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] - 12*b*d^3*e*f^2*x^2*L
og[1 + E^(2*(c + d*x))] - 12*b*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - 4*b*d^3*f^3*x^3*Log[1 + E^(2*(
c + d*x))] - 4*b*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(2*(c + d*x))] - (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyL
og[2, (-I)*E^(c + d*x)] + (12*I)*a*d^2*(1 + E^(2*c))*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)] - 6*b*d^2*e^2*f*P
olyLog[2, -E^(2*(c + d*x))] - 6*b*d^2*e^2*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] - 12*b*d^2*e*f^2*x*PolyLog[2,
-E^(2*(c + d*x))] - 12*b*d^2*e*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 6*b*d^2*f^3*x^2*PolyLog[2, -E^(2*
(c + d*x))] - 6*b*d^2*E^(2*c)*f^3*x^2*PolyLog[2, -E^(2*(c + d*x))] + (24*I)*a*d*e*f^2*PolyLog[3, (-I)*E^(c + d
*x)] + (24*I)*a*d*e*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (24*I)*a*d*f^3*x*PolyLog[3, (-I)*E^(c + d*x)] +
(24*I)*a*d*E^(2*c)*f^3*x*PolyLog[3, (-I)*E^(c + d*x)] - (24*I)*a*d*e*f^2*PolyLog[3, I*E^(c + d*x)] - (24*I)*a
*d*e*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (24*I)*a*d*f^3*x*PolyLog[3, I*E^(c + d*x)] - (24*I)*a*d*E^(2*c)*f
^3*x*PolyLog[3, I*E^(c + d*x)] + 6*b*d*e*f^2*PolyLog[3, -E^(2*(c + d*x))] + 6*b*d*e*E^(2*c)*f^2*PolyLog[3, -E^
(2*(c + d*x))] + 6*b*d*f^3*x*PolyLog[3, -E^(2*(c + d*x))] + 6*b*d*E^(2*c)*f^3*x*PolyLog[3, -E^(2*(c + d*x))] -
(24*I)*a*f^3*PolyLog[4, (-I)*E^(c + d*x)] - (24*I)*a*E^(2*c)*f^3*PolyLog[4, (-I)*E^(c + d*x)] + (24*I)*a*f^3*
PolyLog[4, I*E^(c + d*x)] + (24*I)*a*E^(2*c)*f^3*PolyLog[4, I*E^(c + d*x)] - 3*b*f^3*PolyLog[4, -E^(2*(c + d*x
))] - 3*b*E^(2*c)*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*(a^2 + b^2)*d^4*(1 + E^(2*c))) - (b*(4*e^3*E^(2*c)*x +
6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f^3*x^4 + (4*a*Sqrt[a^2 + b^2]*e^3*ArcTan[(a + b*E^(c + d*
x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (4*a*Sqrt[-a^2 - b^2]*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a
^2 + b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) - (2*e^3*E^(2*c)*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))])/d + (2*e^3
*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2
+ b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d +
(6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 +
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt
[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])]
)/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1
+ (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2
*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*
Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*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)]))])/d^2 - (6*(-1 + E^(2*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)]))])/d^2 - (12*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^
c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 +
b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 +
(12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*e*f^2*PolyLo
g[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(
a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*
c)]))])/d^3 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*
f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 + (12*f^3*PolyLog[4, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4))/(2*(a^2 + b^2)*(-1 + E^(2*c))) + (b*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f
^3*x^3)*Csch[c/2]*Sech[c/2]*Sech[c])/(8*(a^2 + b^2))

Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \operatorname {sech}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1718 vs. \(2 (720) = 1440\).

Time = 0.29 (sec) , antiderivative size = 1718, normalized size of antiderivative = 2.19 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(6*b*f^3*polylog(4, (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) + 6*b*f^3*polylog(4, (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) + 3*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*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*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x
+ b*d^2*e^2*f)*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*(-I*a*d^2*f^3*x^2 + b*d^2*f^3*x^2 - 2*I*a*d^2*e*f^2*x + 2*b*d^2*e*f^2*x - I*a*d^2*e^2*f
+ b*d^2*e^2*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - 3*(I*a*d^2*f^3*x^2 + b*d^2*f^3*x^2 + 2*I*a*d^2*e*f^
2*x + 2*b*d^2*e*f^2*x + I*a*d^2*e^2*f + b*d^2*e^2*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + (b*d^3*e^3 -
3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*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*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*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*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*
e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*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*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f -
3*b*c^2*d*e*f^2 + b*c^3*f^3)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((a^2 + b^2)/b^2) - b)/b) + (I*a*d^3*e^3 - b*d^3*e^3 - 3*I*a*c*d^2*e^2*f + 3*b*c*d^2*e^2*f + 3*I*a*c^2*d*e*f^
2 - 3*b*c^2*d*e*f^2 - I*a*c^3*f^3 + b*c^3*f^3)*log(cosh(d*x + c) + sinh(d*x + c) + I) + (-I*a*d^3*e^3 - b*d^3*
e^3 + 3*I*a*c*d^2*e^2*f + 3*b*c*d^2*e^2*f - 3*I*a*c^2*d*e*f^2 - 3*b*c^2*d*e*f^2 + I*a*c^3*f^3 + b*c^3*f^3)*log
(cosh(d*x + c) + sinh(d*x + c) - I) + (-I*a*d^3*f^3*x^3 - b*d^3*f^3*x^3 - 3*I*a*d^3*e*f^2*x^2 - 3*b*d^3*e*f^2*
x^2 - 3*I*a*d^3*e^2*f*x - 3*b*d^3*e^2*f*x - 3*I*a*c*d^2*e^2*f - 3*b*c*d^2*e^2*f + 3*I*a*c^2*d*e*f^2 + 3*b*c^2*
d*e*f^2 - I*a*c^3*f^3 - b*c^3*f^3)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (I*a*d^3*f^3*x^3 - b*d^3*f^3*x
^3 + 3*I*a*d^3*e*f^2*x^2 - 3*b*d^3*e*f^2*x^2 + 3*I*a*d^3*e^2*f*x - 3*b*d^3*e^2*f*x + 3*I*a*c*d^2*e^2*f - 3*b*c
*d^2*e^2*f - 3*I*a*c^2*d*e*f^2 + 3*b*c^2*d*e*f^2 + I*a*c^3*f^3 - b*c^3*f^3)*log(-I*cosh(d*x + c) - I*sinh(d*x
+ c) + 1) - 6*(-I*a*f^3 + b*f^3)*polylog(4, I*cosh(d*x + c) + I*sinh(d*x + c)) - 6*(I*a*f^3 + b*f^3)*polylog(4
, -I*cosh(d*x + c) - I*sinh(d*x + c)) - 6*(b*d*f^3*x + b*d*e*f^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c
) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*(b*d*f^3*x + b*d*e*f^2)*polylog(3, (a*co
sh(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) - 6*(I*a*d*f^3*x
- b*d*f^3*x + I*a*d*e*f^2 - b*d*e*f^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) - 6*(-I*a*d*f^3*x - b*d*
f^3*x - I*a*d*e*f^2 - b*d*e*f^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x + c)))/((a^2 + b^2)*d^4)

Sympy [F]

\[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \operatorname {sech}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

integrate((f*x+e)**3*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*sech(c + d*x)/(a + b*sinh(c + d*x)), x)

Maxima [F]

\[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

-e^3*(2*a*arctan(e^(-d*x - c))/((a^2 + b^2)*d) - b*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2 + b^2
)*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d)) + integrate(4*f^3*x^3/((b*(e^(d*x + c) - e^(-d*x - c)) + 2
*a)*(e^(d*x + c) + e^(-d*x - c))) + 12*e*f^2*x^2/((b*(e^(d*x + c) - e^(-d*x - c)) + 2*a)*(e^(d*x + c) + e^(-d*
x - c))) + 12*e^2*f*x/((b*(e^(d*x + c) - e^(-d*x - c)) + 2*a)*(e^(d*x + c) + e^(-d*x - c))), x)

Giac [F]

\[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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