3.4.9 \(\int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [309]

Optimal. Leaf size=780 \[ \frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i b f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i b f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {3 a f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}+\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.27, antiderivative size = 780, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5692, 3403, 2296, 2221, 2611, 6744, 2320, 6724, 6874, 4269, 3799, 5559, 4265} \begin {gather*} -\frac {6 b f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac {6 i b f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 i b f^3 \text {Li}_3\left (i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 d^4 \left (a^2+b^2\right )}+\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 a f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (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 )^{3/2}}-\frac {b^2 (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 )^{3/2}}+\frac {a (e+f x)^3 \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^3}{d \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 14.50, size = 1531, normalized size = 1.96 \begin {gather*} -\frac {f \left (-12 a d^3 e^2 e^{2 c} x+12 a d^3 e^2 \left (1+e^{2 c}\right ) x+12 a d^3 e f x^2+4 a d^3 f^2 x^3+12 b d^2 e^2 \left (1+e^{2 c}\right ) \text {ArcTan}\left (e^{c+d x}\right )-6 a d^2 e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+12 i b d e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\text {PolyLog}\left (2,-i e^{c+d x}\right )+\text {PolyLog}\left (2,i e^{c+d x}\right )\right )-6 a d e \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\text {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \text {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \text {PolyLog}\left (2,i e^{c+d x}\right )+2 \text {PolyLog}\left (3,-i e^{c+d x}\right )-2 \text {PolyLog}\left (3,i e^{c+d x}\right )\right )-a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right )-6 d x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}-\frac {b^2 \left (2 d^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} (e+f x)^3 \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )-3 d^2 f (e+f x)^2 \left (d x \left (2 \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 )+\sqrt {-a^2-b^2} e^c \left (-\log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )\right )-\sqrt {-a^2-b^2} e^c \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 )+\sqrt {-a^2-b^2} e^c \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 )\right )+3 d f^2 (e+f x) \left (d^2 x^2 \left (2 \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 )+\sqrt {-a^2-b^2} e^c \left (-\log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )\right )-2 \sqrt {-a^2-b^2} e^c \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+2 \sqrt {-a^2-b^2} e^c \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )-f^3 \left (d^3 x^3 \left (2 \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 )+\sqrt {-a^2-b^2} e^c \left (-\log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )\right )-6 \sqrt {-a^2-b^2} e^c \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 \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 )\right )}{\left (-a^2-b^2\right )^{3/2} d^4 \sqrt {\left (a^2+b^2\right ) e^{2 c}}}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (b e^3 \cosh (c)+3 b e^2 f x \cosh (c)+3 b e f^2 x^2 \cosh (c)+b f^3 x^3 \cosh (c)+a e^3 \sinh (d x)+3 a e^2 f x \sinh (d x)+3 a e f^2 x^2 \sinh (d x)+a f^3 x^3 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(f*(-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^
2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I
)*b*d*e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)]
+ PolyLog[2, I*E^(c + d*x)]) - 6*a*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E
^(2*(c + d*x))]) + (6*I)*b*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c + d*x)]
- 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*Po
lyLog[3, I*E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog
[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))])))/((a^2 + b^2)*d^4*(1 + E^(2*c))) - (b^2*(2*d^3*Sqrt[(
a^2 + b^2)*E^(2*c)]*(e + f*x)^3*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] - 3*d^2*f*(e + f*x)^2*(d*x*(2*Sqr
t[(a^2 + b^2)*E^(2*c)]*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] + Sqrt[-a^2 - b^2]*E^c*(-Log[1 + (b*E^(2*c
+ d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])]
)) - Sqrt[-a^2 - b^2]*E^c*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + Sqrt[-a^2 - b
^2]*E^c*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]) + 3*d*f^2*(e + f*x)*(d^2*x^2*(2*
Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] + Sqrt[-a^2 - b^2]*E^c*(-Log[1 + (b*E^(
2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)
])])) - 2*Sqrt[-a^2 - b^2]*E^c*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 2*Sqrt[-
a^2 - b^2]*E^c*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]) - f^3*(d^3*x^3*(2*Sqrt[(a
^2 + b^2)*E^(2*c)]*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] + Sqrt[-a^2 - b^2]*E^c*(-Log[1 + (b*E^(2*c + d
*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + Log[1 + (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*PolyLog[4, -((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*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])))/((-a^2 - b^2)^(3/2)*d^4*Sqrt[(
a^2 + b^2)*E^(2*c)]) + (Sech[c]*Sech[c + d*x]*(b*e^3*Cosh[c] + 3*b*e^2*f*x*Cosh[c] + 3*b*e*f^2*x^2*Cosh[c] + b
*f^3*x^3*Cosh[c] + a*e^3*Sinh[d*x] + 3*a*e^2*f*x*Sinh[d*x] + 3*a*e*f^2*x^2*Sinh[d*x] + a*f^3*x^3*Sinh[d*x]))/(
(a^2 + b^2)*d)

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Maple [F]
time = 2.46, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b*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*a*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*a*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*a*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*b*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^2*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c)
- a + sqrt(a^2 + b^2)))/((a^2 + b^2)^(3/2)*d) + 2*(b*e^(-d*x - c) + a)/((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2
*c))*d))*e^3 - 6*b*f*arctan(e^(d*x + c))*e^2/((a^2 + b^2)*d^2) - 2*(a*f^3*x^3 + 3*a*f^2*x^2*e + 3*a*f*x*e^2 -
(b*f^3*x^3*e^c + 3*b*f^2*x^2*e^(c + 1) + 3*b*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)) + integrate(-2*(b^2*f^3*x^3*e^c + 3*b^2*f^2*x^2*e^(c + 1) + 3*b^2*f*x*e^(c + 2))*e^(d*x)/(a
^2*b + b^3 - (a^2*b*e^(2*c) + b^3*e^(2*c))*e^(2*d*x) - 2*(a^3*e^c + a*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. 10547 vs. \(2 (726) = 1452\).
time = 0.59, size = 10547, normalized size = 13.52 \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*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{3} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*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\,\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*(a + b*sinh(c + d*x))),x)

[Out]

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

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