3.4.53 \(\int \frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [353]

Optimal. Leaf size=917 \[ \frac {(e+f x)^3}{b d}-\frac {a^2 (e+f x)^3}{b \left (a^2+b^2\right ) d}+\frac {6 a f (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac {a 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 )^{3/2} d}+\frac {a 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 )^{3/2} d}-\frac {3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {3 a^2 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {6 i a 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 a 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 a b 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 b 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 f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}+\frac {3 a^2 f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {6 i a f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 i a f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 a b 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 a b 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 f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^4}-\frac {3 a^2 f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^4}-\frac {6 a b 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 a b 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 {a (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b \left (a^2+b^2\right ) d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.38, antiderivative size = 917, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 14, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5702, 4269, 3799, 2221, 2611, 2320, 6724, 5692, 3403, 2296, 6744, 6874, 5559, 4265} \begin {gather*} \frac {6 i a \text {Li}_3\left (-i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}-\frac {6 i a \text {Li}_3\left (i e^{c+d x}\right ) f^3}{\left (a^2+b^2\right ) d^4}+\frac {3 \text {Li}_3\left (-e^{2 (c+d x)}\right ) f^3}{2 b d^4}-\frac {3 a^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) f^3}{2 b \left (a^2+b^2\right ) d^4}-\frac {6 a b \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^3}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {6 a b \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^3}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 i a (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}+\frac {6 i a (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) f^2}{\left (a^2+b^2\right ) d^3}-\frac {3 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) f^2}{b d^3}+\frac {3 a^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) f^2}{b \left (a^2+b^2\right ) d^3}+\frac {6 a b (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f^2}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {6 a b (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f^2}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 a (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) f}{\left (a^2+b^2\right ) d^2}-\frac {3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) f}{b d^2}+\frac {3 a^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) f}{b \left (a^2+b^2\right ) d^2}-\frac {3 a b (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) f}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {3 a b (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) f}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {(e+f x)^3}{b d}-\frac {a^2 (e+f x)^3}{b \left (a^2+b^2\right ) d}-\frac {a b (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {a b (e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \tanh (c+d x)}{b d}-\frac {a^2 (e+f x)^3 \tanh (c+d x)}{b \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e + f*x)^3/(b*d) - (a^2*(e + f*x)^3)/(b*(a^2 + b^2)*d) + (6*a*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)
*d^2) - (a*b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) + (a*b*(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*f*(e + f*x)^2*Log[1 + E^(2*(c +
d*x))])/(b*d^2) + (3*a^2*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b*(a^2 + b^2)*d^2) - ((6*I)*a*f^2*(e + f*x)
*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^3) + ((6*I)*a*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b
^2)*d^3) - (3*a*b*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*b*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*f^2*
(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*d^3) + (3*a^2*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b*(a^2 +
b^2)*d^3) + ((6*I)*a*f^3*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^4) - ((6*I)*a*f^3*PolyLog[3, I*E^(c + d
*x)])/((a^2 + b^2)*d^4) + (6*a*b*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*a*b*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*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*d^4) - (3*a^2*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b*(a^2 +
b^2)*d^4) - (6*a*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) + (6*a*b*
f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (a*(e + f*x)^3*Sech[c + d*
x])/((a^2 + b^2)*d) + ((e + f*x)^3*Tanh[c + d*x])/(b*d) - (a^2*(e + f*x)^3*Tanh[c + d*x])/(b*(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 5702

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

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1928\) vs. \(2(917)=1834\).
time = 14.52, size = 1928, normalized size = 2.10 \begin {gather*} \frac {f \left (12 b d^3 e^2 e^{2 c} x+12 b d^3 e e^{2 c} f x^2+4 b d^3 e^{2 c} f^2 x^3+12 a d^2 e^2 \text {ArcTan}\left (e^{c+d x}\right )+12 a d^2 e^2 e^{2 c} \text {ArcTan}\left (e^{c+d x}\right )+12 i a d^2 e f x \log \left (1-i e^{c+d x}\right )+12 i a d^2 e e^{2 c} f x \log \left (1-i e^{c+d x}\right )+6 i a d^2 f^2 x^2 \log \left (1-i e^{c+d x}\right )+6 i a d^2 e^{2 c} f^2 x^2 \log \left (1-i e^{c+d x}\right )-12 i a d^2 e f x \log \left (1+i e^{c+d x}\right )-12 i a d^2 e e^{2 c} f x \log \left (1+i e^{c+d x}\right )-6 i a d^2 f^2 x^2 \log \left (1+i e^{c+d x}\right )-6 i a d^2 e^{2 c} f^2 x^2 \log \left (1+i e^{c+d x}\right )-6 b d^2 e^2 \log \left (1+e^{2 (c+d x)}\right )-6 b d^2 e^2 e^{2 c} \log \left (1+e^{2 (c+d x)}\right )-12 b d^2 e f x \log \left (1+e^{2 (c+d x)}\right )-12 b d^2 e e^{2 c} f x \log \left (1+e^{2 (c+d x)}\right )-6 b d^2 f^2 x^2 \log \left (1+e^{2 (c+d x)}\right )-6 b d^2 e^{2 c} f^2 x^2 \log \left (1+e^{2 (c+d x)}\right )-12 i a d \left (1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )+12 i a d \left (1+e^{2 c}\right ) f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )-6 b d e f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-6 b d e e^{2 c} f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-6 b d f^2 x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-6 b d e^{2 c} f^2 x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )+12 i a f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )+12 i a e^{2 c} f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )-12 i a f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )-12 i a e^{2 c} f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )+3 b f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )+3 b e^{2 c} f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}+\frac {a b \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 )}{\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 (-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 )}{\left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(12*b*d^3*e^2*E^(2*c)*x + 12*b*d^3*e*E^(2*c)*f*x^2 + 4*b*d^3*E^(2*c)*f^2*x^3 + 12*a*d^2*e^2*ArcTan[E^(c + d
*x)] + 12*a*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + (12*I)*a*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (12*I)*a*d^2*e*E
^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*a*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (6*I)*a*d^2*E^(2*c)*f^2*x^2*L
og[1 - I*E^(c + d*x)] - (12*I)*a*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (12*I)*a*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c
+ d*x)] - (6*I)*a*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (6*I)*a*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - 6*
b*d^2*e^2*Log[1 + E^(2*(c + d*x))] - 6*b*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 12*b*d^2*e*f*x*Log[1 + E^(
2*(c + d*x))] - 12*b*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] - 6*b*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] - 6
*b*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (12*I)*a*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c +
d*x)] + (12*I)*a*d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)] - 6*b*d*e*f*PolyLog[2, -E^(2*(c + d*x))
] - 6*b*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] - 6*b*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - 6*b*d*E^(2*c)*
f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (12*I)*a*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (12*I)*a*E^(2*c)*f^2*PolyLog[
3, (-I)*E^(c + d*x)] - (12*I)*a*f^2*PolyLog[3, I*E^(c + d*x)] - (12*I)*a*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)]
+ 3*b*f^2*PolyLog[3, -E^(2*(c + d*x))] + 3*b*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))]))/(2*(a^2 + b^2)*d^4*(1
+ E^(2*c))) + (a*b*(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*L
og[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*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]*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 - 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 + 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]*(-(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^2 + b^2)*d)

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Maple [F]
time = 2.28, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sech(d*x+c)*tanh(d*x+c)/(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) - lo
g(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) - (a*b*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*(a*e^(-d*x - c) - b)/((a^2 + b^2 + (a^2 + b^2)*e^(-2*d*x - 2*
c))*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)) - integrate(-2*(a*b*f^3*x^3*e^c + 3*a*b*f^2*x^2*e^(c + 1) + 3*a*b*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. 10811 vs. \(2 (864) = 1728\).
time = 0.57, size = 10811, normalized size = 11.79 \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)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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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} \tanh {\left (c + d x \right )} \operatorname {sech}{\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)*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {tanh}\left (c+d\,x\right )\,{\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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