3.4.14 \(\int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [314]

Optimal. Leaf size=928 \[ \frac {2 a b^2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}+\frac {a (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}+\frac {b f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i a f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i a b^2 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i a f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {b^3 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a b^2 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 i a b^2 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {2 b^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {2 b^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {b^3 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {b (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {b f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \]

[Out]

-I*a*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^2-2*I*a*b^2*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)^2/d^3+2*
a*b^2*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)^2/d-b^3*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)^2/d^2+I*a*
f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^3+a*f*(f*x+e)*sech(d*x+c)/(a^2+b^2)/d^2-b*f*(f*x+e)*tanh(d*x+c)/(a^2+
b^2)/d^2+1/2*a*(f*x+e)^2*sech(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d-I*a*f^2*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^3+a*(
f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)
^2/d^2+2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^2-a*f^2*arctan(sinh(d*x+c))/
(a^2+b^2)/d^3+I*a*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+1/2*b^3*f^2*polylog(3,-exp(2*d*x+2*c))/(a^2+
b^2)^2/d^3+1/2*b*(f*x+e)^2*sech(d*x+c)^2/(a^2+b^2)/d+2*I*a*b^2*f*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d
^2-b^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/(a^2+b^2)^2/d+b*f^2*ln(cosh(d*x+c))/(a^2+b^2)/d^3+b^3*(f*x+e)^2*ln(1+b*e
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^2/
d-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^3-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+
(a^2+b^2)^(1/2)))/(a^2+b^2)^2/d^3+2*I*a*b^2*f^2*polylog(3,-I*exp(d*x+c))/(a^2+b^2)^2/d^3-2*I*a*b^2*f*(f*x+e)*p
olylog(2,-I*exp(d*x+c))/(a^2+b^2)^2/d^2

________________________________________________________________________________________

Rubi [A]
time = 1.35, antiderivative size = 928, normalized size of antiderivative = 1.00, number of steps used = 39, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5692, 5680, 2221, 2611, 2320, 6724, 6874, 4265, 3799, 4271, 3855, 5559, 4269, 3556} \begin {gather*} \frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) b^3}{\left (a^2+b^2\right )^2 d}-\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}+\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) b^3}{2 \left (a^2+b^2\right )^2 d^3}+\frac {2 a (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d}-\frac {2 i a f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^2}+\frac {2 i a f^2 \text {Li}_3\left (-i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}-\frac {2 i a f^2 \text {Li}_3\left (i e^{c+d x}\right ) b^2}{\left (a^2+b^2\right )^2 d^3}+\frac {(e+f x)^2 \text {sech}^2(c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac {f^2 \log (\cosh (c+d x)) b}{\left (a^2+b^2\right ) d^3}-\frac {f (e+f x) \tanh (c+d x) b}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {a f^2 \text {ArcTan}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i a f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {i a f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {i a f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {i a f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {a f (e+f x) \text {sech}(c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {a (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*b^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) + (a*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)
*d) - (a*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2])])/((a^2 + b^2)^2*d) + (b^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2
*d) - (b^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) + (b*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d
^3) - ((2*I)*a*b^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a*f*(e + f*x)*PolyLog[2,
(-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) + ((2*I)*a*b^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2
) + (I*a*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) + (2*b^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (b^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + ((2
*I)*a*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2
+ b^2)*d^3) - ((2*I)*a*b^2*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*a*f^2*PolyLog[3, I*E^(c +
d*x)])/((a^2 + b^2)*d^3) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3
) - (2*b^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (b^3*f^2*PolyLog[3,
-E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^3) + (a*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) + (b*(e + f*x)^2*S
ech[c + d*x]^2)/(2*(a^2 + b^2)*d) - (b*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)*d^2) + (a*(e + f*x)^2*Sech[c +
d*x]*Tanh[c + d*x])/(2*(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 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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

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 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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 4271

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-b^2)*(c + d*x)^m*Cot[e
+ f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))), Int[(c +
d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)^m*(b*Csc[e + f*x])^
(n - 2), x], x] - Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

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 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 6874

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3368\) vs. \(2(928)=1856\).
time = 25.77, size = 3368, normalized size = 3.63 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/6*(-12*b^3*d^3*e^2*E^(2*c)*x + 12*a^2*b*d*E^(2*c)*f^2*x + 12*b^3*d*E^(2*c)*f^2*x - 12*b^3*d^3*e*E^(2*c)*f*x
^2 - 4*b^3*d^3*E^(2*c)*f^2*x^3 - 6*a^3*d^2*e^2*ArcTan[E^(c + d*x)] - 18*a*b^2*d^2*e^2*ArcTan[E^(c + d*x)] - 6*
a^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 18*a*b^2*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 12*a^3*f^2*ArcTan[E^(
c + d*x)] + 12*a*b^2*f^2*ArcTan[E^(c + d*x)] + 12*a^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] + 12*a*b^2*E^(2*c)*f^2*A
rcTan[E^(c + d*x)] - (6*I)*a^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] - (18*I)*a*b^2*d^2*e*f*x*Log[1 - I*E^(c + d*x)
] - (6*I)*a^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] - (18*I)*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)]
- (3*I)*a^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] - (9*I)*a*b^2*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] - (3*I)*a^3*d
^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] - (9*I)*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (6*I)*a^3
*d^2*e*f*x*Log[1 + I*E^(c + d*x)] + (18*I)*a*b^2*d^2*e*f*x*Log[1 + I*E^(c + d*x)] + (6*I)*a^3*d^2*e*E^(2*c)*f*
x*Log[1 + I*E^(c + d*x)] + (18*I)*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] + (3*I)*a^3*d^2*f^2*x^2*Log[1
+ I*E^(c + d*x)] + (9*I)*a*b^2*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] + (3*I)*a^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E
^(c + d*x)] + (9*I)*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*b^3*d^2*e^2*Log[1 + E^(2*(c + d*x))]
+ 6*b^3*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 6*a^2*b*f^2*Log[1 + E^(2*(c + d*x))] - 6*b^3*f^2*Log[1 + E^
(2*(c + d*x))] - 6*a^2*b*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 6*b^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] + 1
2*b^3*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 12*b^3*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*b^3*d^2*f^2*x
^2*Log[1 + E^(2*(c + d*x))] + 6*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + (6*I)*a*(a^2 + 3*b^2)*d*(1
+ E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] - (6*I)*a*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)*PolyLo
g[2, I*E^(c + d*x)] + 6*b^3*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 6*b^3*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x
))] + 6*b^3*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*b^3*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] - (6*I)*
a^3*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (18*I)*a*b^2*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^3*E^(2*c)*f^2*P
olyLog[3, (-I)*E^(c + d*x)] - (18*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^3*f^2*PolyLog[3,
I*E^(c + d*x)] + (18*I)*a*b^2*f^2*PolyLog[3, I*E^(c + d*x)] + (6*I)*a^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)]
+ (18*I)*a*b^2*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*b^3*f^2*PolyLog[3, -E^(2*(c + d*x))] - 3*b^3*E^(2*c)
*f^2*PolyLog[3, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^3*(1 + E^(2*c))) - (b^3*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x
^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 +
b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^
(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d)
+ (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) +
(3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2
*(c + d*x)))])/d + (6*e*f*x*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
*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*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 + (6*e*f*x*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*x
*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E
^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*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 - (6*(-1 + E^(2*c))*f*(e + f*x)*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)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])
/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyL
og[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(
a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*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))/(3*(a^2 + b^2)^2*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*Sech[c + d*x]^2*(-6*a^2*b*e*f - 6
*b^3*e*f + 12*b^3*d^2*e^2*x - 6*a^2*b*f^2*x - 6*b^3*f^2*x + 12*b^3*d^2*e*f*x^2 + 4*b^3*d^2*f^2*x^3 + 6*a^2*b*e
*f*Cosh[2*c] + 6*b^3*e*f*Cosh[2*c] + 6*a^2*b*f^2*x*Cosh[2*c] + 6*b^3*f^2*x*Cosh[2*c] + 6*a^2*b*e*f*Cosh[2*d*x]
+ 6*b^3*e*f*Cosh[2*d*x] + 6*a^2*b*f^2*x*Cosh[2*d*x] + 6*b^3*f^2*x*Cosh[2*d*x] - 3*a^3*d*e^2*Cosh[c - d*x] - 3
*a*b^2*d*e^2*Cosh[c - d*x] - 6*a^3*d*e*f*x*Cosh[c - d*x] - 6*a*b^2*d*e*f*x*Cosh[c - d*x] - 3*a^3*d*f^2*x^2*Cos
h[c - d*x] - 3*a*b^2*d*f^2*x^2*Cosh[c - d*x] + ...

________________________________________________________________________________________

Maple [F]
time = 1.31, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {sech}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*
d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 3*a*b^2*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*
d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x)
+ 2*b^3*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*
c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*b^3*d^2*f*e*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d
^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 2*a^3*d^2*f*integrate(
x*e^(d*x + c + 1)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2
+ 2*a^2*b^2*d^2 + b^4*d^2), x) + 6*a*b^2*d^2*f*integrate(x*e^(d*x + c + 1)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b
^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - a^2*b*f^2*(2*(d*x
+ c)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - b^3*f^2*(2*(d*x
+ c)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - 2*a^3*f^2*arct
an(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) - 2*a*b^2*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3)
+ (b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - b^3*log(e^(-2*d*x - 2*c)
+ 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (a^3 + 3*a*b^2)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (a*e^(-d
*x - c) + 2*b*e^(-2*d*x - 2*c) - a*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2
)*e^(-4*d*x - 4*c))*d))*e^2 + (2*b*f^2*x + 2*b*f*e + (a*d*f^2*x^2*e^(3*c) + 2*a*f*e^(3*c + 1) + 2*(a*f^2*e^(3*
c) + a*d*f*e^(3*c + 1))*x)*e^(3*d*x) + 2*(b*d*f^2*x^2*e^(2*c) + b*f*e^(2*c + 1) + (b*f^2*e^(2*c) + 2*b*d*f*e^(
2*c + 1))*x)*e^(2*d*x) - (a*d*f^2*x^2*e^c - 2*a*f*e^(c + 1) + 2*(a*d*f*e^(c + 1) - a*f^2*e^c)*x)*e^(d*x))/(a^2
*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*
x)) - integrate(2*(b^4*f^2*x^2 + 2*b^4*f*x*e - (a*b^3*f^2*x^2*e^c + 2*a*b^3*f*x*e^(c + 1))*e^(d*x))/(a^4*b + 2
*a^2*b^3 + b^5 - (a^4*b*e^(2*c) + 2*a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 2*a^3*b^2*e^c + a*
b^4*e^c)*e^(d*x)), x)

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 15652 vs. \(2 (871) = 1742\).
time = 0.62, size = 15652, normalized size = 16.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x + c)^4 + 4*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^
3)*c*f^2)*sinh(d*x + c)^4 + 4*(a^2*b + b^3)*c*f^2 - 4*(a^2*b + b^3)*d*f*cosh(1) - 2*((a^3 + a*b^2)*d^2*f^2*x^2
+ 2*(a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d^2*cosh(1)^2 + (a^3 + a*b^2)*d^2*sinh(1)^2 + 2*((a^3 + a*b^2)*d^2*
f*x + (a^3 + a*b^2)*d*f)*cosh(1) + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*d^2*cosh(1) + (a^3 + a*b^2)*d*f)*s
inh(1))*cosh(d*x + c)^3 - 4*(a^2*b + b^3)*d*f*sinh(1) - 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d*f^2*x
+ (a^3 + a*b^2)*d^2*cosh(1)^2 + (a^3 + a*b^2)*d^2*sinh(1)^2 + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*d*f)*c
osh(1) - 8*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*c*f^2)*cosh(d*x + c) + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b
^2)*d^2*cosh(1) + (a^3 + a*b^2)*d*f)*sinh(1))*sinh(d*x + c)^3 - 4*((a^2*b + b^3)*d^2*f^2*x^2 - (a^2*b + b^3)*d
*f^2*x + (a^2*b + b^3)*d^2*cosh(1)^2 + (a^2*b + b^3)*d^2*sinh(1)^2 - 2*(a^2*b + b^3)*c*f^2 + (2*(a^2*b + b^3)*
d^2*f*x + (a^2*b + b^3)*d*f)*cosh(1) + (2*(a^2*b + b^3)*d^2*f*x + 2*(a^2*b + b^3)*d^2*cosh(1) + (a^2*b + b^3)*
d*f)*sinh(1))*cosh(d*x + c)^2 - 2*(2*(a^2*b + b^3)*d^2*f^2*x^2 - 2*(a^2*b + b^3)*d*f^2*x + 2*(a^2*b + b^3)*d^2
*cosh(1)^2 + 2*(a^2*b + b^3)*d^2*sinh(1)^2 - 4*(a^2*b + b^3)*c*f^2 - 12*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)
*c*f^2)*cosh(d*x + c)^2 + 2*(2*(a^2*b + b^3)*d^2*f*x + (a^2*b + b^3)*d*f)*cosh(1) + 3*((a^3 + a*b^2)*d^2*f^2*x
^2 + 2*(a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d^2*cosh(1)^2 + (a^3 + a*b^2)*d^2*sinh(1)^2 + 2*((a^3 + a*b^2)*d^
2*f*x + (a^3 + a*b^2)*d*f)*cosh(1) + 2*((a^3 + a*b^2)*d^2*f*x + (a^3 + a*b^2)*d^2*cosh(1) + (a^3 + a*b^2)*d*f)
*sinh(1))*cosh(d*x + c) + 2*(2*(a^2*b + b^3)*d^2*f*x + 2*(a^2*b + b^3)*d^2*cosh(1) + (a^2*b + b^3)*d*f)*sinh(1
))*sinh(d*x + c)^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 - 2*(a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*d^2*cosh(1)^2 + (a
^3 + a*b^2)*d^2*sinh(1)^2 + 2*((a^3 + a*b^2)*d^2*f*x - (a^3 + a*b^2)*d*f)*cosh(1) + 2*((a^3 + a*b^2)*d^2*f*x +
(a^3 + a*b^2)*d^2*cosh(1) - (a^3 + a*b^2)*d*f)*sinh(1))*cosh(d*x + c) - 4*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^
3*d*f*sinh(1) + (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^4 + 4*(b^3*d*f^2*x + b^3*d*f*c
osh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*si
nh(d*x + c)^4 + 2*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*f
*cosh(1) + b^3*d*f*sinh(1) + 3*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c
)^2 + 4*((b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^3 + (b^3*d*f^2*x + b^3*d*f*cosh(1) +
b^3*d*f*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*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) - 4*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1) + (b^3*
d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^4 + 4*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh
(1))*cosh(d*x + c)*sinh(d*x + c)^3 + (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*sinh(d*x + c)^4 + 2*(b^
3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*si
nh(1) + 3*(b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^3*d*f^2*x
+ b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh(d*x + c)^3 + (b^3*d*f^2*x + b^3*d*f*cosh(1) + b^3*d*f*sinh(1))*cosh
(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt(
(a^2 + b^2)/b^2) - b)/b + 1) + 2*(2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*
f^2*x + (2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x - I*(a^3 + 3*a*b^2)
*d*f*cosh(1) - I*(a^3 + 3*a*b^2)*d*f*sinh(1))*cosh(d*x + c)^4 + 4*(2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d
*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x - I*(a^3 + 3*a*b^2)*d*f*cosh(1) - I*(a^3 + 3*a*b^2)*d*f*sinh(1))*cosh(d
*x + c)*sinh(d*x + c)^3 + (2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x -
I*(a^3 + 3*a*b^2)*d*f*cosh(1) - I*(a^3 + 3*a*b^2)*d*f*sinh(1))*sinh(d*x + c)^4 - I*(a^3 + 3*a*b^2)*d*f*cosh(1
) - I*(a^3 + 3*a*b^2)*d*f*sinh(1) + 2*(2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^
2)*d*f^2*x - I*(a^3 + 3*a*b^2)*d*f*cosh(1) - I*(a^3 + 3*a*b^2)*d*f*sinh(1))*cosh(d*x + c)^2 + 2*(2*b^3*d*f^2*x
+ 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x - I*(a^3 + 3*a*b^2)*d*f*cosh(1) - I*(a^3
+ 3*a*b^2)*d*f*sinh(1) + 3*(2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x
- I*(a^3 + 3*a*b^2)*d*f*cosh(1) - I*(a^3 + 3*a*b^2)*d*f*sinh(1))*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((2*b^3*
d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a^3 + 3*a*b^2)*d*f^2*x - I*(a^3 + 3*a*b^2)*d*f*cosh(1) -
I*(a^3 + 3*a*b^2)*d*f*sinh(1))*cosh(d*x + c)^3 + (2*b^3*d*f^2*x + 2*b^3*d*f*cosh(1) + 2*b^3*d*f*sinh(1) - I*(a
^3 + 3*a*b^2)*d*f^2*x - I*(a^3 + 3*a*b^2)*d*f*c...

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________