3.4.58 \(\int \frac {(e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [358]

Optimal. Leaf size=1176 \[ \frac {(e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \text {ArcTan}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \text {ArcTan}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b 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^2 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b 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^2 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 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 a b^2 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 {a b^2 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 b f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^2 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 b f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^2 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a b^2 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 a b^2 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 {a b^2 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.42, antiderivative size = 1176, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5702, 4271, 3855, 4265, 2611, 2320, 6724, 5692, 5680, 2221, 6874, 3799, 5559, 4269, 3556} \begin {gather*} -\frac {(e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d}-\frac {2 b (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d}+\frac {f^2 \text {ArcTan}(\sinh (c+d x)) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}+\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac {2 i b f^2 \text {Li}_3\left (i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}-\frac {f (e+f x) \text {sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) a^2}{2 b \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \text {sech}^2(c+d x) a}{2 \left (a^2+b^2\right ) d}-\frac {b^2 (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}-\frac {b^2 (e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}+\frac {b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d}-\frac {f^2 \log (\cosh (c+d x)) a}{\left (a^2+b^2\right ) d^3}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac {b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}+\frac {2 b^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}-\frac {b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) a}{2 \left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \tanh (c+d x) a}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {f^2 \text {ArcTan}(\sinh (c+d x))}{b d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(
e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (f^2*ArcTan[Sinh[c + d*x]])/(b*d^3) + (a^2*f^2*ArcTan[Sinh
[c + d*x]])/(b*(a^2 + b^2)*d^3) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b
^2)^2*d) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b^2*(e +
f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (a*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^3) - (I*f*(e +
f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^
2)^2*d^2) + (I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (I*f*(e + f*x)*PolyLog[2, I
*E^(c + d*x)])/(b*d^2) - ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^2*f*(e
+ f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (2*a*b^2*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*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))])/((a^2 + b^2)^2*d^2) + (a*b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (I*f^2*P
olyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - ((2*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I
*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I
)*a^2*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b
^2)*d^3) + (2*a*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (2*a*b^2*f
^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (a*b^2*f^2*PolyLog[3, -E^(2*(c
+ d*x))])/(2*(a^2 + b^2)^2*d^3) + (f*(e + f*x)*Sech[c + d*x])/(b*d^2) - (a^2*f*(e + f*x)*Sech[c + d*x])/(b*(a^
2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)
*d^2) + ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*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 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 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 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}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac {\int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b}-\frac {a \int (e+f x)^2 \text {sech}^3(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)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}-\frac {f^2 \int \text {sech}(c+d x) \, dx}{b d^2}\\ &=\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {(a b) \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 {\left (a b^3\right ) \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 {a \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}{b \left (a^2+b^2\right )}-\frac {(i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b d}+\frac {(i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b d}\\ &=\frac {a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {(a b) \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 {\left (a b^3\right ) \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 {\left (a b^3\right ) \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}^2(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}-\frac {a^2 \int (e+f x)^2 \text {sech}^3(c+d x) \, dx}{b \left (a^2+b^2\right )}+\frac {\left (i f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^2}-\frac {\left (i f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^2}\\ &=\frac {a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}-\frac {a b^2 (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 b^2 (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 {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (a^2 b\right ) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a b^2\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac {a^2 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{2 b \left (a^2+b^2\right )}+\frac {\left (2 a b^2 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 a b^2 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 {(a f) \int (e+f x) \text {sech}^2(c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}-\frac {\left (i f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}+\frac {\left (a^2 f^2\right ) \int \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d^2}\\ &=\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 a b^2 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 a b^2 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 {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac {\left (2 a b^2\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^2 b 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^2 b 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 (i a^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (i a^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}+\frac {\left (2 a b^2 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 a b^2 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 (a f^2\right ) \int \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 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 a b^2 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 {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (2 a b^2 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 a b^2 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 a b^2 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^2 b 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^2 b 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^2 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}+\frac {\left (i a^2 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}\\ &=\frac {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 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 a b^2 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 b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 a b^2 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 a b^2 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 {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (2 i a^2 b 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^2 b 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^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {\left (i a^2 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {\left (a b^2 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 {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 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 a b^2 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 b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a b^2 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 a b^2 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 {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac {\left (a b^2 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 {(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac {a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac {a b^2 (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 b^2 (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 b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac {a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac {i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 b 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^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a b^2 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 a b^2 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 b^2 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 b f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac {i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 b f^2 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac {i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a b^2 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 a b^2 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 b^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac {f (e+f x) \text {sech}(c+d x)}{b d^2}-\frac {a^2 f (e+f x) \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac {a (e+f x)^2 \text {sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac {a^2 (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{2 b \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. \(3264\) vs. \(2(1176)=2352\).
time = 25.00, size = 3264, normalized size = 2.78 \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]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(-12*a*b^2*d^3*e^2*E^(2*c)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2*x - 12*a*b^2*d^3*e*E^(2*c)*f*x^
2 - 4*a*b^2*d^3*E^(2*c)*f^2*x^3 - 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*ArcTan[E^(c + d*x)] - 6*
a^2*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^
(c + d*x)] - 12*b^3*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTan[E^(c + d*x)] - 12*b^3*E^(2*c)*f^2*Ar
cTan[E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)]
- (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] -
(3*I)*a^2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] - (3*I)*a^2*b*d^
2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (6*I)*a^2*b*
d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c + d*x)] + (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*
Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f^2*x^2*Log[1 +
I*E^(c + d*x)] - (3*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c
+ d*x)] - (3*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*a*b^2*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6
*a*b^2*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*f^2*Log[1 + E^(
2*(c + d*x))] - 6*a^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] + 12
*a*b^2*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*a*b^2*d^2*
f^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + (6*I)*b*(a^2 - b^2)*
d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*b*(-a^2 + b^2)*d*(1 + E^(2*c))*f*(e + f*x)*Po
lyLog[2, I*E^(c + d*x)] + 6*a*b^2*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 6*a*b^2*d*e*E^(2*c)*f*PolyLog[2, -E^(2*
(c + d*x))] + 6*a*b^2*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a*b^2*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x
))] - (6*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^2*b*
E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*b^3*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^2*b*f^
2*PolyLog[3, I*E^(c + d*x)] - (6*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)] + (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3, I*E
^(c + d*x)] - (6*I)*b^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a*b^2*f^2*PolyLog[3, -E^(2*(c + d*x))] - 3*a
*b^2*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) + (a*b^2*(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)^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*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[b - 2*a*E^(c + d*x) - b*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*PolyLog[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^3*e*f + 6*a*b^2*e*f - 12*a*b^2*d^2*e^2*x + 6*a^3*f^2*x + 6*a*b^2*f^2*x - 12*a*b^2*d^2*e*
f*x^2 - 4*a*b^2*d^2*f^2*x^3 - 6*a^3*e*f*Cosh[2*c] - 6*a*b^2*e*f*Cosh[2*c] - 6*a^3*f^2*x*Cosh[2*c] - 6*a*b^2*f^
2*x*Cosh[2*c] - 6*a^3*e*f*Cosh[2*d*x] - 6*a*b^2*e*f*Cosh[2*d*x] - 6*a^3*f^2*x*Cosh[2*d*x] - 6*a*b^2*f^2*x*Cosh
[2*d*x] - 3*a^2*b*d*e^2*Cosh[c - d*x] - 3*b^3*d*e^2*Cosh[c - d*x] - 6*a^2*b*d*e*f*x*Cosh[c - d*x] - 6*b^3*d*e*
f*x*Cosh[c - d*x] - 3*a^2*b*d*f^2*x^2*Cosh[c - d*x] - 3*b^3*d*f^2*x^2*Cosh[c - d*x] + 3*a^2*b*d*e^2*Cosh[3*c +
d*x] + 3*b^3*d*e^2*Cosh[3*c + d*x] + 6*a^2*b*d*e*f*x*Cosh[3*c + d*x] + 6*b^3*d*e*f*x*Cosh[3*c + d*x] + 3*a^2*
b*d*f^2*x^2*Cosh[3*c + d*x] + 3*b^3*d*f^2*x^2*C...

________________________________________________________________________________________

Maple [F]
time = 1.26, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \mathrm {sech}\left (d x +c \right )^{2} \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-a^2*b*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) + b^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) -
2*a*b^2*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*a*b^2*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^2*b*d^2*f*integ
rate(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) + 2*b^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) + a^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)) + a*b^2*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^2*b*f^2
*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) - 2*b^3*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^
3) - (a*b^2*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) - a*b^2*log(e^(-2*d*x
- 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (a^2*b - b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) - (b*
e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*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*a*f^2*x + 2*a*f*e - (b*d*f^2*x^2*e^(3*c) + 2*b*f*e^(3*c + 1) + 2*(b*f^2*
e^(3*c) + b*d*f*e^(3*c + 1))*x)*e^(3*d*x) + 2*(a*d*f^2*x^2*e^(2*c) + a*f*e^(2*c + 1) + (a*f^2*e^(2*c) + 2*a*d*
f*e^(2*c + 1))*x)*e^(2*d*x) + (b*d*f^2*x^2*e^c - 2*b*f*e^(c + 1) + 2*(b*d*f*e^(c + 1) - b*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*(a*b^3*f^2*x^2 + 2*a*b^3*f*x*e - (a^2*b^2*f^2*x^2*e^c + 2*a^2*b^2*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. 16498 vs. \(2 (1099) = 2198\).
time = 0.68, size = 16498, normalized size = 14.03 \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)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Integral((e + f*x)**2*tanh(c + d*x)*sech(c + d*x)**2/(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)^2*tanh(d*x+c)/(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 {\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{{\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((tanh(c + d*x)*(e + f*x)^2)/(cosh(c + d*x)^2*(a + b*sinh(c + d*x))),x)

[Out]

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

________________________________________________________________________________________