3.5.7 \(\int \frac {(e+f x)^2 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [407]

Optimal. Leaf size=1067 \[ \frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {2 i a^2 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^4 f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 i a^4 f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {a^3 f (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^4 f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {2 i a^2 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {2 i a^4 f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {a^3 f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d} \]

[Out]

-2*a^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2-2*a^3*f*(f*x+e)*polylog(2,-b*e
xp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2+1/3*a*(f*x+e)^3/b^2/f-2*a^4*(f*x+e)^2*arctan(exp(d*x+c))/b^3/
(a^2+b^2)/d-a*f*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/b^2/d^2+2*I*f^2*polylog(3,I*exp(d*x+c))/b/d^3-1/2*a^3*f^2*p
olylog(3,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^3-2*I*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b/d^2-2*I*a^2*f^2*polylog(3,
I*exp(d*x+c))/b^3/d^3+2*I*a^4*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-2*f*(f*x+e)*cosh(d*x+c)/b/d
^2+(f*x+e)^2*sinh(d*x+c)/b/d-2*(f*x+e)^2*arctan(exp(d*x+c))/b/d+2*I*a^2*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b^3/
d^2+2*I*a^4*f^2*polylog(3,I*exp(d*x+c))/b^3/(a^2+b^2)/d^3-2*I*a^4*f*(f*x+e)*polylog(2,I*exp(d*x+c))/b^3/(a^2+b
^2)/d^2+2*f^2*sinh(d*x+c)/b/d^3+2*a^2*(f*x+e)^2*arctan(exp(d*x+c))/b^3/d+1/2*a*f^2*polylog(3,-exp(2*d*x+2*c))/
b^2/d^3-2*I*f^2*polylog(3,-I*exp(d*x+c))/b/d^3+2*I*f*(f*x+e)*polylog(2,-I*exp(d*x+c))/b/d^2+a^3*f*(f*x+e)*poly
log(2,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^2+2*I*a^2*f^2*polylog(3,-I*exp(d*x+c))/b^3/d^3-2*I*a^2*f*(f*x+e)*polylo
g(2,-I*exp(d*x+c))/b^3/d^2-2*I*a^4*f^2*polylog(3,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^3-a*(f*x+e)^2*ln(1+exp(2*d*x+2
*c))/b^2/d+a^3*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/b^2/(a^2+b^2)/d-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/
2)))/b^2/(a^2+b^2)/d-a^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d+2*a^3*f^2*polylog(3,
-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^3+2*a^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^
2/(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.24, antiderivative size = 1067, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5700, 5557, 3377, 2717, 4265, 2611, 2320, 6724, 3799, 2221, 5686, 5692, 5680, 6874} \begin {gather*} -\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) 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 ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^3}+\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right ) a^2}{b^3 d}-\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right ) a^2}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right ) a^2}{b^3 d^3}+\frac {(e+f x)^3 a}{3 b^2 f}-\frac {(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d}-\frac {f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) a}{b^2 d^2}+\frac {f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) a}{2 b^2 d^3}-\frac {2 (e+f x)^2 \text {ArcTan}\left (e^{c+d x}\right )}{b d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e + f*x)^3)/(3*b^2*f) + (2*a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^2*ArcTan[E^(c + d*x
)])/(b*d) - (2*a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2
) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)^2*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(
b^2*d) + (a^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - ((2*I)*a^2*f*(e + f*x)*PolyLog[2, (-
I)*E^(c + d*x)])/(b^3*d^2) + ((2*I)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^4*f*(e + f*x)
*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + ((2*I)*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*
d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - ((2*I)*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)
])/(b^3*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2
+ b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) -
(a*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^2) + (a^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*
(a^2 + b^2)*d^2) + ((2*I)*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) - ((2*I)*f^2*PolyLog[3, (-I)*E^(c +
d*x)])/(b*d^3) - ((2*I)*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - ((2*I)*a^2*f^2*PolyLog[3
, I*E^(c + d*x)])/(b^3*d^3) + ((2*I)*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I)*a^4*f^2*PolyLog[3, I*E^(c
+ d*x)])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 +
b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (a*f^2*P
olyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (a^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) + (2
*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5557

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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 5686

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[(e + f*x)^m*Sech
[c + d*x]*(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]

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 5700

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(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*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*(Tanh[c + d*x]^n/(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 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sinh (c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {a \int (e+f x)^2 \tanh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int (e+f x)^2 \cosh (c+d x) \, dx}{b}-\frac {\int (e+f x)^2 \text {sech}(c+d x) \, dx}{b}\\ &=\frac {a (e+f x)^3}{3 b^2 f}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}+\frac {a^2 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac {(2 a) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^2}+\frac {(2 i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b d}-\frac {(2 i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b d}-\frac {(2 f) \int (e+f x) \sinh (c+d x) \, dx}{b d}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {a^3 \int (e+f x)^2 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (2 i a^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 d}+\frac {\left (2 i a^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 d}+\frac {(2 a f) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d}-\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (2 i f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (2 f^2\right ) \int \cosh (c+d x) \, dx}{b d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {a^3 (e+f x)^3}{3 b^2 \left (a^2+b^2\right ) f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {a^3 \int \left (a (e+f x)^2 \text {sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (2 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 (2 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 (2 i a^2 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 d^2}-\frac {\left (2 i a^2 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 d^2}+\frac {\left (a f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {a^3 (e+f x)^3}{3 b^2 \left (a^2+b^2\right ) f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {a^4 \int (e+f x)^2 \text {sech}(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {a^3 \int (e+f x)^2 \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (2 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^3 d^3}-\frac {\left (2 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^3 d^3}+\frac {\left (a f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b^2 d^3}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {2 i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {a f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}+\frac {\left (2 a^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (2 i a^4 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}-\frac {\left (2 i a^4 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^4 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^4 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {2 i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {a f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {\left (2 a^3 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (2 a^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 )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {\left (2 a^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 )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {\left (2 i a^4 f^2\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}+\frac {\left (2 i a^4 f^2\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^4 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^4 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {a^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {\left (2 i a^4 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {\left (2 i a^4 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {\left (a^3 f^2\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^4 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^4 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {a^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^4 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {2 i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 i a^4 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}-\frac {\left (a^3 f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^3}{3 b^2 f}+\frac {2 a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {2 f (e+f x) \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {2 i a^2 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {2 i f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^4 f (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {2 i a^4 f (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {a f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 d^2}+\frac {a^3 f (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {2 i a^2 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {2 i f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^4 f^2 \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {2 i a^2 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {2 i f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {2 i a^4 f^2 \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 a^3 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {a f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {a^3 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}+\frac {2 f^2 \sinh (c+d x)}{b d^3}+\frac {(e+f x)^2 \sinh (c+d x)}{b d}\\ \end {align*}

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Mathematica [A]
time = 7.83, size = 1948, normalized size = 1.83 \begin {gather*} \frac {e^{-c} \left (6 a^3 d^3 e^2 e^c x+6 a b^2 d^3 e^2 e^c x+6 a^3 d^3 e e^c f x^2+6 a b^2 d^3 e e^c f x^2+2 a^3 d^3 e^c f^2 x^3+2 a b^2 d^3 e^c f^2 x^3-12 b^3 d^2 e^2 e^c \text {ArcTan}\left (e^{c+d x}\right )-3 a^2 b d^2 e^2 \cosh (d x)-3 b^3 d^2 e^2 \cosh (d x)+3 a^2 b d^2 e^2 e^{2 c} \cosh (d x)+3 b^3 d^2 e^2 e^{2 c} \cosh (d x)-6 a^2 b d e f \cosh (d x)-6 b^3 d e f \cosh (d x)-6 a^2 b d e e^{2 c} f \cosh (d x)-6 b^3 d e e^{2 c} f \cosh (d x)-6 a^2 b f^2 \cosh (d x)-6 b^3 f^2 \cosh (d x)+6 a^2 b e^{2 c} f^2 \cosh (d x)+6 b^3 e^{2 c} f^2 \cosh (d x)-6 a^2 b d^2 e f x \cosh (d x)-6 b^3 d^2 e f x \cosh (d x)+6 a^2 b d^2 e e^{2 c} f x \cosh (d x)+6 b^3 d^2 e e^{2 c} f x \cosh (d x)-6 a^2 b d f^2 x \cosh (d x)-6 b^3 d f^2 x \cosh (d x)-6 a^2 b d e^{2 c} f^2 x \cosh (d x)-6 b^3 d e^{2 c} f^2 x \cosh (d x)-3 a^2 b d^2 f^2 x^2 \cosh (d x)-3 b^3 d^2 f^2 x^2 \cosh (d x)+3 a^2 b d^2 e^{2 c} f^2 x^2 \cosh (d x)+3 b^3 d^2 e^{2 c} f^2 x^2 \cosh (d x)-12 i b^3 d^2 e e^c f x \log \left (1-i e^{c+d x}\right )-6 i b^3 d^2 e^c f^2 x^2 \log \left (1-i e^{c+d x}\right )+12 i b^3 d^2 e e^c f x \log \left (1+i e^{c+d x}\right )+6 i b^3 d^2 e^c f^2 x^2 \log \left (1+i e^{c+d x}\right )-6 a b^2 d^2 e^2 e^c \log \left (1+e^{2 (c+d x)}\right )-12 a b^2 d^2 e e^c f x \log \left (1+e^{2 (c+d x)}\right )-6 a b^2 d^2 e^c f^2 x^2 \log \left (1+e^{2 (c+d x)}\right )-6 a^3 d^2 e^2 e^c \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )-12 a^3 d^2 e 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 )-6 a^3 d^2 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 )-12 a^3 d^2 e 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 )-6 a^3 d^2 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 )+12 i b^3 d e^c f (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )-12 i b^3 d e^c f (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )-6 a b^2 d e e^c f \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-6 a b^2 d e^c f^2 x \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )-12 a^3 d e e^c f \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 )-12 a^3 d e^c f^2 x \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 )-12 a^3 d e e^c f \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 )-12 a^3 d e^c f^2 x \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 )-12 i b^3 e^c f^2 \text {PolyLog}\left (3,-i e^{c+d x}\right )+12 i b^3 e^c f^2 \text {PolyLog}\left (3,i e^{c+d x}\right )+3 a b^2 e^c f^2 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )+12 a^3 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 )+12 a^3 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 )+3 a^2 b d^2 e^2 \sinh (d x)+3 b^3 d^2 e^2 \sinh (d x)+3 a^2 b d^2 e^2 e^{2 c} \sinh (d x)+3 b^3 d^2 e^2 e^{2 c} \sinh (d x)+6 a^2 b d e f \sinh (d x)+6 b^3 d e f \sinh (d x)-6 a^2 b d e e^{2 c} f \sinh (d x)-6 b^3 d e e^{2 c} f \sinh (d x)+6 a^2 b f^2 \sinh (d x)+6 b^3 f^2 \sinh (d x)+6 a^2 b e^{2 c} f^2 \sinh (d x)+6 b^3 e^{2 c} f^2 \sinh (d x)+6 a^2 b d^2 e f x \sinh (d x)+6 b^3 d^2 e f x \sinh (d x)+6 a^2 b d^2 e e^{2 c} f x \sinh (d x)+6 b^3 d^2 e e^{2 c} f x \sinh (d x)+6 a^2 b d f^2 x \sinh (d x)+6 b^3 d f^2 x \sinh (d x)-6 a^2 b d e^{2 c} f^2 x \sinh (d x)-6 b^3 d e^{2 c} f^2 x \sinh (d x)+3 a^2 b d^2 f^2 x^2 \sinh (d x)+3 b^3 d^2 f^2 x^2 \sinh (d x)+3 a^2 b d^2 e^{2 c} f^2 x^2 \sinh (d x)+3 b^3 d^2 e^{2 c} f^2 x^2 \sinh (d x)\right )}{6 b^2 \left (a^2+b^2\right ) d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 4.86, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\sinh ^{2}\left (d x +c \right )\right ) \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^4)*d) - 4*b*arctan(e^(-d*x - c))/((a
^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(
-d*x - c)/(b*d))*e^2 - 1/6*(2*a*d^3*f^2*x^3*e^c + 6*a*d^3*f*x^2*e^(c + 1) - 3*(b*d^2*f^2*x^2*e^(2*c) + 2*b*f^2
*e^(2*c) - 2*b*d*f*e^(2*c + 1) - 2*(b*d*f^2*e^(2*c) - b*d^2*f*e^(2*c + 1))*x)*e^(d*x) + 3*(b*d^2*f^2*x^2 + 2*b
*d*f*e + 2*b*f^2 + 2*(b*d^2*f*e + b*d*f^2)*x)*e^(-d*x))*e^(-c)/(b^2*d^3) + integrate(2*(a^3*b*f^2*x^2 + 2*a^3*
b*f*x*e - (a^4*f^2*x^2*e^c + 2*a^4*f*x*e^(c + 1))*e^(d*x))/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^
(2*d*x) - 2*(a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x) - integrate(-2*(a*f^2*x^2 + 2*a*f*x*e - (b*f^2*x^2*e^c + 2*
b*f*x*e^(c + 1))*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*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. 4087 vs. \(2 (990) = 1980\).
time = 0.44, size = 4087, normalized size = 3.83 \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*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(3*(a^2*b + b^3)*d^2*f^2*x^2 + 6*(a^2*b + b^3)*d*f^2*x + 3*(a^2*b + b^3)*d^2*cosh(1)^2 + 3*(a^2*b + b^3)*
d^2*sinh(1)^2 + 6*(a^2*b + b^3)*f^2 - 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)*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 + (a^2*b + b^3)*d^2*cosh(1) - (a^2*b + b^3)*d*f)*sinh(1))*cosh(d*x + c)^
2 - 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*s
inh(1)^2 + 2*(a^2*b + b^3)*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 + (a^2*b + b^3)*d^2*cosh(1) - (a^2*b + b^3)*d*f)*sinh(1))*sinh(d*x + c)^2 + 6*((a^2*b + b^3)*d^2*f*x + (a^
2*b + b^3)*d*f)*cosh(1) - 2*((a^3 + a*b^2)*d^3*f^2*x^3 + 2*(a^3 + a*b^2)*c^3*f^2 + 3*((a^3 + a*b^2)*d^3*x + 2*
(a^3 + a*b^2)*c*d^2)*cosh(1)^2 + 3*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*sinh(1)^2 + 3*((a^3 + a*b^2)*
d^3*f*x^2 - 2*(a^3 + a*b^2)*c^2*d*f)*cosh(1) + 3*((a^3 + a*b^2)*d^3*f*x^2 - 2*(a^3 + a*b^2)*c^2*d*f + 2*((a^3
+ a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*cosh(1))*sinh(1))*cosh(d*x + c) + 12*((a^3*d*f^2*x + a^3*d*f*cosh(1) +
a^3*d*f*sinh(1))*cosh(d*x + c) + (a^3*d*f^2*x + a^3*d*f*cosh(1) + a^3*d*f*sinh(1))*sinh(d*x + c))*dilog((a*co
sh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*((a
^3*d*f^2*x + a^3*d*f*cosh(1) + a^3*d*f*sinh(1))*cosh(d*x + c) + (a^3*d*f^2*x + a^3*d*f*cosh(1) + a^3*d*f*sinh(
1))*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) + 12*((a*b^2*d*f^2*x + I*b^3*d*f^2*x + a*b^2*d*f*cosh(1) + I*b^3*d*f*cosh(1) + a*b^2*d*f
*sinh(1) + I*b^3*d*f*sinh(1))*cosh(d*x + c) + (a*b^2*d*f^2*x + I*b^3*d*f^2*x + a*b^2*d*f*cosh(1) + I*b^3*d*f*c
osh(1) + a*b^2*d*f*sinh(1) + I*b^3*d*f*sinh(1))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 12*(
(a*b^2*d*f^2*x - I*b^3*d*f^2*x + a*b^2*d*f*cosh(1) - I*b^3*d*f*cosh(1) + a*b^2*d*f*sinh(1) - I*b^3*d*f*sinh(1)
)*cosh(d*x + c) + (a*b^2*d*f^2*x - I*b^3*d*f^2*x + a*b^2*d*f*cosh(1) - I*b^3*d*f*cosh(1) + a*b^2*d*f*sinh(1) -
I*b^3*d*f*sinh(1))*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 6*((a^3*c^2*f^2 - 2*a^3*c*d*f*c
osh(1) + a^3*d^2*cosh(1)^2 + a^3*d^2*sinh(1)^2 - 2*(a^3*c*d*f - a^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c) + (a^3
*c^2*f^2 - 2*a^3*c*d*f*cosh(1) + a^3*d^2*cosh(1)^2 + a^3*d^2*sinh(1)^2 - 2*(a^3*c*d*f - a^3*d^2*cosh(1))*sinh(
1))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 6*((a^3*c^2*
f^2 - 2*a^3*c*d*f*cosh(1) + a^3*d^2*cosh(1)^2 + a^3*d^2*sinh(1)^2 - 2*(a^3*c*d*f - a^3*d^2*cosh(1))*sinh(1))*c
osh(d*x + c) + (a^3*c^2*f^2 - 2*a^3*c*d*f*cosh(1) + a^3*d^2*cosh(1)^2 + a^3*d^2*sinh(1)^2 - 2*(a^3*c*d*f - a^3
*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) +
2*a) + 6*((a^3*d^2*f^2*x^2 - a^3*c^2*f^2 + 2*(a^3*d^2*f*x + a^3*c*d*f)*cosh(1) + 2*(a^3*d^2*f*x + a^3*c*d*f)*s
inh(1))*cosh(d*x + c) + (a^3*d^2*f^2*x^2 - a^3*c^2*f^2 + 2*(a^3*d^2*f*x + a^3*c*d*f)*cosh(1) + 2*(a^3*d^2*f*x
+ a^3*c*d*f)*sinh(1))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x +
c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 6*((a^3*d^2*f^2*x^2 - a^3*c^2*f^2 + 2*(a^3*d^2*f*x + a^3*c*d*f)*cosh(1) +
2*(a^3*d^2*f*x + a^3*c*d*f)*sinh(1))*cosh(d*x + c) + (a^3*d^2*f^2*x^2 - a^3*c^2*f^2 + 2*(a^3*d^2*f*x + a^3*c*
d*f)*cosh(1) + 2*(a^3*d^2*f*x + a^3*c*d*f)*sinh(1))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (
b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 6*((a*b^2*c^2*f^2 + I*b^3*c^2*f^2 - 2*a*b^2
*c*d*f*cosh(1) - 2*I*b^3*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 + I*b^3*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 + I*b
^3*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1) - 2*I*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*co
sh(d*x + c) + (a*b^2*c^2*f^2 + I*b^3*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) - 2*I*b^3*c*d*f*cosh(1) + a*b^2*d^2*cosh(
1)^2 + I*b^3*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 + I*b^3*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*s
inh(1) - 2*I*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) + 6*
((a*b^2*c^2*f^2 - I*b^3*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1) + 2*I*b^3*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 - I*b^3*
d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - I*b^3*d^2*sinh(1)^2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1) + 2*I*
(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*cosh(d*x + c) + (a*b^2*c^2*f^2 - I*b^3*c^2*f^2 - 2*a*b^2*c*d*f*cosh(1)
+ 2*I*b^3*c*d*f*cosh(1) + a*b^2*d^2*cosh(1)^2 - I*b^3*d^2*cosh(1)^2 + a*b^2*d^2*sinh(1)^2 - I*b^3*d^2*sinh(1)^
2 - 2*(a*b^2*c*d*f - a*b^2*d^2*cosh(1))*sinh(1) + 2*I*(b^3*c*d*f - b^3*d^2*cosh(1))*sinh(1))*sinh(d*x + c))*lo
g(cosh(d*x + c) + sinh(d*x + c) - I) + 6*((a*b^2*d^2*f^2*x^2 - I*b^3*d^2*f^2*x^2 - a*b^2*c^2*f^2 + I*b^3*c^2*f
^2 + 2*(a*b^2*d^2*f*x + a*b^2*c*d*f)*cosh(1) - ...

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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