\(\int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [383]

Optimal result

Integrand size = 28, antiderivative size = 772 \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {a^3 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d} \]

[Out]

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

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5686, 5559, 4265, 2317, 2438, 5702, 4269, 3799, 2221, 5692, 3403, 2296, 2611, 2320, 6724, 6874} \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b d \left (a^2+b^2\right )}-\frac {a^3 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3 \left (a^2+b^2\right )}-\frac {2 a^3 f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2 \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {a^3 (e+f x)^2}{b^2 d \left (a^2+b^2\right )}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}+\frac {2 a f (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{b^2 d^2}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}-\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d} \]

[In]

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

[Out]

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3403

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

Rule 3799

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

Rule 4265

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

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5559

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 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 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*} \text {integral}& = \frac {\int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {(2 f) \int (e+f x) \text {sech}(c+d x) \, dx}{b d} \\ & = \frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^2 \int \frac {(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}+\frac {a^2 \int (e+f x)^2 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(2 a f) \int (e+f x) \tanh (c+d x) \, dx}{b^2 d}-\frac {\left (2 i f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (2 i f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b d^2} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2\right ) \int \frac {e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}+\frac {a^2 \int \left (a (e+f x)^2 \text {sech}^2(c+d x)-b (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {(4 a f) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^2 d}-\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-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 {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}-\frac {\left (2 a^2 b\right ) \int \frac {e^{c+d x} (e+f x)^2}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^{3/2}}+\frac {a^3 \int (e+f x)^2 \text {sech}^2(c+d x) \, dx}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \int (e+f x)^2 \text {sech}(c+d x) \tanh (c+d x) \, dx}{b \left (a^2+b^2\right )}-\frac {\left (2 a f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d^2} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}+\frac {a^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 )^{3/2} d}-\frac {a^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 )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}+\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d}-\frac {\left (2 a^3 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \text {sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d}-\frac {\left (a f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^2 d^3} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^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 )^{3/2} d}-\frac {a^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 )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (4 a^3 f\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {\left (2 i a^2 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}-\frac {\left (2 i a^2 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^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 )^{3/2} d}-\frac {a^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 )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {\left (2 i a^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {\left (2 i a^2 f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {\left (2 a^3 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^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 )^{3/2} d}-\frac {a^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 )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (a^3 f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3} \\ & = -\frac {a (e+f x)^2}{b^2 d}+\frac {a^3 (e+f x)^2}{b^2 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \arctan \left (e^{c+d x}\right )}{b d^2}-\frac {4 a^2 f (e+f x) \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {a^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 )^{3/2} d}-\frac {a^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 )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 d^2}-\frac {2 a^3 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {2 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {a f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 d^3}-\frac {a^3 f^2 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x)}{b d}+\frac {a^2 (e+f x)^2 \text {sech}(c+d x)}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \tanh (c+d x)}{b^2 d}+\frac {a^3 (e+f x)^2 \tanh (c+d x)}{b^2 \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 633, normalized size of antiderivative = 0.82 \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {f \left (4 a d^2 e e^{2 c} x-4 a d^2 e \left (1+e^{2 c}\right ) x+2 a d^2 e^{2 c} f x^2-2 a d^2 \left (1+e^{2 c}\right ) f x^2-4 b d e \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )+2 a d e \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+2 i b \left (1+e^{2 c}\right ) f \left (d x \left (-\log \left (1-i e^{c+d x}\right )+\log \left (1+i e^{c+d x}\right )\right )+\operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-\operatorname {PolyLog}\left (2,i e^{c+d x}\right )\right )+a \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )\right )\right )}{\left (a^2+b^2\right ) \left (1+e^{2 c}\right )}+\frac {a^2 \left (2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\left (a^2+b^2\right )^{3/2}}+\frac {d^2 (e+f x)^2 \text {sech}(c+d x) (b+a \text {sech}(c) \sinh (d x))}{a^2+b^2}}{d^3} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \tanh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3661 vs. \(2 (712) = 1424\).

Time = 0.34 (sec) , antiderivative size = 3661, normalized size of antiderivative = 4.74 \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*(a^3 + a*b^2)*d^2*e^2 - 4*(a^3 + a*b^2)*c*d*e*f + 2*(a^3 + a*b^2)*c^2*f^2 - 2*((a^3 + a*b^2)*d^2*f^2*x^2 +
2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2 - 2*((a^3 + a*b^2
)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*sinh(d*x + c)^2 +
2*(a^2*b*d*f^2*x + a^2*b*d*e*f + (a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^2*x + a^2*b*d*e
*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d*f^2*x + a^2*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((
a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*
(a^2*b*d*f^2*x + a^2*b*d*e*f + (a^2*b*d*f^2*x + a^2*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^2*x + a^2*b*d*e*f)
*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d*f^2*x + a^2*b*d*e*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - (a^2*
b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^
2 + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^2*e^2 - 2*a^2*b
*c*d*e*f + a^2*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2
*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a^2*b*d^2*e^2 - 2*a^2*b*
c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*cosh(d*x + c)*s
inh(d*x + c) + (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*
b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*
x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2 + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2
)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)*
sinh(d*x + c) + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*sinh(d*x + c)^2)*sqr
t((a^2 + b^2)/b^2)*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) - (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2 + (a^2*b*d^2*f^2*x
^2 + 2*a^2*b*d^2*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2
*e*f*x + 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e*f*x
+ 2*a^2*b*c*d*e*f - a^2*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - 2*(a^2*b*f^2*cosh(d*x + c)^2 + 2*a^
2*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*b*f^2*sinh(d*x + c)^2 + a^2*b*f^2)*sqrt((a^2 + b^2)/b^2)*polylog(3,
(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) + 2*(a^2*b*
f^2*cosh(d*x + c)^2 + 2*a^2*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*b*f^2*sinh(d*x + c)^2 + a^2*b*f^2)*sqrt((a
^2 + b^2)/b^2)*polylog(3, (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) - 2*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + (a^2*b + b^3)*d^2*e^2)*cosh(d*x +
c) + 2*((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)*f^2 + ((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)*f^2)*cosh(d*x + c)^2 +
2*((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*f^2 + I*(a^2*b + b^3)
*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 2*((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2 +
((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2)*cosh(
d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*f^2 - I*(a^2*b + b^3)*f^2)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I
*sinh(d*x + c)) + 2*((a^3 + a*b^2)*d*e*f + I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2
+ ((a^3 + a*b^2)*d*e*f + I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2
+ 2*((a^3 + a*b^2)*d*e*f + I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)
*sinh(d*x + c) + ((a^3 + a*b^2)*d*e*f + I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*s
inh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) + 2*((a^3 + a*b^2)*d*e*f - I*(a^2*b + b^3)*d*e*f - (a^3
+ a*b^2)*c*f^2 + I*(a^2*b + b^3)*c*f^2 + ((a^3 + a*b^2)*d*e*f - I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2 +
I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*d*e*f - I*(a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*c*f^2
+ I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*d*e*f - I*(a^2*b + b^3)*d*e*f - (a^3 +
a*b^2)*c*f^2 + I*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) + 2*((a^3 + a*b^
2)*d*f^2*x - I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2 + ((a^3 + a*b^2)*d*f^2*x -
I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*d*f^
2*x - I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a
^3 + a*b^2)*d*f^2*x - I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2 - I*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*
log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + 2*((a^3 + a*b^2)*d*f^2*x + I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2
)*c*f^2 + I*(a^2*b + b^3)*c*f^2 + ((a^3 + a*b^2)*d*f^2*x + I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2 + I*(
a^2*b + b^3)*c*f^2)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*d*f^2*x + I*(a^2*b + b^3)*d*f^2*x + (a^3 + a*b^2)*c*f^2
+ I*(a^2*b + b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*d*f^2*x + I*(a^2*b + b^3)*d*f^2*x + (a^
3 + a*b^2)*c*f^2 + I*(a^2*b + b^3)*c*f^2)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 2*((a
^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + (a^2*b + b^3)*d^2*e^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2
*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))/((a^
4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)*sinh(d*x + c) + (a^4 +
2*a^2*b^2 + b^4)*d^3*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \tanh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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