\(\int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [349]

Optimal result

Integrand size = 26, antiderivative size = 716 \[ \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}-\frac {2 a^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}-\frac {a (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i a^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac {2 i a^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac {2 a f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}-\frac {2 a f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^2}+\frac {a f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac {2 a f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}+\frac {2 a f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right ) d^3}-\frac {a f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5686, 4265, 2611, 2320, 6724, 5692, 5680, 2221, 6874, 3799} \[ \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 a^2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d \left (a^2+b^2\right )}-\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}+\frac {2 i a^2 f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3 \left (a^2+b^2\right )}+\frac {2 a f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )}+\frac {2 a f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )}-\frac {a f^2 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d^3 \left (a^2+b^2\right )}+\frac {2 i a^2 f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}-\frac {2 i a^2 f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2 \left (a^2+b^2\right )}-\frac {2 a f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}-\frac {2 a f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )}+\frac {a f (e+f x) \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{d^2 \left (a^2+b^2\right )}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )}-\frac {a (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{d \left (a^2+b^2\right )}+\frac {2 (e+f x)^2 \arctan \left (e^{c+d x}\right )}{b d}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{b d^2} \]

[In]

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

[Out]

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

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1640\) vs. \(2(716)=1432\).

Time = 10.38 (sec) , antiderivative size = 1640, normalized size of antiderivative = 2.29 \[ \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {-12 a d^3 e^2 e^{2 c} x+12 a d^3 e^2 \left (1+e^{2 c}\right ) x+12 a d^3 e f x^2+4 a d^3 f^2 x^3+12 b d^2 e^2 \left (1+e^{2 c}\right ) \arctan \left (e^{c+d x}\right )-6 a d^2 e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right )+12 i b d e \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 )-6 a d e \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 )+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 x^2 \log \left (1-i e^{c+d x}\right )-d^2 x^2 \log \left (1+i e^{c+d x}\right )-2 d x \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+2 d x \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+2 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )-2 \operatorname {PolyLog}\left (3,i e^{c+d x}\right )\right )-a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right )-6 d x \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )+3 \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )}{6 \left (a^2+b^2\right ) d^3 \left (1+e^{2 c}\right )}+\frac {a \left (6 e^2 e^{2 c} x+6 e e^{2 c} f x^2+2 e^{2 c} f^2 x^3+\frac {6 a \sqrt {a^2+b^2} e^2 \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {3 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}-\frac {3 e^2 e^{2 c} \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}+\frac {6 e 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 )}{d}-\frac {6 e e^{2 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 )}{d}+\frac {3 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 )}{d}-\frac {3 e^{2 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 )}{d}+\frac {6 e 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 )}{d}-\frac {6 e e^{2 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 )}{d}+\frac {3 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 )}{d}-\frac {3 e^{2 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 )}{d}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}\right )}{3 \left (a^2+b^2\right ) \left (-1+e^{2 c}\right )}-\frac {a x \left (3 e^2+3 e f x+f^2 x^2\right ) \text {csch}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}\right ) \text {sech}(c)}{6 \left (a^2+b^2\right )} \]

[In]

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

[Out]

(-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2*(1
+ E^(2*c))*ArcTan[E^(c + d*x)] - 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*d*e*
(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)] + PolyLo
g[2, I*E^(c + d*x)]) - 6*a*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c +
d*x))]) + (6*I)*b*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c + d*x)] - 2*d*x*
PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog[3,
I*E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, -E^(
2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))]))/(6*(a^2 + b^2)*d^3*(1 + E^(2*c))) + (a*(6*e^2*E^(2*c)*x + 6*e
*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(S
qrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(
(a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^
2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2
)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) +
b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6
*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*
c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt
[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e
*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(
a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b
^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*
E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*
c)*f^2*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)*(-1 + E^(2*c))) - (a*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Csch[c/
2]*Sech[c/2]*Sech[c])/(6*(a^2 + b^2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1083, normalized size of antiderivative = 1.51 \[ \int \frac {(e+f x)^2 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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