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

Optimal. Leaf size=904 \[ -\frac {(e+f x)^2 a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}+\frac {f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b \left (a^2+b^2\right )^{3/2} 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 \left (a^2+b^2\right )^{3/2} d}-\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\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 \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} 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 \left (a^2+b^2\right )^{3/2} 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 \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 a^2}{b^3 d}-\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^2}{b^3 d^2}-\frac {f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^2}{b^3 d^3}+\frac {(e+f x)^2 \tanh (c+d x) a^2}{b^3 d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d^2}+\frac {2 i f^2 \text {Li}_2\left (-i e^{c+d x}\right ) a}{b^2 d^3}-\frac {2 i f^2 \text {Li}_2\left (i e^{c+d x}\right ) a}{b^2 d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) a}{b^2 d}+\frac {(e+f x)^3}{3 b f}-\frac {(e+f x)^2}{b d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {f^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{b d^3}-\frac {(e+f x)^2 \tanh (c+d x)}{b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.82, antiderivative size = 904, normalized size of antiderivative = 1.00, number of steps used = 44, number of rules used = 19, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.559, Rules used = {5581, 3720, 3718, 2190, 2279, 2391, 32, 5567, 5451, 4180, 5583, 4184, 5573, 3322, 2264, 2531, 2282, 6589, 6742} \[ -\frac {(e+f x)^2 a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}+\frac {f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac {(e+f x)^2 \tanh (c+d x) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {(e+f x)^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b \left (a^2+b^2\right )^{3/2} 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 \left (a^2+b^2\right )^{3/2} d}-\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f (e+f x) \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^2}+\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {(e+f x)^2 \text {sech}(c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^2 a^2}{b^3 d}-\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) a^2}{b^3 d^2}-\frac {f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^2}{b^3 d^3}+\frac {(e+f x)^2 \tanh (c+d x) a^2}{b^3 d}-\frac {4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) a}{b^2 d^2}+\frac {2 i f^2 \text {PolyLog}\left (2,-i e^{c+d x}\right ) a}{b^2 d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,i e^{c+d x}\right ) a}{b^2 d^3}+\frac {(e+f x)^2 \text {sech}(c+d x) a}{b^2 d}+\frac {(e+f x)^3}{3 b f}-\frac {(e+f x)^2}{b d}+\frac {2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{b d^2}+\frac {f^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{b d^3}-\frac {(e+f x)^2 \tanh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2264

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 2279

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 2282

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 2391

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 2531

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 3322

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 3718

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 3720

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

Rule 4180

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 4184

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 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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 5567

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*Sec
h[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 5573

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 5581

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 5583

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 6589

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 6742

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

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Mathematica [A]  time = 8.40, size = 935, normalized size = 1.03 \[ \frac {\left (2 e^2 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^2-f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^2+f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^2-2 f (e+f x) \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+2 f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+2 f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) a^3}{b \left (a^2+b^2\right )^{3/2} d^3}-\frac {2 f^2 \left (-\frac {2 \tan ^{-1}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right ) \tanh ^{-1}(\coth (c))}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}-\frac {i \text {csch}(c) \left (i \left (d x+\tanh ^{-1}(\coth (c))\right ) \left (\log \left (1-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\log \left (1+e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )+i \left (\text {Li}_2\left (-e^{-d x-\tanh ^{-1}(\coth (c))}\right )-\text {Li}_2\left (e^{-d x-\tanh ^{-1}(\coth (c))}\right )\right )\right )}{\sqrt {1-\coth ^2(c)}}\right ) a}{\left (a^2+b^2\right ) d^3}-\frac {4 e f \tan ^{-1}\left (\frac {\sinh (c)+\cosh (c) \tanh \left (\frac {d x}{2}\right )}{\sqrt {\cosh ^2(c)-\sinh ^2(c)}}\right ) a}{\left (a^2+b^2\right ) d^2 \sqrt {\cosh ^2(c)-\sinh ^2(c)}}+\frac {x \left (3 e^2+3 f x e+f^2 x^2\right )}{3 b}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (a \cosh (c) e^2-b \sinh (d x) e^2+2 a f x \cosh (c) e-2 b f x \sinh (d x) e+a f^2 x^2 \cosh (c)-b f^2 x^2 \sinh (d x)\right )}{\left (a^2+b^2\right ) d}+\frac {b f^2 \text {csch}(c) \left (d^2 e^{-\tanh ^{-1}(\coth (c))} x^2-\frac {i \coth (c) \left (-d x \left (2 i \tanh ^{-1}(\coth (c))-\pi \right )-\pi \log \left (1+e^{2 d x}\right )-2 \left (i d x+i \tanh ^{-1}(\coth (c))\right ) \log \left (1-e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )+\pi \log (\cosh (d x))+2 i \tanh ^{-1}(\coth (c)) \log \left (i \sinh \left (d x+\tanh ^{-1}(\coth (c))\right )\right )+i \text {Li}_2\left (e^{2 i \left (i d x+i \tanh ^{-1}(\coth (c))\right )}\right )\right )}{\sqrt {1-\coth ^2(c)}}\right ) \text {sech}(c)}{\left (a^2+b^2\right ) d^3 \sqrt {\text {csch}^2(c) \left (\sinh ^2(c)-\cosh ^2(c)\right )}}+\frac {2 b e f \text {sech}(c) (\cosh (c) \log (\cosh (c) \cosh (d x)+\sinh (c) \sinh (d x))-d x \sinh (c))}{\left (a^2+b^2\right ) d^2 \left (\cosh ^2(c)-\sinh ^2(c)\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(3*e^2 + 3*e*f*x + f^2*x^2))/(3*b) + (a^3*(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 + Sqr
t[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]))]))/(b*(a^2 + b^2)^(3/2)*d^3) + (2*b*e*f*Sech[c]*(Cosh[c]
*Log[Cosh[c]*Cosh[d*x] + Sinh[c]*Sinh[d*x]] - d*x*Sinh[c]))/((a^2 + b^2)*d^2*(Cosh[c]^2 - Sinh[c]^2)) - (4*a*e
*f*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]])/((a^2 + b^2)*d^2*Sqrt[Cosh[c]^2 - Si
nh[c]^2]) + (b*f^2*Csch[c]*((d^2*x^2)/E^ArcTanh[Coth[c]] - (I*Coth[c]*(-(d*x*(-Pi + (2*I)*ArcTanh[Coth[c]])) -
Pi*Log[1 + E^(2*d*x)] - 2*(I*d*x + I*ArcTanh[Coth[c]])*Log[1 - E^((2*I)*(I*d*x + I*ArcTanh[Coth[c]]))] + Pi*L
og[Cosh[d*x]] + (2*I)*ArcTanh[Coth[c]]*Log[I*Sinh[d*x + ArcTanh[Coth[c]]]] + I*PolyLog[2, E^((2*I)*(I*d*x + I*
ArcTanh[Coth[c]]))]))/Sqrt[1 - Coth[c]^2])*Sech[c])/((a^2 + b^2)*d^3*Sqrt[Csch[c]^2*(-Cosh[c]^2 + Sinh[c]^2)])
- (2*a*f^2*(((-I)*Csch[c]*(I*(d*x + ArcTanh[Coth[c]])*(Log[1 - E^(-(d*x) - ArcTanh[Coth[c]])] - Log[1 + E^(-(
d*x) - ArcTanh[Coth[c]])]) + I*(PolyLog[2, -E^(-(d*x) - ArcTanh[Coth[c]])] - PolyLog[2, E^(-(d*x) - ArcTanh[Co
th[c]])])))/Sqrt[1 - Coth[c]^2] - (2*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]]*Arc
Tanh[Coth[c]])/Sqrt[Cosh[c]^2 - Sinh[c]^2]))/((a^2 + b^2)*d^3) + (Sech[c]*Sech[c + d*x]*(a*e^2*Cosh[c] + 2*a*e
*f*x*Cosh[c] + a*f^2*x^2*Cosh[c] - b*e^2*Sinh[d*x] - 2*b*e*f*x*Sinh[d*x] - b*f^2*x^2*Sinh[d*x]))/((a^2 + b^2)*
d)

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fricas [C]  time = 0.66, size = 4195, normalized size = 4.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*(a^4 + 2*a^2*b^2 + b^4)*d^3*f^2*x^3 + 6*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f*x^2 + 6*(a^4 + 2*a^2*b^2 + b^4)
*d^3*e^2*x + 12*(a^2*b^2 + b^4)*d^2*e^2 - 24*(a^2*b^2 + b^4)*c*d*e*f + 12*(a^2*b^2 + b^4)*c^2*f^2 + 2*((a^4 +
2*a^2*b^2 + b^4)*d^3*f^2*x^3 - 12*(a^2*b^2 + b^4)*c*d*e*f + 6*(a^2*b^2 + b^4)*c^2*f^2 + 3*((a^4 + 2*a^2*b^2 +
b^4)*d^3*e*f - 2*(a^2*b^2 + b^4)*d^2*f^2)*x^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2 - 4*(a^2*b^2 + b^4)*d^2*e*f
)*x)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^2*x^3 - 12*(a^2*b^2 + b^4)*c*d*e*f + 6*(a^2*b^2 + b^4)
*c^2*f^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e*f - 2*(a^2*b^2 + b^4)*d^2*f^2)*x^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^
3*e^2 - 4*(a^2*b^2 + b^4)*d^2*e*f)*x)*sinh(d*x + c)^2 - 12*(a^3*b*d*f^2*x + a^3*b*d*e*f + (a^3*b*d*f^2*x + a^3
*b*d*e*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f^2*x + a^3
*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) + 12*(a^3*b*d*f^2*x + a^3*b*d*e*f + (a^3*b*d*f^2*x + a^3*b
*d*e*f)*cosh(d*x + c)^2 + 2*(a^3*b*d*f^2*x + a^3*b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d*f^2*x + a^3*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) + 6*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2 + (a^3*
b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*
f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*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) - 6*(a^3*b*d^2*e^2
- 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2 + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3
*b*d^2*e^2 - 2*a^3*b*c*d*e*f + a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*e^2 - 2*a^3*b*c*d*e*f +
a^3*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) - 6*(a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2 + (a^3*b*d^
2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*f^2*x^2 + 2*a^
3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^
2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*si
nh(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*b*d^2*f^2*x^2 + 2*a^3
*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2 + (a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^
3*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*b*c^2*f^2)*cos
h(d*x + c)*sinh(d*x + c) + (a^3*b*d^2*f^2*x^2 + 2*a^3*b*d^2*e*f*x + 2*a^3*b*c*d*e*f - a^3*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))*sq
rt((a^2 + b^2)/b^2) - b)/b) + 12*(a^3*b*f^2*cosh(d*x + c)^2 + 2*a^3*b*f^2*cosh(d*x + c)*sinh(d*x + c) + a^3*b*
f^2*sinh(d*x + c)^2 + a^3*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) - 12*(a^3*b*f^2*cosh(d*x + c)^2 + 2*a^3*b*f^2*cosh(d*x
+ c)*sinh(d*x + c) + a^3*b*f^2*sinh(d*x + c)^2 + a^3*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) + 12*((a^3*b + a*b^3)*d^2*f^
2*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f*x + (a^3*b + a*b^3)*d^2*e^2)*cosh(d*x + c) + (-12*I*(a^3*b + a*b^3)*f^2 + 12
*(a^2*b^2 + b^4)*f^2 + (-12*I*(a^3*b + a*b^3)*f^2 + 12*(a^2*b^2 + b^4)*f^2)*cosh(d*x + c)^2 + (-24*I*(a^3*b +
a*b^3)*f^2 + 24*(a^2*b^2 + b^4)*f^2)*cosh(d*x + c)*sinh(d*x + c) + (-12*I*(a^3*b + a*b^3)*f^2 + 12*(a^2*b^2 +
b^4)*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (12*I*(a^3*b + a*b^3)*f^2 + 12*(a^2*b^2
+ b^4)*f^2 + (12*I*(a^3*b + a*b^3)*f^2 + 12*(a^2*b^2 + b^4)*f^2)*cosh(d*x + c)^2 + (24*I*(a^3*b + a*b^3)*f^2 +
24*(a^2*b^2 + b^4)*f^2)*cosh(d*x + c)*sinh(d*x + c) + (12*I*(a^3*b + a*b^3)*f^2 + 12*(a^2*b^2 + b^4)*f^2)*sin
h(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + (-12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^2 + b^4)*d*
e*f + 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2 + b^4)*c*f^2 + (-12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^2 + b^4
)*d*e*f + 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)^2 + (-24*I*(a^3*b + a*b^3)*d*e*
f + 24*(a^2*b^2 + b^4)*d*e*f + 24*I*(a^3*b + a*b^3)*c*f^2 - 24*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)*sinh(d*x +
c) + (-12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^2 + b^4)*d*e*f + 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2 + b^4
)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) + (12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^2 +
b^4)*d*e*f - 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2 + b^4)*c*f^2 + (12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^
2 + b^4)*d*e*f - 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)^2 + (24*I*(a^3*b + a*b^3
)*d*e*f + 24*(a^2*b^2 + b^4)*d*e*f - 24*I*(a^3*b + a*b^3)*c*f^2 - 24*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)*sinh
(d*x + c) + (12*I*(a^3*b + a*b^3)*d*e*f + 12*(a^2*b^2 + b^4)*d*e*f - 12*I*(a^3*b + a*b^3)*c*f^2 - 12*(a^2*b^2
+ b^4)*c*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) + (12*I*(a^3*b + a*b^3)*d*f^2*x + 12*(a^
2*b^2 + b^4)*d*f^2*x + 12*I*(a^3*b + a*b^3)*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2 + (12*I*(a^3*b + a*b^3)*d*f^2*x +
12*(a^2*b^2 + b^4)*d*f^2*x + 12*I*(a^3*b + a*b^3)*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)^2 + (24*I*(
a^3*b + a*b^3)*d*f^2*x + 24*(a^2*b^2 + b^4)*d*f^2*x + 24*I*(a^3*b + a*b^3)*c*f^2 + 24*(a^2*b^2 + b^4)*c*f^2)*c
osh(d*x + c)*sinh(d*x + c) + (12*I*(a^3*b + a*b^3)*d*f^2*x + 12*(a^2*b^2 + b^4)*d*f^2*x + 12*I*(a^3*b + a*b^3)
*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (-12*I*(a^3*b
+ a*b^3)*d*f^2*x + 12*(a^2*b^2 + b^4)*d*f^2*x - 12*I*(a^3*b + a*b^3)*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2 + (-12*
I*(a^3*b + a*b^3)*d*f^2*x + 12*(a^2*b^2 + b^4)*d*f^2*x - 12*I*(a^3*b + a*b^3)*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2
)*cosh(d*x + c)^2 + (-24*I*(a^3*b + a*b^3)*d*f^2*x + 24*(a^2*b^2 + b^4)*d*f^2*x - 24*I*(a^3*b + a*b^3)*c*f^2 +
24*(a^2*b^2 + b^4)*c*f^2)*cosh(d*x + c)*sinh(d*x + c) + (-12*I*(a^3*b + a*b^3)*d*f^2*x + 12*(a^2*b^2 + b^4)*d
*f^2*x - 12*I*(a^3*b + a*b^3)*c*f^2 + 12*(a^2*b^2 + b^4)*c*f^2)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh
(d*x + c) + 1) + 4*(3*(a^3*b + a*b^3)*d^2*f^2*x^2 + 6*(a^3*b + a*b^3)*d^2*e*f*x + 3*(a^3*b + a*b^3)*d^2*e^2 +
((a^4 + 2*a^2*b^2 + b^4)*d^3*f^2*x^3 - 12*(a^2*b^2 + b^4)*c*d*e*f + 6*(a^2*b^2 + b^4)*c^2*f^2 + 3*((a^4 + 2*a^
2*b^2 + b^4)*d^3*e*f - 2*(a^2*b^2 + b^4)*d^2*f^2)*x^2 + 3*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2 - 4*(a^2*b^2 + b^4)
*d^2*e*f)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b + 2*a^2*b^3 + b^5)*d^3*cosh(d*x + c)^2 + 2*(a^4*b + 2*a^2*b
^3 + b^5)*d^3*cosh(d*x + c)*sinh(d*x + c) + (a^4*b + 2*a^2*b^3 + b^5)*d^3*sinh(d*x + c)^2 + (a^4*b + 2*a^2*b^3
+ b^5)*d^3)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \sinh \left (d x +c \right ) \left (\tanh ^{2}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -2 \, b e f {\left (\frac {2 \, {\left (d x + c\right )}}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d^{2}}\right )} - 4 \, a f^{2} \int \frac {x e^{\left (d x + c\right )}}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} - 4 \, b f^{2} \int \frac {x}{a^{2} d e^{\left (2 \, d x + 2 \, c\right )} + b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + a^{2} d + b^{2} d}\,{d x} - {\left (\frac {a^{3} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b + b^{3}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {2 \, {\left (a e^{\left (-d x - c\right )} - b\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} - \frac {d x + c}{b d}\right )} e^{2} - \frac {4 \, a e f \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} + \frac {12 \, b^{2} e f x + {\left (a^{2} d f^{2} + b^{2} d f^{2}\right )} x^{3} + 3 \, {\left (a^{2} d e f + {\left (d e f + 2 \, f^{2}\right )} b^{2}\right )} x^{2} + {\left ({\left (a^{2} d f^{2} e^{\left (2 \, c\right )} + b^{2} d f^{2} e^{\left (2 \, c\right )}\right )} x^{3} + 3 \, {\left (a^{2} d e f e^{\left (2 \, c\right )} + b^{2} d e f e^{\left (2 \, c\right )}\right )} x^{2}\right )} e^{\left (2 \, d x\right )} + 6 \, {\left (a b f^{2} x^{2} e^{c} + 2 \, a b e f x e^{c}\right )} e^{\left (d x\right )}}{3 \, {\left (a^{2} b d + b^{3} d + {\left (a^{2} b d e^{\left (2 \, c\right )} + b^{3} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} - \int -\frac {2 \, {\left (a^{3} f^{2} x^{2} e^{c} + 2 \, a^{3} e f x e^{c}\right )} e^{\left (d x\right )}}{a^{2} b^{2} + b^{4} - {\left (a^{2} b^{2} e^{\left (2 \, c\right )} + b^{4} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} b e^{c} + a b^{3} e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b*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*a*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*b*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) - (a^3*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 + b^3)*sqrt(a^2 + b^2)*d) - 2*(a*e^(-d*x - c) - b)/((a^2 + b^2 +
(a^2 + b^2)*e^(-2*d*x - 2*c))*d) - (d*x + c)/(b*d))*e^2 - 4*a*e*f*arctan(e^(d*x + c))/((a^2 + b^2)*d^2) + 1/3
*(12*b^2*e*f*x + (a^2*d*f^2 + b^2*d*f^2)*x^3 + 3*(a^2*d*e*f + (d*e*f + 2*f^2)*b^2)*x^2 + ((a^2*d*f^2*e^(2*c) +
b^2*d*f^2*e^(2*c))*x^3 + 3*(a^2*d*e*f*e^(2*c) + b^2*d*e*f*e^(2*c))*x^2)*e^(2*d*x) + 6*(a*b*f^2*x^2*e^c + 2*a*
b*e*f*x*e^c)*e^(d*x))/(a^2*b*d + b^3*d + (a^2*b*d*e^(2*c) + b^3*d*e^(2*c))*e^(2*d*x)) - integrate(-2*(a^3*f^2*
x^2*e^c + 2*a^3*e*f*x*e^c)*e^(d*x)/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x) - 2*(a^3*b*e^c +
a*b^3*e^c)*e^(d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {sinh}\left (c+d\,x\right )\,{\mathrm {tanh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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