3.309 \(\int \frac {(e+f x)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=780 \[ -\frac {6 i b f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 i b f^3 \text {Li}_3\left (i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 d^4 \left (a^2+b^2\right )}+\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 a f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{d^2 \left (a^2+b^2\right )}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a (e+f x)^3 \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^3}{d \left (a^2+b^2\right )} \]

[Out]

a*(f*x+e)^3/(a^2+b^2)/d-6*b*f*(f*x+e)^2*arctan(exp(d*x+c))/(a^2+b^2)/d^2-3*a*f*(f*x+e)^2*ln(1+exp(2*d*x+2*c))/
(a^2+b^2)/d^2+b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d-b^2*(f*x+e)^3*ln(1+b*exp(
d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d+6*I*b*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/(a^2+b^2)/d^3-6*I*b*f
^2*(f*x+e)*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^3-3*a*f^2*(f*x+e)*polylog(2,-exp(2*d*x+2*c))/(a^2+b^2)/d^3+3*b^
2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-3*b^2*f*(f*x+e)^2*polylog(2,-b*
exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^2-6*I*b*f^3*polylog(3,-I*exp(d*x+c))/(a^2+b^2)/d^4+6*I*b*f^3
*polylog(3,I*exp(d*x+c))/(a^2+b^2)/d^4+3/2*a*f^3*polylog(3,-exp(2*d*x+2*c))/(a^2+b^2)/d^4-6*b^2*f^2*(f*x+e)*po
lylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^3+6*b^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a
^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^3+6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^4
-6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/(a^2+b^2)^(3/2)/d^4+b*(f*x+e)^3*sech(d*x+c)/(a^2+b^2)/
d+a*(f*x+e)^3*tanh(d*x+c)/(a^2+b^2)/d

________________________________________________________________________________________

Rubi [A]  time = 1.69, antiderivative size = 780, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {5573, 3322, 2264, 2190, 2531, 6609, 2282, 6589, 6742, 4184, 3718, 5451, 4180} \[ \frac {6 i b f^2 (e+f x) \text {PolyLog}\left (2,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 i b f^2 (e+f x) \text {PolyLog}\left (2,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac {3 a f^2 (e+f x) \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{d^3 \left (a^2+b^2\right )}-\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {6 b^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^3 \left (a^2+b^2\right )^{3/2}}+\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^2 \left (a^2+b^2\right )^{3/2}}-\frac {6 i b f^3 \text {PolyLog}\left (3,-i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {6 i b f^3 \text {PolyLog}\left (3,i e^{c+d x}\right )}{d^4 \left (a^2+b^2\right )}+\frac {3 a f^3 \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 d^4 \left (a^2+b^2\right )}+\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}-\frac {6 b^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{d^4 \left (a^2+b^2\right )^{3/2}}-\frac {3 a f (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{d^2 \left (a^2+b^2\right )}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{d \left (a^2+b^2\right )^{3/2}}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{d \left (a^2+b^2\right )^{3/2}}+\frac {a (e+f x)^3 \tanh (c+d x)}{d \left (a^2+b^2\right )}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{d \left (a^2+b^2\right )}+\frac {a (e+f x)^3}{d \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

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

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 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 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 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 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 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 6609

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

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)^3 \text {sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2+b^2}+\frac {b^2 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {\int \left (a (e+f x)^3 \text {sech}^2(c+d x)-b (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^2+b^2}+\frac {\left (2 b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2+b^2}\\ &=\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{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 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{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 \int (e+f x)^3 \text {sech}^2(c+d x) \, dx}{a^2+b^2}-\frac {b \int (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}\\ &=\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \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 (3 b^2 f\right ) \int (e+f x)^2 \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 {(3 a f) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}-\frac {(3 b f) \int (e+f x)^2 \text {sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d}\\ &=\frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac {(6 a f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right ) d}-\frac {\left (6 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-\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 (6 b^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-\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 (6 i b f^2\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}-\frac {\left (6 i b f^2\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (6 a f^2\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac {\left (6 b^2 f^3\right ) \int \text {Li}_3\left (-\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^3}-\frac {\left (6 b^2 f^3\right ) \int \text {Li}_3\left (-\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^3}-\frac {\left (6 i b f^3\right ) \int \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}+\frac {\left (6 i b f^3\right ) \int \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (6 b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {\left (6 b^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {\left (6 i b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {\left (6 i b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {\left (3 a f^3\right ) \int \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i b f^3 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac {\left (3 a f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}\\ &=\frac {a (e+f x)^3}{\left (a^2+b^2\right ) d}-\frac {6 b f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}+\frac {b^2 (e+f x)^3 \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 {b^2 (e+f x)^3 \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 {3 a f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {6 i b f^2 (e+f x) \text {Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 b^2 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^2}-\frac {3 a f^2 (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^3}-\frac {6 i b f^3 \text {Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}+\frac {6 i b f^3 \text {Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^4}-\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {6 b^2 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^3}+\frac {3 a f^3 \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right ) d^4}+\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {6 b^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d^4}+\frac {b (e+f x)^3 \text {sech}(c+d x)}{\left (a^2+b^2\right ) d}+\frac {a (e+f x)^3 \tanh (c+d x)}{\left (a^2+b^2\right ) d}\\ \end {align*}

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Mathematica [A]  time = 13.46, size = 1143, normalized size = 1.47 \[ \frac {\left (-2 e^3 \tanh ^{-1}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right ) d^3+f^3 x^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3+3 e^2 f x \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) d^3-f^3 x^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e f^2 x^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3-3 e^2 f x \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) d^3+3 f (e+f x)^2 \text {Li}_2\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d^2-3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d^2-6 e f^2 \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d-6 f^3 x \text {Li}_3\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right ) d+6 e f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 x \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) d+6 f^3 \text {Li}_4\left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}-a}\right )-6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right ) b^2}{\left (a^2+b^2\right )^{3/2} d^4}-\frac {f \left (4 a f^2 x^3 d^3+12 a e f x^2 d^3-12 a e^2 e^{2 c} x d^3+12 a e^2 \left (1+e^{2 c}\right ) x d^3+12 b e^2 \left (1+e^{2 c}\right ) \tan ^{-1}\left (e^{c+d x}\right ) d^2-6 a e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right ) d^2+12 i b 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 )-\text {Li}_2\left (-i e^{c+d x}\right )+\text {Li}_2\left (i e^{c+d x}\right )\right ) d-6 a 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 )-\text {Li}_2\left (-e^{2 (c+d x)}\right )\right ) d+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 \log \left (1-i e^{c+d x}\right ) x^2-d^2 \log \left (1+i e^{c+d x}\right ) x^2-2 d \text {Li}_2\left (-i e^{c+d x}\right ) x+2 d \text {Li}_2\left (i e^{c+d x}\right ) x+2 \text {Li}_3\left (-i e^{c+d x}\right )-2 \text {Li}_3\left (i e^{c+d x}\right )\right )-a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right ) x^2-6 d \text {Li}_2\left (-e^{2 (c+d x)}\right ) x+3 \text {Li}_3\left (-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}+\frac {\text {sech}(c) \text {sech}(c+d x) \left (b \cosh (c) e^3+a \sinh (d x) e^3+3 b f x \cosh (c) e^2+3 a f x \sinh (d x) e^2+3 b f^2 x^2 \cosh (c) e+3 a f^2 x^2 \sinh (d x) e+b f^3 x^3 \cosh (c)+a f^3 x^3 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^3*Sech[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*(f*(-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)]
+ PolyLog[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*Po
lyLog[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))])))/((a^2 + b^2)*d^4*(1 + E^(2*c))) + (b^2*(-2*d^3*e^3*A
rcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3
*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2])] - 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2])] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x
)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2]))] - 6*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*f^3*x*PolyLog[3, (b*E
^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*e*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*f^3
*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^
2])] - 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/((a^2 + b^2)^(3/2)*d^4) + (Sech[c]*Sech[c
+ d*x]*(b*e^3*Cosh[c] + 3*b*e^2*f*x*Cosh[c] + 3*b*e*f^2*x^2*Cosh[c] + b*f^3*x^3*Cosh[c] + a*e^3*Sinh[d*x] + 3*
a*e^2*f*x*Sinh[d*x] + 3*a*e*f^2*x^2*Sinh[d*x] + a*f^3*x^3*Sinh[d*x]))/((a^2 + b^2)*d)

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fricas [C]  time = 0.77, size = 6397, normalized size = 8.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(4*(a^3 + a*b^2)*d^3*e^3 - 12*(a^3 + a*b^2)*c*d^2*e^2*f + 12*(a^3 + a*b^2)*c^2*d*e*f^2 - 4*(a^3 + a*b^2)*
c^3*f^3 - 4*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3*(a^3
+ a*b^2)*c*d^2*e^2*f - 3*(a^3 + a*b^2)*c^2*d*e*f^2 + (a^3 + a*b^2)*c^3*f^3)*cosh(d*x + c)^2 - 4*((a^3 + a*b^2)
*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3*(a^3 + a*b^2)*c*d^2*e^2*f - 3*(
a^3 + a*b^2)*c^2*d*e*f^2 + (a^3 + a*b^2)*c^3*f^3)*sinh(d*x + c)^2 - 6*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b
^3*d^2*e^2*f + (b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(b^3*d^2*f^3*x^2 + 2*
b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*
e^2*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*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f + (b
^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x +
b^3*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^2*f^3*x^2 + 2*b^3*d^2*e*f^2*x + b^3*d^2*e^2*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))*sq
rt((a^2 + b^2)/b^2) - b)/b + 1) + 2*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3 + (b^3*
d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^3*d^3*e^3 - 3*b^3*c*d^2*
e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^
3*c^2*d*e*f^2 - b^3*c^3*f^3)*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) - 2*(b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3 + (b
^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^3*d^3*e^3 - 3*b^3*c*d
^2*e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^3*e^3 - 3*b^3*c*d^2*e^2*f + 3
*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*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) - 2*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c
*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3 + (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*
b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2
+ 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^3*
d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*s
inh(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*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d
^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3 + (b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^
3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^3*d^3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 +
3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d^
3*f^3*x^3 + 3*b^3*d^3*e*f^2*x^2 + 3*b^3*d^3*e^2*f*x + 3*b^3*c*d^2*e^2*f - 3*b^3*c^2*d*e*f^2 + b^3*c^3*f^3)*sin
h(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) - 12*(b^3*f^3*cosh(d*x + c)^2 + 2*b^3*f^3*cosh(d*x + c)*sinh(d*x + c) + b^3*
f^3*sinh(d*x + c)^2 + b^3*f^3)*sqrt((a^2 + b^2)/b^2)*polylog(4, (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*(b^3*f^3*cosh(d*x + c)^2 + 2*b^3*f^3*cosh(d*x + c)*s
inh(d*x + c) + b^3*f^3*sinh(d*x + c)^2 + b^3*f^3)*sqrt((a^2 + b^2)/b^2)*polylog(4, (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*(b^3*d*f^3*x + b^3*d*e*f^2 + (b^3
*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)^2 + 2*(b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d*f
^3*x + b^3*d*e*f^2)*sinh(d*x + c)^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*(b^3*d*f^3*x + b^3*d*e*f^2 + (b^3*d*f^3*x + b^
3*d*e*f^2)*cosh(d*x + c)^2 + 2*(b^3*d*f^3*x + b^3*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (b^3*d*f^3*x + b^3*d*
e*f^2)*sinh(d*x + c)^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) - 4*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 +
3*(a^2*b + b^3)*d^3*e^2*f*x + (a^2*b + b^3)*d^3*e^3)*cosh(d*x + c) + (12*(a^3 + a*b^2)*d*f^3*x + 12*I*(a^2*b
+ b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b + b^3)*d*e*f^2 + (12*(a^3 + a*b^2)*d*f^3*x + 12*I*(a^2
*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)^2 + (24*(a^3 + a*b^2)
*d*f^3*x + 24*I*(a^2*b + b^3)*d*f^3*x + 24*(a^3 + a*b^2)*d*e*f^2 + 24*I*(a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)*s
inh(d*x + c) + (12*(a^3 + a*b^2)*d*f^3*x + 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b
+ b^3)*d*e*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (12*(a^3 + a*b^2)*d*f^3*x - 12*I*
(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I*(a^2*b + b^3)*d*e*f^2 + (12*(a^3 + a*b^2)*d*f^3*x - 12
*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I*(a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)^2 + (24*(a^3 +
a*b^2)*d*f^3*x - 24*I*(a^2*b + b^3)*d*f^3*x + 24*(a^3 + a*b^2)*d*e*f^2 - 24*I*(a^2*b + b^3)*d*e*f^2)*cosh(d*x
+ c)*sinh(d*x + c) + (12*(a^3 + a*b^2)*d*f^3*x - 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I
*(a^2*b + b^3)*d*e*f^2)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + (6*(a^3 + a*b^2)*d^2*e^2*
f + 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*
c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*e^2*f + 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b
^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x +
c)^2 + (12*(a^3 + a*b^2)*d^2*e^2*f + 12*I*(a^2*b + b^3)*d^2*e^2*f - 24*(a^3 + a*b^2)*c*d*e*f^2 - 24*I*(a^2*b
+ b^3)*c*d*e*f^2 + 12*(a^3 + a*b^2)*c^2*f^3 + 12*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*(a^
3 + a*b^2)*d^2*e^2*f + 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2
+ 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I
) + (6*(a^3 + a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)
*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b
^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2
*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3 + a*b^2)*d^2*e^2*f - 12*I*(a^2*b + b^3)*d^2*e^2*f - 24*(a^3 + a*
b^2)*c*d*e*f^2 + 24*I*(a^2*b + b^3)*c*d*e*f^2 + 12*(a^3 + a*b^2)*c^2*f^3 - 12*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*
x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 +
12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(cosh
(d*x + c) + sinh(d*x + c) - I) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 - 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^
2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 -
6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*f^3*x^2 - 6*I*(a^2*b + b^3)*d^2*f^3
*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*
b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3 + a*b^2)*
d^2*f^3*x^2 - 12*I*(a^2*b + b^3)*d^2*f^3*x^2 + 24*(a^3 + a*b^2)*d^2*e*f^2*x - 24*I*(a^2*b + b^3)*d^2*e*f^2*x +
24*(a^3 + a*b^2)*c*d*e*f^2 - 24*I*(a^2*b + b^3)*c*d*e*f^2 - 12*(a^3 + a*b^2)*c^2*f^3 + 12*I*(a^2*b + b^3)*c^2
*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 - 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a
*b^2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2
- 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c)
+ 1) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x + 12*I*(a^
2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 -
6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d
^2*e*f^2*x + 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a
^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3 + a*b^2)*d^2*f^3*x^2 + 12*I*(a^2*b
+ b^3)*d^2*f^3*x^2 + 24*(a^3 + a*b^2)*d^2*e*f^2*x + 24*I*(a^2*b + b^3)*d^2*e*f^2*x + 24*(a^3 + a*b^2)*c*d*e*f
^2 + 24*I*(a^2*b + b^3)*c*d*e*f^2 - 12*(a^3 + a*b^2)*c^2*f^3 - 12*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)*sinh(
d*x + c) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x + 12*I*
(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^
3 - 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 12*((a^3 + a*b^2
)*f^3 + I*(a^2*b + b^3)*f^3 + ((a^3 + a*b^2)*f^3 + I*(a^2*b + b^3)*f^3)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*f^3
+ I*(a^2*b + b^3)*f^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*f^3 + I*(a^2*b + b^3)*f^3)*sinh(d*x + c)^
2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) - 12*((a^3 + a*b^2)*f^3 - I*(a^2*b + b^3)*f^3 + ((a^3 + a*b^2
)*f^3 - I*(a^2*b + b^3)*f^3)*cosh(d*x + c)^2 + 2*((a^3 + a*b^2)*f^3 - I*(a^2*b + b^3)*f^3)*cosh(d*x + c)*sinh(
d*x + c) + ((a^3 + a*b^2)*f^3 - I*(a^2*b + b^3)*f^3)*sinh(d*x + c)^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x
+ c)) - 4*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + (a^2*b +
b^3)*d^3*e^3 + 2*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3
*(a^3 + a*b^2)*c*d^2*e^2*f - 3*(a^3 + a*b^2)*c^2*d*e*f^2 + (a^3 + a*b^2)*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c)
)/((a^4 + 2*a^2*b^2 + b^4)*d^4*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^4*cosh(d*x + c)*sinh(d*x + c) + (
a^4 + 2*a^2*b^2 + b^4)*d^4*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^4)

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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)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 1.93, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \mathrm {sech}\left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*sech(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 \[ 3 \, a e^{2} 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 )} - 6 \, b f^{3} \int \frac {x^{2} 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} + 6 \, a f^{3} \int \frac {x^{2}}{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} - 12 \, b e 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} + 12 \, a e 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} + e^{3} {\left (\frac {b^{2} \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^{2}\right )}^{\frac {3}{2}} d} + \frac {2 \, {\left (b e^{\left (-d x - c\right )} + a\right )}}{{\left (a^{2} + b^{2} + {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d}\right )} - \frac {6 \, b e^{2} f \arctan \left (e^{\left (d x + c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d^{2}} - \frac {2 \, {\left (a f^{3} x^{3} + 3 \, a e f^{2} x^{2} + 3 \, a e^{2} f x - {\left (b f^{3} x^{3} e^{c} + 3 \, b e f^{2} x^{2} e^{c} + 3 \, b e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} d + b^{2} d + {\left (a^{2} d e^{\left (2 \, c\right )} + b^{2} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + \int -\frac {2 \, {\left (b^{2} f^{3} x^{3} e^{c} + 3 \, b^{2} e f^{2} x^{2} e^{c} + 3 \, b^{2} e^{2} f x e^{c}\right )} e^{\left (d x\right )}}{a^{2} b + b^{3} - {\left (a^{2} b e^{\left (2 \, c\right )} + b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} e^{c} + a b^{2} 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)^3*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)^3/(cosh(c + d*x)^2*(a + b*sinh(c + d*x))),x)

[Out]

int((e + f*x)^3/(cosh(c + d*x)^2*(a + b*sinh(c + d*x))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{3} \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*sech(c + d*x)**2/(a + b*sinh(c + d*x)), x)

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