3.329 \(\int \frac{(e+f x)^3 \cosh (c+d x)}{(a+b \sinh (c+d x))^3} \, dx\)

Optimal. Leaf size=631 \[ \frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac{3 f^3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )}+\frac{3 f^3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^4 \left (a^2+b^2\right )}-\frac{3 a f^3 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac{3 a f^3 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac{3 f^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac{3 f^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac{3 a f (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{3 a f (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac{3 f (e+f x)^2}{2 b d^2 \left (a^2+b^2\right )}-\frac{(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2} \]

[Out]

(-3*f*(e + f*x)^2)/(2*b*(a^2 + b^2)*d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b
*(a^2 + b^2)*d^3) + (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*
d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) - (3*a*f*(e + f*x)
^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^3*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqr
t[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(
a^2 + b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2
)*d^3) - (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (3*a*f^3*P
olyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sin
h[c + d*x])^2) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.09479, antiderivative size = 631, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423, Rules used = {5464, 3324, 3322, 2264, 2190, 2531, 2282, 6589, 5561, 2279, 2391} \[ \frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}-\frac{3 a f^2 (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^3 \left (a^2+b^2\right )^{3/2}}+\frac{3 f^3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )}+\frac{3 f^3 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^4 \left (a^2+b^2\right )}-\frac{3 a f^3 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac{3 a f^3 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{b d^4 \left (a^2+b^2\right )^{3/2}}+\frac{3 f^2 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac{3 f^2 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{b d^3 \left (a^2+b^2\right )}+\frac{3 a f (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{3 a f (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{2 b d^2 \left (a^2+b^2\right )^{3/2}}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{2 d^2 \left (a^2+b^2\right ) (a+b \sinh (c+d x))}-\frac{3 f (e+f x)^2}{2 b d^2 \left (a^2+b^2\right )}-\frac{(e+f x)^3}{2 b d (a+b \sinh (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*f*(e + f*x)^2)/(2*b*(a^2 + b^2)*d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b
*(a^2 + b^2)*d^3) + (3*a*f*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*
d^2) + (3*f^2*(e + f*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*(a^2 + b^2)*d^3) - (3*a*f*(e + f*x)
^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(2*b*(a^2 + b^2)^(3/2)*d^2) + (3*f^3*PolyLog[2, -((b*E^(c +
d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqr
t[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^3) + (3*f^3*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(
a^2 + b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2
)*d^3) - (3*a*f^3*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) + (3*a*f^3*P
olyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*(a^2 + b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*Sin
h[c + d*x])^2) - (3*f*(e + f*x)^2*Cosh[c + d*x])/(2*(a^2 + b^2)*d^2*(a + b*Sinh[c + d*x]))

Rule 5464

Int[Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)])^(n_.), x_Symbo
l] :> Simp[((e + f*x)^m*(a + b*Sinh[c + d*x])^(n + 1))/(b*d*(n + 1)), x] - Dist[(f*m)/(b*d*(n + 1)), Int[(e +
f*x)^(m - 1)*(a + b*Sinh[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n,
-1]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[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 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 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 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 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 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 5561

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 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 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]

Rubi steps

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

Mathematica [B]  time = 24.6796, size = 5753, normalized size = 9.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.43, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\cosh \left ( dx+c \right ) }{ \left ( a+b\sinh \left ( dx+c \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 4.17084, size = 24868, normalized size = 39.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(6*(a^2*b^2 + b^4)*d^2*e^2*f - 12*(a^2*b^2 + b^4)*c*d*e*f^2 + 6*(a^2*b^2 + b^4)*c^2*f^3 - 6*((a^2*b^2 + b^
4)*d^2*f^3*x^2 + 2*(a^2*b^2 + b^4)*d^2*e*f^2*x + 2*(a^2*b^2 + b^4)*c*d*e*f^2 - (a^2*b^2 + b^4)*c^2*f^3)*cosh(d
*x + c)^4 - 6*((a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^2*b^2 + b^4)*d^2*e*f^2*x + 2*(a^2*b^2 + b^4)*c*d*e*f^2 - (a^
2*b^2 + b^4)*c^2*f^3)*sinh(d*x + c)^4 - 6*(3*(a^3*b + a*b^3)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*e*f^2*x - (a^
3*b + a*b^3)*d^2*e^2*f + 8*(a^3*b + a*b^3)*c*d*e*f^2 - 4*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c)^3 - 6*(3*(a^3*
b + a*b^3)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*e*f^2*x - (a^3*b + a*b^3)*d^2*e^2*f + 8*(a^3*b + a*b^3)*c*d*e*f
^2 - 4*(a^3*b + a*b^3)*c^2*f^3 + 4*((a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^2*b^2 + b^4)*d^2*e*f^2*x + 2*(a^2*b^2 +
b^4)*c*d*e*f^2 - (a^2*b^2 + b^4)*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d^3*f
^3*x^3 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(2*a^4 + a^2*b^2 - b^4)*d^2*e^2*f + 12*(2*a^4 + a^2*b^2 - b^4)*
c*d*e*f^2 - 6*(2*a^4 + a^2*b^2 - b^4)*c^2*f^3 + 3*(2*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2 + (2*a^4 + a^2*b^2 - b^
4)*d^2*f^3)*x^2 + 6*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f + (2*a^4 + a^2*b^2 - b^4)*d^2*e*f^2)*x)*cosh(d*x + c)^2
- 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(2*a^4 + a^2*b^2 - b^4)*d^
2*e^2*f + 12*(2*a^4 + a^2*b^2 - b^4)*c*d*e*f^2 - 6*(2*a^4 + a^2*b^2 - b^4)*c^2*f^3 + 3*(2*(a^4 + 2*a^2*b^2 + b
^4)*d^3*e*f^2 + (2*a^4 + a^2*b^2 - b^4)*d^2*f^3)*x^2 + 18*((a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^2*b^2 + b^4)*d^2
*e*f^2*x + 2*(a^2*b^2 + b^4)*c*d*e*f^2 - (a^2*b^2 + b^4)*c^2*f^3)*cosh(d*x + c)^2 + 6*((a^4 + 2*a^2*b^2 + b^4)
*d^3*e^2*f + (2*a^4 + a^2*b^2 - b^4)*d^2*e*f^2)*x + 9*(3*(a^3*b + a*b^3)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*e
*f^2*x - (a^3*b + a*b^3)*d^2*e^2*f + 8*(a^3*b + a*b^3)*c*d*e*f^2 - 4*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c))*s
inh(d*x + c)^2 - 6*(a*b^3*f^3*cosh(d*x + c)^4 + a*b^3*f^3*sinh(d*x + c)^4 + 4*a^2*b^2*f^3*cosh(d*x + c)^3 - 4*
a^2*b^2*f^3*cosh(d*x + c) + a*b^3*f^3 + 2*(2*a^3*b - a*b^3)*f^3*cosh(d*x + c)^2 + 4*(a*b^3*f^3*cosh(d*x + c) +
a^2*b^2*f^3)*sinh(d*x + c)^3 + 2*(3*a*b^3*f^3*cosh(d*x + c)^2 + 6*a^2*b^2*f^3*cosh(d*x + c) + (2*a^3*b - a*b^
3)*f^3)*sinh(d*x + c)^2 + 4*(a*b^3*f^3*cosh(d*x + c)^3 + 3*a^2*b^2*f^3*cosh(d*x + c)^2 - a^2*b^2*f^3 + (2*a^3*
b - a*b^3)*f^3*cosh(d*x + c))*sinh(d*x + c))*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) + 6*(a*b^3*f^3*cosh(d*x + c)^4 + a*b^3*f^3*
sinh(d*x + c)^4 + 4*a^2*b^2*f^3*cosh(d*x + c)^3 - 4*a^2*b^2*f^3*cosh(d*x + c) + a*b^3*f^3 + 2*(2*a^3*b - a*b^3
)*f^3*cosh(d*x + c)^2 + 4*(a*b^3*f^3*cosh(d*x + c) + a^2*b^2*f^3)*sinh(d*x + c)^3 + 2*(3*a*b^3*f^3*cosh(d*x +
c)^2 + 6*a^2*b^2*f^3*cosh(d*x + c) + (2*a^3*b - a*b^3)*f^3)*sinh(d*x + c)^2 + 4*(a*b^3*f^3*cosh(d*x + c)^3 + 3
*a^2*b^2*f^3*cosh(d*x + c)^2 - a^2*b^2*f^3 + (2*a^3*b - a*b^3)*f^3*cosh(d*x + c))*sinh(d*x + c))*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) + 6*((a^3*b + a*b^3)*d^2*f^3*x^2 + 2*(a^3*b + a*b^3)*d^2*e*f^2*x - 3*(a^3*b + a*b^3)*d^2*e^2*f + 8*(a
^3*b + a*b^3)*c*d*e*f^2 - 4*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c) + 6*((a^2*b^2 + b^4)*f^3*cosh(d*x + c)^4 +
(a^2*b^2 + b^4)*f^3*sinh(d*x + c)^4 + 4*(a^3*b + a*b^3)*f^3*cosh(d*x + c)^3 + 2*(2*a^4 + a^2*b^2 - b^4)*f^3*co
sh(d*x + c)^2 - 4*(a^3*b + a*b^3)*f^3*cosh(d*x + c) + (a^2*b^2 + b^4)*f^3 + 4*((a^2*b^2 + b^4)*f^3*cosh(d*x +
c) + (a^3*b + a*b^3)*f^3)*sinh(d*x + c)^3 + 2*(3*(a^2*b^2 + b^4)*f^3*cosh(d*x + c)^2 + 6*(a^3*b + a*b^3)*f^3*c
osh(d*x + c) + (2*a^4 + a^2*b^2 - b^4)*f^3)*sinh(d*x + c)^2 + 4*((a^2*b^2 + b^4)*f^3*cosh(d*x + c)^3 + 3*(a^3*
b + a*b^3)*f^3*cosh(d*x + c)^2 + (2*a^4 + a^2*b^2 - b^4)*f^3*cosh(d*x + c) - (a^3*b + a*b^3)*f^3)*sinh(d*x + c
) + (a*b^3*d*f^3*x + a*b^3*d*e*f^2 + (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c)^4 + (a*b^3*d*f^3*x + a*b^3*
d*e*f^2)*sinh(d*x + c)^4 + 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c)^3 + 4*(a^2*b^2*d*f^3*x + a^2*b^
2*d*e*f^2 + (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d*f^3*x + (2
*a^3*b - a*b^3)*d*e*f^2)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d*f^3*x + (2*a^3*b - a*b^3)*d*e*f^2 + 3*(a*b^3
*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c)^2 + 6*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c)
^2 - 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c) - 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2 - (a*b^3*d*f^3
*x + a*b^3*d*e*f^2)*cosh(d*x + c)^3 - 3*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^
3)*d*f^3*x + (2*a^3*b - a*b^3)*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c))*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^2*b^2
+ b^4)*f^3*cosh(d*x + c)^4 + (a^2*b^2 + b^4)*f^3*sinh(d*x + c)^4 + 4*(a^3*b + a*b^3)*f^3*cosh(d*x + c)^3 + 2*(
2*a^4 + a^2*b^2 - b^4)*f^3*cosh(d*x + c)^2 - 4*(a^3*b + a*b^3)*f^3*cosh(d*x + c) + (a^2*b^2 + b^4)*f^3 + 4*((a
^2*b^2 + b^4)*f^3*cosh(d*x + c) + (a^3*b + a*b^3)*f^3)*sinh(d*x + c)^3 + 2*(3*(a^2*b^2 + b^4)*f^3*cosh(d*x + c
)^2 + 6*(a^3*b + a*b^3)*f^3*cosh(d*x + c) + (2*a^4 + a^2*b^2 - b^4)*f^3)*sinh(d*x + c)^2 + 4*((a^2*b^2 + b^4)*
f^3*cosh(d*x + c)^3 + 3*(a^3*b + a*b^3)*f^3*cosh(d*x + c)^2 + (2*a^4 + a^2*b^2 - b^4)*f^3*cosh(d*x + c) - (a^3
*b + a*b^3)*f^3)*sinh(d*x + c) - (a*b^3*d*f^3*x + a*b^3*d*e*f^2 + (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c
)^4 + (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*sinh(d*x + c)^4 + 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c)^3
+ 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2 + (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((
2*a^3*b - a*b^3)*d*f^3*x + (2*a^3*b - a*b^3)*d*e*f^2)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d*f^3*x + (2*a^3*
b - a*b^3)*d*e*f^2 + 3*(a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c)^2 + 6*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)
*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(d*x + c) - 4*(a^2*b^2*d*f^3*x + a
^2*b^2*d*e*f^2 - (a*b^3*d*f^3*x + a*b^3*d*e*f^2)*cosh(d*x + c)^3 - 3*(a^2*b^2*d*f^3*x + a^2*b^2*d*e*f^2)*cosh(
d*x + c)^2 - ((2*a^3*b - a*b^3)*d*f^3*x + (2*a^3*b - a*b^3)*d*e*f^2)*cosh(d*x + c))*sinh(d*x + c))*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) + 3*(2*(a^2*b^2 + b^4)*d*e*f^2 - 2*(a^2*b^2 + b^4)*c*f^3 + 2*((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^
2 + b^4)*c*f^3)*cosh(d*x + c)^4 + 2*((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*sinh(d*x + c)^4 + 8*((a^
3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^3 + 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c
*f^3 + ((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*((2*a^4 + a^2*b^2
- b^4)*d*e*f^2 - (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c)^2 + 4*((2*a^4 + a^2*b^2 - b^4)*d*e*f^2 - (2*a^4
+ a^2*b^2 - b^4)*c*f^3 + 3*((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^2 + 6*((a^3*b + a*b
^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*
b^3)*c*f^3)*cosh(d*x + c) - 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3 - ((a^2*b^2 + b^4)*d*e*f^2 - (a
^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^3 - 3*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^2 - (
(2*a^4 + a^2*b^2 - b^4)*d*e*f^2 - (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c) - (a*b^3*d^2*e^2
*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3 + (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x + c)^4
+ (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*sinh(d*x + c)^4 + 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*
d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c)^3 + 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3 + (a
*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d^2*
e^2*f - 2*(2*a^3*b - a*b^3)*c*d*e*f^2 + (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d^2*
e^2*f - 2*(2*a^3*b - a*b^3)*c*d*e*f^2 + (2*a^3*b - a*b^3)*c^2*f^3 + 3*(a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a
*b^3*c^2*f^3)*cosh(d*x + c)^2 + 6*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c))*s
inh(d*x + c)^2 - 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c) - 4*(a^2*b^2*d^2*
e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3 - (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x
+ c)^3 - 3*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d
^2*e^2*f - 2*(2*a^3*b - a*b^3)*c*d*e*f^2 + (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c))*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) + 3*(2*(a^2*b^2 + b^
4)*d*e*f^2 - 2*(a^2*b^2 + b^4)*c*f^3 + 2*((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^4 + 2
*((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*sinh(d*x + c)^4 + 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b
^3)*c*f^3)*cosh(d*x + c)^3 + 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3 + ((a^2*b^2 + b^4)*d*e*f^2 - (
a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*((2*a^4 + a^2*b^2 - b^4)*d*e*f^2 - (2*a^4 + a^2*b^2 -
b^4)*c*f^3)*cosh(d*x + c)^2 + 4*((2*a^4 + a^2*b^2 - b^4)*d*e*f^2 - (2*a^4 + a^2*b^2 - b^4)*c*f^3 + 3*((a^2*b^
2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^2 + 6*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3
)*cosh(d*x + c))*sinh(d*x + c)^2 - 8*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c) - 8*((a^3
*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3 - ((a^2*b^2 + b^4)*d*e*f^2 - (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^
3 - 3*((a^3*b + a*b^3)*d*e*f^2 - (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^2 - ((2*a^4 + a^2*b^2 - b^4)*d*e*f^2 - (
2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c) + (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*
f^3 + (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x + c)^4 + (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e
*f^2 + a*b^3*c^2*f^3)*sinh(d*x + c)^4 + 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x
+ c)^3 + 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3 + (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2
+ a*b^3*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d^2*e^2*f - 2*(2*a^3*b - a*b^3)*c*d*e*f
^2 + (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d^2*e^2*f - 2*(2*a^3*b - a*b^3)*c*d*e*f
^2 + (2*a^3*b - a*b^3)*c^2*f^3 + 3*(a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x + c)^2 + 6*(
a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(a^2*b^2*d^2*e^2
*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c) - 4*(a^2*b^2*d^2*e^2*f - 2*a^2*b^2*c*d*e*f^2 + a^2*b
^2*c^2*f^3 - (a*b^3*d^2*e^2*f - 2*a*b^3*c*d*e*f^2 + a*b^3*c^2*f^3)*cosh(d*x + c)^3 - 3*(a^2*b^2*d^2*e^2*f - 2*
a^2*b^2*c*d*e*f^2 + a^2*b^2*c^2*f^3)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d^2*e^2*f - 2*(2*a^3*b - a*b^3)*c*d*
e*f^2 + (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c))*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) + 3*(2*(a^2*b^2 + b^4)*d*f^3*x + 2*(a^2*b^2 + b^4)*c*f^
3 + 2*((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^4 + 2*((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^
2 + b^4)*c*f^3)*sinh(d*x + c)^4 + 8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^3 + 8*((a^
3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3 + ((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)
)*sinh(d*x + c)^3 + 4*((2*a^4 + a^2*b^2 - b^4)*d*f^3*x + (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c)^2 + 4*((
2*a^4 + a^2*b^2 - b^4)*d*f^3*x + (2*a^4 + a^2*b^2 - b^4)*c*f^3 + 3*((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*
c*f^3)*cosh(d*x + c)^2 + 6*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 -
8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c) - 8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^
3)*c*f^3 - ((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^3 - 3*((a^3*b + a*b^3)*d*f^3*x + (a
^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^2 - ((2*a^4 + a^2*b^2 - b^4)*d*f^3*x + (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(
d*x + c))*sinh(d*x + c) + (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3 + (a*b^
3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c)^4 + (a*b^3*d^2*f^3*x^2
+ 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*sinh(d*x + c)^4 + 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^
2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c)^3 + 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^
2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3 + (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f
^2 - a*b^3*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d^2*f^3*x^2 + 2*(2*a^3*b - a*b^3)*d^
2*e*f^2*x + 2*(2*a^3*b - a*b^3)*c*d*e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*
d^2*f^3*x^2 + 2*(2*a^3*b - a*b^3)*d^2*e*f^2*x + 2*(2*a^3*b - a*b^3)*c*d*e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3 + 3*
(a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c)^2 + 6*(a^2*b^2*d^2
*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(
a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c) - 4*(a^2*b^
2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d
^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c)^3 - 3*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2
*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d^2*f^3*x^2 + 2*(2*a^3*b - a*
b^3)*d^2*e*f^2*x + 2*(2*a^3*b - a*b^3)*c*d*e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c))*sq
rt((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) + 3*(2*(a^2*b^2 + b^4)*d*f^3*x + 2*(a^2*b^2 + b^4)*c*f^3 + 2*((a^2*b^2 + b^4)*d*f^3*x + (a^
2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^4 + 2*((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*sinh(d*x + c)^4 + 8*
((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^3 + 8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^
3)*c*f^3 + ((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*((2*a^4 + a^2*
b^2 - b^4)*d*f^3*x + (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c)^2 + 4*((2*a^4 + a^2*b^2 - b^4)*d*f^3*x + (2*
a^4 + a^2*b^2 - b^4)*c*f^3 + 3*((a^2*b^2 + b^4)*d*f^3*x + (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^2 + 6*((a^3*b +
a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b
+ a*b^3)*c*f^3)*cosh(d*x + c) - 8*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3 - ((a^2*b^2 + b^4)*d*f^3*x
+ (a^2*b^2 + b^4)*c*f^3)*cosh(d*x + c)^3 - 3*((a^3*b + a*b^3)*d*f^3*x + (a^3*b + a*b^3)*c*f^3)*cosh(d*x + c)^2
- ((2*a^4 + a^2*b^2 - b^4)*d*f^3*x + (2*a^4 + a^2*b^2 - b^4)*c*f^3)*cosh(d*x + c))*sinh(d*x + c) - (a*b^3*d^2
*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3 + (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x
+ 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c)^4 + (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*
e*f^2 - a*b^3*c^2*f^3)*sinh(d*x + c)^4 + 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2
- a^2*b^2*c^2*f^3)*cosh(d*x + c)^3 + 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^
2*b^2*c^2*f^3 + (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c))*s
inh(d*x + c)^3 + 2*((2*a^3*b - a*b^3)*d^2*f^3*x^2 + 2*(2*a^3*b - a*b^3)*d^2*e*f^2*x + 2*(2*a^3*b - a*b^3)*c*d*
e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + 2*((2*a^3*b - a*b^3)*d^2*f^3*x^2 + 2*(2*a^3*b - a*b^3)*d^
2*e*f^2*x + 2*(2*a^3*b - a*b^3)*c*d*e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3 + 3*(a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f
^2*x + 2*a*b^3*c*d*e*f^2 - a*b^3*c^2*f^3)*cosh(d*x + c)^2 + 6*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2
*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*
e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3)*cosh(d*x + c) - 4*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*
x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c^2*f^3 - (a*b^3*d^2*f^3*x^2 + 2*a*b^3*d^2*e*f^2*x + 2*a*b^3*c*d*e*f^2 - a*b
^3*c^2*f^3)*cosh(d*x + c)^3 - 3*(a^2*b^2*d^2*f^3*x^2 + 2*a^2*b^2*d^2*e*f^2*x + 2*a^2*b^2*c*d*e*f^2 - a^2*b^2*c
^2*f^3)*cosh(d*x + c)^2 - ((2*a^3*b - a*b^3)*d^2*f^3*x^2 + 2*(2*a^3*b - a*b^3)*d^2*e*f^2*x + 2*(2*a^3*b - a*b^
3)*c*d*e*f^2 - (2*a^3*b - a*b^3)*c^2*f^3)*cosh(d*x + c))*sinh(d*x + c))*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*(3*(a^3*b + a
*b^3)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*e*f^2*x - 9*(a^3*b + a*b^3)*d^2*e^2*f + 24*(a^3*b + a*b^3)*c*d*e*f^2
- 12*(a^3*b + a*b^3)*c^2*f^3 - 12*((a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^2*b^2 + b^4)*d^2*e*f^2*x + 2*(a^2*b^2 +
b^4)*c*d*e*f^2 - (a^2*b^2 + b^4)*c^2*f^3)*cosh(d*x + c)^3 - 9*(3*(a^3*b + a*b^3)*d^2*f^3*x^2 + 6*(a^3*b + a*b
^3)*d^2*e*f^2*x - (a^3*b + a*b^3)*d^2*e^2*f + 8*(a^3*b + a*b^3)*c*d*e*f^2 - 4*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*
x + c)^2 - 2*(2*(a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(2*a^4 + a^2*b^2 -
b^4)*d^2*e^2*f + 12*(2*a^4 + a^2*b^2 - b^4)*c*d*e*f^2 - 6*(2*a^4 + a^2*b^2 - b^4)*c^2*f^3 + 3*(2*(a^4 + 2*a^2
*b^2 + b^4)*d^3*e*f^2 + (2*a^4 + a^2*b^2 - b^4)*d^2*f^3)*x^2 + 6*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f + (2*a^4 +
a^2*b^2 - b^4)*d^2*e*f^2)*x)*cosh(d*x + c))*sinh(d*x + c))/((a^4*b^3 + 2*a^2*b^5 + b^7)*d^4*cosh(d*x + c)^4 +
(a^4*b^3 + 2*a^2*b^5 + b^7)*d^4*sinh(d*x + c)^4 + 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4*cosh(d*x + c)^3 + 2*(2*
a^6*b + 3*a^4*b^3 - b^7)*d^4*cosh(d*x + c)^2 - 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4*cosh(d*x + c) + (a^4*b^3 +
2*a^2*b^5 + b^7)*d^4 + 4*((a^4*b^3 + 2*a^2*b^5 + b^7)*d^4*cosh(d*x + c) + (a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4)*s
inh(d*x + c)^3 + 2*(3*(a^4*b^3 + 2*a^2*b^5 + b^7)*d^4*cosh(d*x + c)^2 + 6*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4*co
sh(d*x + c) + (2*a^6*b + 3*a^4*b^3 - b^7)*d^4)*sinh(d*x + c)^2 + 4*((a^4*b^3 + 2*a^2*b^5 + b^7)*d^4*cosh(d*x +
c)^3 + 3*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4*cosh(d*x + c)^2 + (2*a^6*b + 3*a^4*b^3 - b^7)*d^4*cosh(d*x + c) -
(a^5*b^2 + 2*a^3*b^4 + a*b^6)*d^4)*sinh(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )}{{\left (b \sinh \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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