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

Optimal. Leaf size=642 \[ \frac {6 f^3 \left (a^2+b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 f^3 \left (a^2+b^2\right ) \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d} \]

[Out]

3/8*f^3*x/b/d^3+1/4*(f*x+e)^3/b/d-1/4*(a^2+b^2)*(f*x+e)^4/b^3/f+6*a*f^3*cosh(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*co
sh(d*x+c)/b^2/d^2+(a^2+b^2)*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d+(a^2+b^2)*(f*x+e)^3*ln(1+b*
exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3
/d^2+3*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^2-6*(a^2+b^2)*f^2*(f*x+e)*poly
log(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^3-6*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^
(1/2)))/b^3/d^3+6*(a^2+b^2)*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^3/d^4+6*(a^2+b^2)*f^3*polylog(4
,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^3/d^4-6*a*f^2*(f*x+e)*sinh(d*x+c)/b^2/d^3-a*(f*x+e)^3*sinh(d*x+c)/b^2/d-
3/8*f^3*cosh(d*x+c)*sinh(d*x+c)/b/d^4-3/4*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/b/d^2+3/4*f^2*(f*x+e)*sinh(d*x+c
)^2/b/d^3+1/2*(f*x+e)^3*sinh(d*x+c)^2/b/d

________________________________________________________________________________________

Rubi [A]  time = 0.78, antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5565, 3296, 2638, 5446, 3311, 32, 2635, 8, 5561, 2190, 2531, 6609, 2282, 6589} \[ -\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 f^2 \left (a^2+b^2\right ) (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^3}+\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 f \left (a^2+b^2\right ) (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^2}+\frac {6 f^3 \left (a^2+b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 f^3 \left (a^2+b^2\right ) \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}\right )}{b^3 d^4}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \sinh (c+d x) \cosh (c+d x)}{4 b d^2}-\frac {3 f^3 \sinh (c+d x) \cosh (c+d x)}{8 b d^4}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) - ((a^2 + b^2)*(e + f*x)^4)/(4*b^3*f) + (6*a*f^3*Cosh[c + d*x])/(b^2
*d^4) + (3*a*f*(e + f*x)^2*Cosh[c + d*x])/(b^2*d^2) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sq
rt[a^2 + b^2])])/(b^3*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3*d) +
(3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*(a^2 + b^2)*
f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) - (6*(a^2 + b^2)*f^2*(e + f*x)*P
olyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^
2 + b^2]))])/(b^3*d^4) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) -
(6*a*f^2*(e + f*x)*Sinh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Sinh[c + d*x])/(b^2*d) - (3*f^3*Cosh[c + d*x]*Sin
h[c + d*x])/(8*b*d^4) - (3*f*(e + f*x)^2*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^2) + (3*f^2*(e + f*x)*Sinh[c + d*
x]^2)/(4*b*d^3) + ((e + f*x)^3*Sinh[c + d*x]^2)/(2*b*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

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

Rule 3311

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

Rule 5446

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

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 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 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]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=-\frac {a \int (e+f x)^3 \cosh (c+d x) \, dx}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{b}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}\\ &=-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(3 a f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b^2 d}-\frac {(3 f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 b d}\\ &=-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {(3 f) \int (e+f x)^2 \, dx}{4 b d}-\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (3 \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (6 a f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b^2 d^2}-\frac {\left (3 f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 b d^3}\\ &=\frac {(e+f x)^3}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}-\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (6 \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}+\frac {\left (6 a f^3\right ) \int \sinh (c+d x) \, dx}{b^2 d^3}+\frac {\left (3 f^3\right ) \int 1 \, dx}{8 b d^3}\\ &=\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3}+\frac {\left (6 \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3}\\ &=\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}+\frac {\left (6 \left (a^2+b^2\right ) 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 )}{b^3 d^4}+\frac {\left (6 \left (a^2+b^2\right ) 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 )}{b^3 d^4}\\ &=\frac {3 f^3 x}{8 b d^3}+\frac {(e+f x)^3}{4 b d}-\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 b^3 f}+\frac {6 a f^3 \cosh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \cosh (c+d x)}{b^2 d^2}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {\left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}-\frac {6 a f^2 (e+f x) \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \sinh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh (c+d x) \sinh (c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^3 \sinh ^2(c+d x)}{2 b d}\\ \end {align*}

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Mathematica [B]  time = 23.92, size = 10263, normalized size = 15.99 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [C]  time = 0.67, size = 4371, normalized size = 6.81 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 + 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 + 3*b^2*f^3 + (4*b^2*d^3*f^3*x^3 + 4
*b^2*d^3*e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*
d^3*e^2*f - 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c)^4 + (4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 - 6*b^2*d^2*e
^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f - 2*b^2*d^2*e*f^
2 + b^2*d*f^3)*x)*sinh(d*x + c)^4 - 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*
b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x +
c)^3 - 4*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 - 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 - 24*a*b*f^3 + 12*(a*b*d^3*e*f
^2 - a*b*d^2*f^3)*x^2 + 12*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x - (4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*
e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f
- 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(2*b^2*d^3*e*f^2 + b^2*d^2*f^3)*x^2 - 8*
((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*e*f^2*x^3 + 6*(a^2 + b^2)*d^4*e^2*f*x^2 + 4*(a^2 + b^2)*d^4*e^3*x
+ 8*(a^2 + b^2)*c*d^3*e^3 - 12*(a^2 + b^2)*c^2*d^2*e^2*f + 8*(a^2 + b^2)*c^3*d*e*f^2 - 2*(a^2 + b^2)*c^4*f^3)
*cosh(d*x + c)^2 - 2*(4*(a^2 + b^2)*d^4*f^3*x^4 + 16*(a^2 + b^2)*d^4*e*f^2*x^3 + 24*(a^2 + b^2)*d^4*e^2*f*x^2
+ 16*(a^2 + b^2)*d^4*e^3*x + 32*(a^2 + b^2)*c*d^3*e^3 - 48*(a^2 + b^2)*c^2*d^2*e^2*f + 32*(a^2 + b^2)*c^3*d*e*
f^2 - 8*(a^2 + b^2)*c^4*f^3 - 3*(4*b^2*d^3*f^3*x^3 + 4*b^2*d^3*e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f
^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*e^2*f - 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c
)^2 + 24*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b
*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 6*(2*b^2
*d^3*e^2*f + 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x + 16*(a*b*d^3*f^3*x^3 + a*b*d^3*e^3 + 3*a*b*d^2*e^2*f + 6*a*b*d*e*
f^2 + 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*c
osh(d*x + c) + 96*(((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*cosh(d*x + c)
^2 + 2*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x +
c) + ((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*c
osh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 96*((
(a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*
d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d^
2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sin
h(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 32*(((a^2 + b^2)*d^3*e^3
- 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*cosh(d*x + c)^2 + 2*((a^2 + b^2
)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*cosh(d*x + c)*sinh(d*
x + c) + ((a^2 + b^2)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*s
inh(d*x + c)^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) + 32*(((a^2 + b^2
)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*cosh(d*x + c)^2 + 2*(
(a^2 + b^2)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*c^3*f^3)*cosh(d*x +
c)*sinh(d*x + c) + ((a^2 + b^2)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^2 - (a^2 + b^2)*
c^3*f^3)*sinh(d*x + c)^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) + 32*((
(a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f
- 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c^3*f^3)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2
)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b
^2)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b
^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c^3*f^3)*sinh(d*x + c)^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)
+ 32*(((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2
*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c^3*f^3)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^
2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (
a^2 + b^2)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(
a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c^3*f^3)*sinh(d*x
+ c)^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) + 192*((a^2 + b^2)*f^3*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^
3*sinh(d*x + c)^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) + 192*((a^2 + b^2)*f^3*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a
^2 + b^2)*f^3*sinh(d*x + c)^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) - 192*(((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)^2 + 2*((a^2 +
b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*
sinh(d*x + c)^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) - 192*(((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d*f^3*x +
(a^2 + b^2)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*sinh(d*x + c)^
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*(4*a*b*d^3*f^3*x^3 + 4*a*b*d^3*e^3 + 12*a*b*d^2*e^2*f + 24*a*b*d*e*f^2 + 24*a*b*f^3 + (4*b^2*d^3*f^3*x^
3 + 4*b^2*d^3*e^3 - 6*b^2*d^2*e^2*f + 6*b^2*d*e*f^2 - 3*b^2*f^3 + 6*(2*b^2*d^3*e*f^2 - b^2*d^2*f^3)*x^2 + 6*(2
*b^2*d^3*e^2*f - 2*b^2*d^2*e*f^2 + b^2*d*f^3)*x)*cosh(d*x + c)^3 + 12*(a*b*d^3*e*f^2 + a*b*d^2*f^3)*x^2 - 12*(
a*b*d^3*f^3*x^3 + a*b*d^3*e^3 - 3*a*b*d^2*e^2*f + 6*a*b*d*e*f^2 - 6*a*b*f^3 + 3*(a*b*d^3*e*f^2 - a*b*d^2*f^3)*
x^2 + 3*(a*b*d^3*e^2*f - 2*a*b*d^2*e*f^2 + 2*a*b*d*f^3)*x)*cosh(d*x + c)^2 + 12*(a*b*d^3*e^2*f + 2*a*b*d^2*e*f
^2 + 2*a*b*d*f^3)*x - 4*((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*e*f^2*x^3 + 6*(a^2 + b^2)*d^4*e^2*f*x^2 +
4*(a^2 + b^2)*d^4*e^3*x + 8*(a^2 + b^2)*c*d^3*e^3 - 12*(a^2 + b^2)*c^2*d^2*e^2*f + 8*(a^2 + b^2)*c^3*d*e*f^2
- 2*(a^2 + b^2)*c^4*f^3)*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^4*cosh(d*x + c)^2 + 2*b^3*d^4*cosh(d*x + c)*sinh
(d*x + c) + b^3*d^4*sinh(d*x + c)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cosh ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{8} \, e^{3} {\left (\frac {{\left (4 \, a e^{\left (-d x - c\right )} - b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{b^{2} d} - \frac {8 \, {\left (a^{2} + b^{2}\right )} {\left (d x + c\right )}}{b^{3} d} - \frac {4 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2} d} - \frac {8 \, {\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{b^{3} d}\right )} + \frac {{\left (8 \, {\left (a^{2} d^{4} f^{3} e^{\left (2 \, c\right )} + b^{2} d^{4} f^{3} e^{\left (2 \, c\right )}\right )} x^{4} + 32 \, {\left (a^{2} d^{4} e f^{2} e^{\left (2 \, c\right )} + b^{2} d^{4} e f^{2} e^{\left (2 \, c\right )}\right )} x^{3} + 48 \, {\left (a^{2} d^{4} e^{2} f e^{\left (2 \, c\right )} + b^{2} d^{4} e^{2} f e^{\left (2 \, c\right )}\right )} x^{2} + {\left (4 \, b^{2} d^{3} f^{3} x^{3} e^{\left (4 \, c\right )} + 6 \, {\left (2 \, d^{3} e f^{2} - d^{2} f^{3}\right )} b^{2} x^{2} e^{\left (4 \, c\right )} + 6 \, {\left (2 \, d^{3} e^{2} f - 2 \, d^{2} e f^{2} + d f^{3}\right )} b^{2} x e^{\left (4 \, c\right )} - 3 \, {\left (2 \, d^{2} e^{2} f - 2 \, d e f^{2} + f^{3}\right )} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 16 \, {\left (a b d^{3} f^{3} x^{3} e^{\left (3 \, c\right )} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} a b x^{2} e^{\left (3 \, c\right )} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a b x e^{\left (3 \, c\right )} - 3 \, {\left (d^{2} e^{2} f - 2 \, d e f^{2} + 2 \, f^{3}\right )} a b e^{\left (3 \, c\right )}\right )} e^{\left (d x\right )} + 16 \, {\left (a b d^{3} f^{3} x^{3} e^{c} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} a b x^{2} e^{c} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} a b x e^{c} + 3 \, {\left (d^{2} e^{2} f + 2 \, d e f^{2} + 2 \, f^{3}\right )} a b e^{c}\right )} e^{\left (-d x\right )} + {\left (4 \, b^{2} d^{3} f^{3} x^{3} + 6 \, {\left (2 \, d^{3} e f^{2} + d^{2} f^{3}\right )} b^{2} x^{2} + 6 \, {\left (2 \, d^{3} e^{2} f + 2 \, d^{2} e f^{2} + d f^{3}\right )} b^{2} x + 3 \, {\left (2 \, d^{2} e^{2} f + 2 \, d e f^{2} + f^{3}\right )} b^{2}\right )} e^{\left (-2 \, d x\right )}\right )} e^{\left (-2 \, c\right )}}{32 \, b^{3} d^{4}} - \int -\frac {2 \, {\left ({\left (a^{2} b f^{3} + b^{3} f^{3}\right )} x^{3} + 3 \, {\left (a^{2} b e f^{2} + b^{3} e f^{2}\right )} x^{2} + 3 \, {\left (a^{2} b e^{2} f + b^{3} e^{2} f\right )} x - {\left ({\left (a^{3} f^{3} e^{c} + a b^{2} f^{3} e^{c}\right )} x^{3} + 3 \, {\left (a^{3} e f^{2} e^{c} + a b^{2} e f^{2} e^{c}\right )} x^{2} + 3 \, {\left (a^{3} e^{2} f e^{c} + a b^{2} e^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}\right )}}{b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{3} e^{\left (d x + c\right )} - b^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*e^3*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) - 8*(a^2 + b^2)*(d*x + c)/(b^3*d) - (4*a*e^(-d*x - c)
+ b*e^(-2*d*x - 2*c))/(b^2*d) - 8*(a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^3*d)) + 1/32
*(8*(a^2*d^4*f^3*e^(2*c) + b^2*d^4*f^3*e^(2*c))*x^4 + 32*(a^2*d^4*e*f^2*e^(2*c) + b^2*d^4*e*f^2*e^(2*c))*x^3 +
48*(a^2*d^4*e^2*f*e^(2*c) + b^2*d^4*e^2*f*e^(2*c))*x^2 + (4*b^2*d^3*f^3*x^3*e^(4*c) + 6*(2*d^3*e*f^2 - d^2*f^
3)*b^2*x^2*e^(4*c) + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 + d*f^3)*b^2*x*e^(4*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*b
^2*e^(4*c))*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) + 3*(d^3*e*f^2 - d^2*f^3)*a*b*x^2*e^(3*c) + 3*(d^3*e^2*f -
2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^(3*c) - 3*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b*e^(3*c))*e^(d*x) + 16*(a*b*d^3*f
^3*x^3*e^c + 3*(d^3*e*f^2 + d^2*f^3)*a*b*x^2*e^c + 3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b*x*e^c + 3*(d^2*e^
2*f + 2*d*e*f^2 + 2*f^3)*a*b*e^c)*e^(-d*x) + (4*b^2*d^3*f^3*x^3 + 6*(2*d^3*e*f^2 + d^2*f^3)*b^2*x^2 + 6*(2*d^3
*e^2*f + 2*d^2*e*f^2 + d*f^3)*b^2*x + 3*(2*d^2*e^2*f + 2*d*e*f^2 + f^3)*b^2)*e^(-2*d*x))*e^(-2*c)/(b^3*d^4) -
integrate(-2*((a^2*b*f^3 + b^3*f^3)*x^3 + 3*(a^2*b*e*f^2 + b^3*e*f^2)*x^2 + 3*(a^2*b*e^2*f + b^3*e^2*f)*x - ((
a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3*e*f^2*e^c + a*b^2*e*f^2*e^c)*x^2 + 3*(a^3*e^2*f*e^c + a*b^2*e^2*f*e^
c)*x)*e^(d*x))/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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*cosh(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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