3.1.26 \(\int \frac {(e+f x)^3 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [26]

Optimal. Leaf size=864 \[ -\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d} \]

[Out]

-3/8*b*f^3*x/a^2/d^3+1/4*b*(a^2+b^2)*(f*x+e)^4/a^4/f-6*b^2*f^3*cosh(d*x+c)/a^3/d^4-1/3*f*(f*x+e)^2*cosh(d*x+c)
^3/a/d^2+1/3*(f*x+e)^3*cosh(d*x+c)^2*sinh(d*x+c)/a/d-1/2*b*(f*x+e)^3*sinh(d*x+c)^2/a^2/d-40/9*f^3*cosh(d*x+c)/
a/d^4-1/4*b*(f*x+e)^3/a^2/d-2/27*f^3*cosh(d*x+c)^3/a/d^4+2/3*(f*x+e)^3*sinh(d*x+c)/a/d-2*f*(f*x+e)^2*cosh(d*x+
c)/a/d^2+40/9*f^2*(f*x+e)*sinh(d*x+c)/a/d^3+b^2*(f*x+e)^3*sinh(d*x+c)/a^3/d-3*b*(a^2+b^2)*f*(f*x+e)^2*polylog(
2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^2-3*b*(a^2+b^2)*f*(f*x+e)^2*polylog(2,-a*exp(d*x+c)/(b+(a^2+b^2)^(1
/2)))/a^4/d^2+6*b*(a^2+b^2)*f^2*(f*x+e)*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^3+6*b*(a^2+b^2)*f^2
*(f*x+e)*polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^3+3/4*b*f*(f*x+e)^2*cosh(d*x+c)*sinh(d*x+c)/a^2/d^
2-b*(a^2+b^2)*(f*x+e)^3*ln(1+a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*(f*x+e)^3*ln(1+a*exp(d*x+c)/(
b+(a^2+b^2)^(1/2)))/a^4/d-6*b*(a^2+b^2)*f^3*polylog(4,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^4-6*b*(a^2+b^2)
*f^3*polylog(4,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^4-3*b^2*f*(f*x+e)^2*cosh(d*x+c)/a^3/d^2+6*b^2*f^2*(f*x
+e)*sinh(d*x+c)/a^3/d^3+3/8*b*f^3*cosh(d*x+c)*sinh(d*x+c)/a^2/d^4+2/9*f^2*(f*x+e)*cosh(d*x+c)^2*sinh(d*x+c)/a/
d^3-3/4*b*f^2*(f*x+e)*sinh(d*x+c)^2/a^2/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.85, antiderivative size = 864, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 17, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {5713, 5698, 3392, 3377, 2718, 3391, 5684, 5554, 32, 2715, 8, 5680, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {b \sinh ^2(c+d x) (e+f x)^3}{2 a^2 d}-\frac {b \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} a}{b-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}-\frac {b \left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} a}{b+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}+\frac {\cosh ^2(c+d x) \sinh (c+d x) (e+f x)^3}{3 a d}+\frac {b^2 \sinh (c+d x) (e+f x)^3}{a^3 d}+\frac {2 \sinh (c+d x) (e+f x)^3}{3 a d}-\frac {b (e+f x)^3}{4 a^2 d}-\frac {f \cosh ^3(c+d x) (e+f x)^2}{3 a d^2}-\frac {3 b^2 f \cosh (c+d x) (e+f x)^2}{a^3 d^2}-\frac {2 f \cosh (c+d x) (e+f x)^2}{a d^2}-\frac {3 b \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}+\frac {3 b f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{4 a^2 d^2}-\frac {3 b f^2 \sinh ^2(c+d x) (e+f x)}{4 a^2 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac {6 b^2 f^2 \sinh (c+d x) (e+f x)}{a^3 d^3}+\frac {40 f^2 \sinh (c+d x) (e+f x)}{9 a d^3}+\frac {2 f^2 \cosh ^2(c+d x) \sinh (c+d x) (e+f x)}{9 a d^3}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*f^3*x)/(8*a^2*d^3) - (b*(e + f*x)^3)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^4)/(4*a^4*f) - (40*f^3*Cosh[c
+ d*x])/(9*a*d^4) - (6*b^2*f^3*Cosh[c + d*x])/(a^3*d^4) - (2*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) - (3*b^2*f*(
e + f*x)^2*Cosh[c + d*x])/(a^3*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*a*d^4) - (f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*
a*d^2) - (b*(a^2 + b^2)*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(
e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2,
-((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((a*E^(c +
d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2]))])/(a^4*d^3) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]
))])/(a^4*d^3) - (6*b*(a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^4) - (6*b*(
a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^4) + (40*f^2*(e + f*x)*Sinh[c + d*
x])/(9*a*d^3) + (6*b^2*f^2*(e + f*x)*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^3*Sinh[c + d*x])/(3*a*d) + (b^2*(
e + f*x)^3*Sinh[c + d*x])/(a^3*d) + (3*b*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d^4) + (3*b*f*(e + f*x)^2*Cos
h[c + d*x]*Sinh[c + d*x])/(4*a^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*a*d^3) + ((e + f*x)
^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (3*b*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^3*S
inh[c + d*x]^2)/(2*a^2*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 2221

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

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3391

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

Rule 3392

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 5554

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 5680

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

Rule 5684

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 5698

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

Rule 5713

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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 \text {csch}(c+d x)} \, dx &=\int \frac {(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac {\int (e+f x)^3 \cosh ^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}+\frac {2 \int (e+f x)^3 \cosh (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^3 \cosh (c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac {\left (2 f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {(2 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}+\frac {(3 b f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a^2 d}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{a^3 d}+\frac {\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 a d^2}\\ &=\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}+\frac {4 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {(3 b f) \int (e+f x)^2 \, dx}{4 a^2 d}+\frac {\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a^3 d^2}-\frac {\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{9 a d^3}+\frac {\left (3 b f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a^2 d^3}\\ &=-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {4 f^3 \cosh (c+d x)}{9 a d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}+\frac {\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2}+\frac {\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2}-\frac {\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}-\frac {\left (3 b f^3\right ) \int 1 \, dx}{8 a^2 d^3}-\frac {\left (6 b^2 f^3\right ) \int \sinh (c+d x) \, dx}{a^3 d^3}\\ &=-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^3}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^3}\\ &=-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4}-\frac {\left (6 b \left (a^2+b^2\right ) f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4}\\ &=-\frac {3 b f^3 x}{8 a^2 d^3}-\frac {b (e+f x)^3}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac {40 f^3 \cosh (c+d x)}{9 a d^4}-\frac {6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac {2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac {3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac {2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac {f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {3 b \left (a^2+b^2\right ) f (e+f x)^2 \text {Li}_2\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {6 b \left (a^2+b^2\right ) f^2 (e+f x) \text {Li}_3\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^4}-\frac {6 b \left (a^2+b^2\right ) f^3 \text {Li}_4\left (-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^4}+\frac {40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac {6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac {3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac {3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac {2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.06, size = 5462, normalized size = 6.32 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 4.58, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cosh ^{3}\left (d x +c \right )\right )}{a +b \,\mathrm {csch}\left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b + b
^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^3*
d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4*d))*e^3 - 1/864*(216*(a^2*b*d^4*f^3
*e^(3*c) + b^3*d^4*f^3*e^(3*c))*x^4 + 864*(a^2*b*d^4*f^2*e^(3*c) + b^3*d^4*f^2*e^(3*c))*x^3*e + 1296*(a^2*b*d^
4*f*e^(3*c) + b^3*d^4*f*e^(3*c))*x^2*e^2 - 4*(9*a^3*d^3*f^3*x^3*e^(6*c) - 2*a^3*f^3*e^(6*c) - 9*a^3*d^2*f*e^(6
*c + 2) + 6*a^3*d*f^2*e^(6*c + 1) - 9*(a^3*d^2*f^3*e^(6*c) - 3*a^3*d^3*f^2*e^(6*c + 1))*x^2 + 3*(2*a^3*d*f^3*e
^(6*c) + 9*a^3*d^3*f*e^(6*c + 2) - 6*a^3*d^2*f^2*e^(6*c + 1))*x)*e^(3*d*x) + 27*(4*a^2*b*d^3*f^3*x^3*e^(5*c) -
3*a^2*b*f^3*e^(5*c) - 6*a^2*b*d^2*f*e^(5*c + 2) + 6*a^2*b*d*f^2*e^(5*c + 1) - 6*(a^2*b*d^2*f^3*e^(5*c) - 2*a^
2*b*d^3*f^2*e^(5*c + 1))*x^2 + 6*(a^2*b*d*f^3*e^(5*c) + 2*a^2*b*d^3*f*e^(5*c + 2) - 2*a^2*b*d^2*f^2*e^(5*c + 1
))*x)*e^(2*d*x) + 108*(18*a^3*f^3*e^(4*c) + 24*a*b^2*f^3*e^(4*c) - (3*a^3*d^3*f^3*e^(4*c) + 4*a*b^2*d^3*f^3*e^
(4*c))*x^3 + 3*(3*a^3*d^2*f^3*e^(4*c) + 4*a*b^2*d^2*f^3*e^(4*c) - (3*a^3*d^3*f^2*e^(4*c) + 4*a*b^2*d^3*f^2*e^(
4*c))*e)*x^2 - 3*(6*a^3*d*f^3*e^(4*c) + 8*a*b^2*d*f^3*e^(4*c) + (3*a^3*d^3*f*e^(4*c) + 4*a*b^2*d^3*f*e^(4*c))*
e^2 - 2*(3*a^3*d^2*f^2*e^(4*c) + 4*a*b^2*d^2*f^2*e^(4*c))*e)*x + 3*(3*a^3*d^2*f*e^(4*c) + 4*a*b^2*d^2*f*e^(4*c
))*e^2 - 6*(3*a^3*d*f^2*e^(4*c) + 4*a*b^2*d*f^2*e^(4*c))*e)*e^(d*x) + 108*(18*a^3*f^3*e^(2*c) + 24*a*b^2*f^3*e
^(2*c) + (3*a^3*d^3*f^3*e^(2*c) + 4*a*b^2*d^3*f^3*e^(2*c))*x^3 + 3*(3*a^3*d^2*f^3*e^(2*c) + 4*a*b^2*d^2*f^3*e^
(2*c) + (3*a^3*d^3*f^2*e^(2*c) + 4*a*b^2*d^3*f^2*e^(2*c))*e)*x^2 + 3*(6*a^3*d*f^3*e^(2*c) + 8*a*b^2*d*f^3*e^(2
*c) + (3*a^3*d^3*f*e^(2*c) + 4*a*b^2*d^3*f*e^(2*c))*e^2 + 2*(3*a^3*d^2*f^2*e^(2*c) + 4*a*b^2*d^2*f^2*e^(2*c))*
e)*x + 3*(3*a^3*d^2*f*e^(2*c) + 4*a*b^2*d^2*f*e^(2*c))*e^2 + 6*(3*a^3*d*f^2*e^(2*c) + 4*a*b^2*d*f^2*e^(2*c))*e
)*e^(-d*x) + 27*(4*a^2*b*d^3*f^3*x^3*e^c + 6*a^2*b*d^2*f*e^(c + 2) + 6*a^2*b*d*f^2*e^(c + 1) + 3*a^2*b*f^3*e^c
+ 6*(2*a^2*b*d^3*f^2*e^(c + 1) + a^2*b*d^2*f^3*e^c)*x^2 + 6*(2*a^2*b*d^3*f*e^(c + 2) + 2*a^2*b*d^2*f^2*e^(c +
1) + a^2*b*d*f^3*e^c)*x)*e^(-2*d*x) + 4*(9*a^3*d^3*f^3*x^3 + 9*a^3*d^2*f*e^2 + 6*a^3*d*f^2*e + 2*a^3*f^3 + 9*
(3*a^3*d^3*f^2*e + a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*f*e^2 + 6*a^3*d^2*f^2*e + 2*a^3*d*f^3)*x)*e^(-3*d*x))*e^(-3
*c)/(a^4*d^4) + integrate(-2*((a^3*b*f^3 + a*b^3*f^3)*x^3 + 3*(a^3*b*f^2 + a*b^3*f^2)*x^2*e + 3*(a^3*b*f + a*b
^3*f)*x*e^2 - ((a^2*b^2*f^3*e^c + b^4*f^3*e^c)*x^3 + 3*(a^2*b^2*f^2*e^c + b^4*f^2*e^c)*x^2*e + 3*(a^2*b^2*f*e^
c + b^4*f*e^c)*x*e^2)*e^(d*x))/(a^5*e^(2*d*x + 2*c) + 2*a^4*b*e^(d*x + c) - a^5), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 15138 vs. \(2 (830) = 1660\).
time = 0.71, size = 15138, normalized size = 17.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*(36*a^3*d^3*f^3*x^3 + 36*a^3*d^2*f^3*x^2 + 36*a^3*d^3*cosh(1)^3 + 36*a^3*d^3*sinh(1)^3 + 24*a^3*d*f^3*x
- 4*(9*a^3*d^3*f^3*x^3 - 9*a^3*d^2*f^3*x^2 + 9*a^3*d^3*cosh(1)^3 + 9*a^3*d^3*sinh(1)^3 + 6*a^3*d*f^3*x - 2*a^
3*f^3 + 9*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1)^2 + 9*(3*a^3*d^3*f*x + 3*a^3*d^3*cosh(1) - a^3*d^2*f)*sinh(1)^2
+ 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 2*a^3*d*f^2)*cosh(1) + 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 9*a
^3*d^3*cosh(1)^2 + 2*a^3*d*f^2 + 6*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^6 - 4*(9*a^3*d^
3*f^3*x^3 - 9*a^3*d^2*f^3*x^2 + 9*a^3*d^3*cosh(1)^3 + 9*a^3*d^3*sinh(1)^3 + 6*a^3*d*f^3*x - 2*a^3*f^3 + 9*(3*a
^3*d^3*f*x - a^3*d^2*f)*cosh(1)^2 + 9*(3*a^3*d^3*f*x + 3*a^3*d^3*cosh(1) - a^3*d^2*f)*sinh(1)^2 + 3*(9*a^3*d^3
*f^2*x^2 - 6*a^3*d^2*f^2*x + 2*a^3*d*f^2)*cosh(1) + 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 9*a^3*d^3*cosh(1)
^2 + 2*a^3*d*f^2 + 6*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^6 + 8*a^3*f^3 + 27*(4*a^2*b*d
^3*f^3*x^3 - 6*a^2*b*d^2*f^3*x^2 + 4*a^2*b*d^3*cosh(1)^3 + 4*a^2*b*d^3*sinh(1)^3 + 6*a^2*b*d*f^3*x - 3*a^2*b*f
^3 + 6*(2*a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1)^2 + 6*(2*a^2*b*d^3*f*x + 2*a^2*b*d^3*cosh(1) - a^2*b*d^2*f)*sin
h(1)^2 + 6*(2*a^2*b*d^3*f^2*x^2 - 2*a^2*b*d^2*f^2*x + a^2*b*d*f^2)*cosh(1) + 6*(2*a^2*b*d^3*f^2*x^2 - 2*a^2*b*
d^2*f^2*x + 2*a^2*b*d^3*cosh(1)^2 + a^2*b*d*f^2 + 2*(2*a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1))*sinh(1))*cosh(d*x
+ c)^5 + 3*(36*a^2*b*d^3*f^3*x^3 - 54*a^2*b*d^2*f^3*x^2 + 36*a^2*b*d^3*cosh(1)^3 + 36*a^2*b*d^3*sinh(1)^3 + 5
4*a^2*b*d*f^3*x - 27*a^2*b*f^3 + 54*(2*a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1)^2 + 54*(2*a^2*b*d^3*f*x + 2*a^2*b*
d^3*cosh(1) - a^2*b*d^2*f)*sinh(1)^2 + 54*(2*a^2*b*d^3*f^2*x^2 - 2*a^2*b*d^2*f^2*x + a^2*b*d*f^2)*cosh(1) - 8*
(9*a^3*d^3*f^3*x^3 - 9*a^3*d^2*f^3*x^2 + 9*a^3*d^3*cosh(1)^3 + 9*a^3*d^3*sinh(1)^3 + 6*a^3*d*f^3*x - 2*a^3*f^3
+ 9*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1)^2 + 9*(3*a^3*d^3*f*x + 3*a^3*d^3*cosh(1) - a^3*d^2*f)*sinh(1)^2 + 3*(
9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 2*a^3*d*f^2)*cosh(1) + 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 9*a^3*d^
3*cosh(1)^2 + 2*a^3*d*f^2 + 6*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 54*(2*a^2*b*d^3*f^
2*x^2 - 2*a^2*b*d^2*f^2*x + 2*a^2*b*d^3*cosh(1)^2 + a^2*b*d*f^2 + 2*(2*a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1))*s
inh(1))*sinh(d*x + c)^5 - 108*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 - 3*(3*a^3 + 4*a*b^2)*d^2*f^3*x^2 + (3*a^3 + 4*a*
b^2)*d^3*cosh(1)^3 + (3*a^3 + 4*a*b^2)*d^3*sinh(1)^3 + 6*(3*a^3 + 4*a*b^2)*d*f^3*x - 6*(3*a^3 + 4*a*b^2)*f^3 +
3*((3*a^3 + 4*a*b^2)*d^3*f*x - (3*a^3 + 4*a*b^2)*d^2*f)*cosh(1)^2 + 3*((3*a^3 + 4*a*b^2)*d^3*f*x + (3*a^3 + 4
*a*b^2)*d^3*cosh(1) - (3*a^3 + 4*a*b^2)*d^2*f)*sinh(1)^2 + 3*((3*a^3 + 4*a*b^2)*d^3*f^2*x^2 - 2*(3*a^3 + 4*a*b
^2)*d^2*f^2*x + 2*(3*a^3 + 4*a*b^2)*d*f^2)*cosh(1) + 3*((3*a^3 + 4*a*b^2)*d^3*f^2*x^2 - 2*(3*a^3 + 4*a*b^2)*d^
2*f^2*x + (3*a^3 + 4*a*b^2)*d^3*cosh(1)^2 + 2*(3*a^3 + 4*a*b^2)*d*f^2 + 2*((3*a^3 + 4*a*b^2)*d^3*f*x - (3*a^3
+ 4*a*b^2)*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^4 - 3*(36*(3*a^3 + 4*a*b^2)*d^3*f^3*x^3 - 108*(3*a^3 + 4*a*b
^2)*d^2*f^3*x^2 + 36*(3*a^3 + 4*a*b^2)*d^3*cosh(1)^3 + 36*(3*a^3 + 4*a*b^2)*d^3*sinh(1)^3 + 216*(3*a^3 + 4*a*b
^2)*d*f^3*x - 216*(3*a^3 + 4*a*b^2)*f^3 + 108*((3*a^3 + 4*a*b^2)*d^3*f*x - (3*a^3 + 4*a*b^2)*d^2*f)*cosh(1)^2
+ 20*(9*a^3*d^3*f^3*x^3 - 9*a^3*d^2*f^3*x^2 + 9*a^3*d^3*cosh(1)^3 + 9*a^3*d^3*sinh(1)^3 + 6*a^3*d*f^3*x - 2*a^
3*f^3 + 9*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1)^2 + 9*(3*a^3*d^3*f*x + 3*a^3*d^3*cosh(1) - a^3*d^2*f)*sinh(1)^2
+ 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 2*a^3*d*f^2)*cosh(1) + 3*(9*a^3*d^3*f^2*x^2 - 6*a^3*d^2*f^2*x + 9*a
^3*d^3*cosh(1)^2 + 2*a^3*d*f^2 + 6*(3*a^3*d^3*f*x - a^3*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 108*((3*a^3
+ 4*a*b^2)*d^3*f*x + (3*a^3 + 4*a*b^2)*d^3*cosh(1) - (3*a^3 + 4*a*b^2)*d^2*f)*sinh(1)^2 + 108*((3*a^3 + 4*a*b
^2)*d^3*f^2*x^2 - 2*(3*a^3 + 4*a*b^2)*d^2*f^2*x + 2*(3*a^3 + 4*a*b^2)*d*f^2)*cosh(1) - 45*(4*a^2*b*d^3*f^3*x^3
- 6*a^2*b*d^2*f^3*x^2 + 4*a^2*b*d^3*cosh(1)^3 + 4*a^2*b*d^3*sinh(1)^3 + 6*a^2*b*d*f^3*x - 3*a^2*b*f^3 + 6*(2*
a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1)^2 + 6*(2*a^2*b*d^3*f*x + 2*a^2*b*d^3*cosh(1) - a^2*b*d^2*f)*sinh(1)^2 + 6
*(2*a^2*b*d^3*f^2*x^2 - 2*a^2*b*d^2*f^2*x + a^2*b*d*f^2)*cosh(1) + 6*(2*a^2*b*d^3*f^2*x^2 - 2*a^2*b*d^2*f^2*x
+ 2*a^2*b*d^3*cosh(1)^2 + a^2*b*d*f^2 + 2*(2*a^2*b*d^3*f*x - a^2*b*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 10
8*((3*a^3 + 4*a*b^2)*d^3*f^2*x^2 - 2*(3*a^3 + 4*a*b^2)*d^2*f^2*x + (3*a^3 + 4*a*b^2)*d^3*cosh(1)^2 + 2*(3*a^3
+ 4*a*b^2)*d*f^2 + 2*((3*a^3 + 4*a*b^2)*d^3*f*x - (3*a^3 + 4*a*b^2)*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^4 -
216*((a^2*b + b^3)*d^4*f^3*x^4 - 2*(a^2*b + b^3)*c^4*f^3 + 4*((a^2*b + b^3)*d^4*x + 2*(a^2*b + b^3)*c*d^3)*co
sh(1)^3 + 4*((a^2*b + b^3)*d^4*x + 2*(a^2*b + b^3)*c*d^3)*sinh(1)^3 + 6*((a^2*b + b^3)*d^4*f*x^2 - 2*(a^2*b +
b^3)*c^2*d^2*f)*cosh(1)^2 + 6*((a^2*b + b^3)*d^4*f*x^2 - 2*(a^2*b + b^3)*c^2*d^2*f + 2*((a^2*b + b^3)*d^4*x +
2*(a^2*b + b^3)*c*d^3)*cosh(1))*sinh(1)^2 + 4*(...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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