3.5.25 \(\int \frac {(e+f x)^3 \cosh (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx\) [425]

Optimal. Leaf size=638 \[ \frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^4} \]

[Out]

1/4*(f*x+e)^4/b/f-2*(f*x+e)^3*arctanh(exp(d*x+c))/a/d-3*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2+3*f*(f*x+e)^2
*polylog(2,exp(d*x+c))/a/d^2+6*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-6*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/
d^3-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*polylog(4,exp(d*x+c))/a/d^4-(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b
^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d+(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d-3*f*(
f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^2+3*f*(f*x+e)^2*polylog(2,-b*exp(d
*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^2+6*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*
(a^2+b^2)^(1/2)/a/b/d^3-6*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^3-6*f
^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^4+6*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+
b^2)^(1/2)))*(a^2+b^2)^(1/2)/a/b/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.91, antiderivative size = 638, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 14, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5704, 5558, 3377, 2717, 4267, 2611, 6744, 2320, 6724, 5684, 32, 3403, 2296, 2221} \begin {gather*} -\frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^4}+\frac {6 f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^4}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a b d}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac {(e+f x)^4}{4 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e + f*x)^4/(4*b*f) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*b*d) + (Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2
+ b^2])])/(a*b*d) - (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d
*x)])/(a*d^2) - (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^
2) + (3*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^2) + (6*f^2
*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*Sqrt[a^2
+ b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*b*d^3) - (6*Sqrt[a^2 + b^2]*f^2
*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*b*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/
(a*d^4) + (6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) - (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - S
qrt[a^2 + b^2]))])/(a*b*d^4) + (6*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a
*b*d^4)

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 2296

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 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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3403

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 4267

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(Ar
cTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*
fz*x)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5558

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 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 5704

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Cosh[c + d*x]^(p + 1)*(Coth[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 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 (c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh (c+d x) \coth (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{a}+\frac {\int (e+f x)^3 \, dx}{b}-\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a b}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\left (2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a b}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a}+\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a}+\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}-\frac {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d}-\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d^2}-\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d^2}-\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac {\left (6 f^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d^3}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a b d^3}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \text {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 )}{a b d^4}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \text {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 )}{a b d^4}\\ &=\frac {(e+f x)^4}{4 b f}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{c+d x}\right )}{a d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^3}-\frac {6 f^3 \text {Li}_4\left (-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {Li}_4\left (e^{c+d x}\right )}{a d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a b d^4}+\frac {6 \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a b d^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1374\) vs. \(2(638)=1276\).
time = 4.02, size = 1374, normalized size = 2.15 \begin {gather*} \frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )}{4 b}+\frac {-2 d^3 e^3 \tanh ^{-1}\left (e^{c+d x}\right )+3 d^3 e^2 f x \log \left (1-e^{c+d x}\right )+3 d^3 e f^2 x^2 \log \left (1-e^{c+d x}\right )+d^3 f^3 x^3 \log \left (1-e^{c+d x}\right )-3 d^3 e^2 f x \log \left (1+e^{c+d x}\right )-3 d^3 e f^2 x^2 \log \left (1+e^{c+d x}\right )-d^3 f^3 x^3 \log \left (1+e^{c+d x}\right )-3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )+3 d^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )+6 d e f^2 \text {PolyLog}\left (3,-e^{c+d x}\right )+6 d f^3 x \text {PolyLog}\left (3,-e^{c+d x}\right )-6 d e f^2 \text {PolyLog}\left (3,e^{c+d x}\right )-6 d f^3 x \text {PolyLog}\left (3,e^{c+d x}\right )-6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )+6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {\sqrt {-a^2-b^2} \left (2 d^3 e^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \text {ArcTan}\left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )+3 \sqrt {-a^2-b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {-a^2-b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^3 e^2 e^c f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^3 e e^c f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-\sqrt {-a^2-b^2} d^3 e^c f^3 x^3 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-3 \sqrt {-a^2-b^2} d^2 e^c f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} d e e^c f^2 \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} d e^c f^3 x \text {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )+6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )-6 \sqrt {-a^2-b^2} e^c f^3 \text {PolyLog}\left (4,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )\right )}{a b d^4 \sqrt {\left (a^2+b^2\right ) e^{2 c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(4*b) + (-2*d^3*e^3*ArcTanh[E^(c + d*x)] + 3*d^3*e^2*f*x*Log[1
- E^(c + d*x)] + 3*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] + d^3*f^3*x^3*Log[1 - E^(c + d*x)] - 3*d^3*e^2*f*x*Log[
1 + E^(c + d*x)] - 3*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] - d^3*f^3*x^3*Log[1 + E^(c + d*x)] - 3*d^2*f*(e + f*x)
^2*PolyLog[2, -E^(c + d*x)] + 3*d^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)] + 6*d*e*f^2*PolyLog[3, -E^(c + d*x)]
+ 6*d*f^3*x*PolyLog[3, -E^(c + d*x)] - 6*d*e*f^2*PolyLog[3, E^(c + d*x)] - 6*d*f^3*x*PolyLog[3, E^(c + d*x)]
- 6*f^3*PolyLog[4, -E^(c + d*x)] + 6*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (Sqrt[-a^2 - b^2]*(2*d^3*e^3*Sqrt[
(a^2 + b^2)*E^(2*c)]*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]] + 3*Sqrt[-a^2 - b^2]*d^3*e^2*E^c*f*x*Log[1 +
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 3*Sqrt[-a^2 - b^2]*d^3*e*E^c*f^2*x^2*Log[1 + (b*E^(2
*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + Sqrt[-a^2 - b^2]*d^3*E^c*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(
a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 3*Sqrt[-a^2 - b^2]*d^3*e^2*E^c*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sq
rt[(a^2 + b^2)*E^(2*c)])] - 3*Sqrt[-a^2 - b^2]*d^3*e*E^c*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2
+ b^2)*E^(2*c)])] - Sqrt[-a^2 - b^2]*d^3*E^c*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*
c)])] + 3*Sqrt[-a^2 - b^2]*d^2*E^c*f*(e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2
*c)]))] - 3*Sqrt[-a^2 - b^2]*d^2*E^c*f*(e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^
(2*c)]))] - 6*Sqrt[-a^2 - b^2]*d*e*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))
] - 6*Sqrt[-a^2 - b^2]*d*E^c*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sq
rt[-a^2 - b^2]*d*e*E^c*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[-a^2
- b^2]*d*E^c*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*Sqrt[-a^2 - b^2]*E
^c*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 6*Sqrt[-a^2 - b^2]*E^c*f^3*PolyL
og[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(a*b*d^4*Sqrt[(a^2 + b^2)*E^(2*c)])

________________________________________________________________________________________

Maple [F]
time = 3.24, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right ) \coth \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

((d*x + c)/(b*d) - log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d) - sqrt(a^2 + b^2)*log((b*e^(-d*x
- c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(a*b*d))*e^3 + 1/4*(f^3*x^4 + 4*f^2*x^3*e
+ 6*f*x^2*e^2)/b - 3*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*f*e^2/(a*d^2) + 3*(d*x*log(-e^(d*x + c)
+ 1) + dilog(e^(d*x + c)))*f*e^2/(a*d^2) - 3*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*pol
ylog(3, -e^(d*x + c)))*f^2*e/(a*d^3) + 3*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog
(3, e^(d*x + c)))*f^2*e/(a*d^3) - (d^3*x^3*log(e^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylo
g(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilo
g(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*f^3/(a*d^4) - integrate(2*((a^2*f^
3*e^c + b^2*f^3*e^c)*x^3 + 3*(a^2*f^2*e^c + b^2*f^2*e^c)*x^2*e + 3*(a^2*f*e^c + b^2*f*e^c)*x*e^2)*e^(d*x)/(a*b
^2*e^(2*d*x + 2*c) + 2*a^2*b*e^(d*x + c) - a*b^2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2193 vs. \(2 (597) = 1194\).
time = 0.46, size = 2193, normalized size = 3.44 \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)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(a*d^4*f^3*x^4 + 4*a*d^4*f^2*x^3*cosh(1) + 6*a*d^4*f*x^2*cosh(1)^2 + 4*a*d^4*x*cosh(1)^3 + 4*a*d^4*x*sinh(
1)^3 - 24*b*f^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sin
h(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24*b*f^3*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*
x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 24*b*f^3*polylog(4, cosh(d*x + c) + s
inh(d*x + c)) - 24*b*f^3*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) + 6*(a*d^4*f*x^2 + 2*a*d^4*x*cosh(1))*sinh
(1)^2 - 12*(b*d^2*f^3*x^2 + 2*b*d^2*f^2*x*cosh(1) + b*d^2*f*cosh(1)^2 + b*d^2*f*sinh(1)^2 + 2*(b*d^2*f^2*x + b
*d^2*f*cosh(1))*sinh(1))*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b
*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*(b*d^2*f^3*x^2 + 2*b*d^2*f^2*x*cosh(1) + b*d^2*f*cosh(1
)^2 + b*d^2*f*sinh(1)^2 + 2*(b*d^2*f^2*x + b*d^2*f*cosh(1))*sinh(1))*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) - 4*(b*c^3*f^3 -
3*b*c^2*d*f^2*cosh(1) + 3*b*c*d^2*f*cosh(1)^2 - b*d^3*cosh(1)^3 - b*d^3*sinh(1)^3 + 3*(b*c*d^2*f - b*d^3*cosh
(1))*sinh(1)^2 - 3*(b*c^2*d*f^2 - 2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*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) + 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*cosh(1)
+ 3*b*c*d^2*f*cosh(1)^2 - b*d^3*cosh(1)^3 - b*d^3*sinh(1)^3 + 3*(b*c*d^2*f - b*d^3*cosh(1))*sinh(1)^2 - 3*(b*c
^2*d*f^2 - 2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*s
inh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 4*(b*d^3*f^3*x^3 + b*c^3*f^3 + 3*(b*d^3*f*x + b*c*d^2*f)*cos
h(1)^2 + 3*(b*d^3*f*x + b*c*d^2*f)*sinh(1)^2 + 3*(b*d^3*f^2*x^2 - b*c^2*d*f^2)*cosh(1) + 3*(b*d^3*f^2*x^2 - b*
c^2*d*f^2 + 2*(b*d^3*f*x + b*c*d^2*f)*cosh(1))*sinh(1))*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) + 4*(b*d^3*f^3*x^3 + b*c^3*f^3 + 3
*(b*d^3*f*x + b*c*d^2*f)*cosh(1)^2 + 3*(b*d^3*f*x + b*c*d^2*f)*sinh(1)^2 + 3*(b*d^3*f^2*x^2 - b*c^2*d*f^2)*cos
h(1) + 3*(b*d^3*f^2*x^2 - b*c^2*d*f^2 + 2*(b*d^3*f*x + b*c*d^2*f)*cosh(1))*sinh(1))*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) + 24*(
b*d*f^3*x + b*d*f^2*cosh(1) + b*d*f^2*sinh(1))*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) - 24*(b*d*f^3*x + b*d*f^2*cosh(1) + b*d*f
^2*sinh(1))*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*
x + c))*sqrt((a^2 + b^2)/b^2))/b) + 12*(b*d^2*f^3*x^2 + 2*b*d^2*f^2*x*cosh(1) + b*d^2*f*cosh(1)^2 + b*d^2*f*si
nh(1)^2 + 2*(b*d^2*f^2*x + b*d^2*f*cosh(1))*sinh(1))*dilog(cosh(d*x + c) + sinh(d*x + c)) - 12*(b*d^2*f^3*x^2
+ 2*b*d^2*f^2*x*cosh(1) + b*d^2*f*cosh(1)^2 + b*d^2*f*sinh(1)^2 + 2*(b*d^2*f^2*x + b*d^2*f*cosh(1))*sinh(1))*d
ilog(-cosh(d*x + c) - sinh(d*x + c)) - 4*(b*d^3*f^3*x^3 + 3*b*d^3*f^2*x^2*cosh(1) + 3*b*d^3*f*x*cosh(1)^2 + b*
d^3*cosh(1)^3 + b*d^3*sinh(1)^3 + 3*(b*d^3*f*x + b*d^3*cosh(1))*sinh(1)^2 + 3*(b*d^3*f^2*x^2 + 2*b*d^3*f*x*cos
h(1) + b*d^3*cosh(1)^2)*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 4*(b*c^3*f^3 - 3*b*c^2*d*f^2*cosh(1)
+ 3*b*c*d^2*f*cosh(1)^2 - b*d^3*cosh(1)^3 - b*d^3*sinh(1)^3 + 3*(b*c*d^2*f - b*d^3*cosh(1))*sinh(1)^2 - 3*(b*
c^2*d*f^2 - 2*b*c*d^2*f*cosh(1) + b*d^3*cosh(1)^2)*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(b*d^3*
f^3*x^3 + b*c^3*f^3 + 3*(b*d^3*f*x + b*c*d^2*f)*cosh(1)^2 + 3*(b*d^3*f*x + b*c*d^2*f)*sinh(1)^2 + 3*(b*d^3*f^2
*x^2 - b*c^2*d*f^2)*cosh(1) + 3*(b*d^3*f^2*x^2 - b*c^2*d*f^2 + 2*(b*d^3*f*x + b*c*d^2*f)*cosh(1))*sinh(1))*log
(-cosh(d*x + c) - sinh(d*x + c) + 1) - 24*(b*d*f^3*x + b*d*f^2*cosh(1) + b*d*f^2*sinh(1))*polylog(3, cosh(d*x
+ c) + sinh(d*x + c)) + 24*(b*d*f^3*x + b*d*f^2*cosh(1) + b*d*f^2*sinh(1))*polylog(3, -cosh(d*x + c) - sinh(d*
x + c)) + 4*(a*d^4*f^2*x^3 + 3*a*d^4*f*x^2*cosh(1) + 3*a*d^4*x*cosh(1)^2)*sinh(1))/(a*b*d^4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [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)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________