3.5.86 \(\int \frac {(e+f x)^3 \coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [486]

Optimal. Leaf size=972 \[ -\frac {3 f (e+f x)^2}{2 a d^2}+\frac {(e+f x)^3}{2 a d}-\frac {(e+f x)^4}{4 a f}-\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}+\frac {6 b f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {3 f (e+f x)^2 \coth (c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x)^3 \text {csch}(c+d x)}{a^2 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 )}{a^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 )}{a^3 d}+\frac {3 f^2 (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^3}+\frac {(e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a d}+\frac {b^2 (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {6 b f^2 (e+f x) \text {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {6 b f^2 (e+f x) \text {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {3 f^3 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^2}+\frac {3 b^2 f (e+f x)^2 \text {PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^3 d^2}-\frac {6 b f^3 \text {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \text {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac {3 b^2 f^2 (e+f x) \text {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^4}-\frac {6 \left (a^2+b^2\right ) f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^4}+\frac {3 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4}+\frac {3 b^2 f^3 \text {PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.57, antiderivative size = 972, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 23, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {5688, 3801, 3797, 2221, 2317, 2438, 32, 2611, 6744, 2320, 6724, 5704, 5558, 3377, 2718, 5560, 4267, 5554, 3392, 2715, 8, 5684, 5680} \begin {gather*} -\frac {b^2 (e+f x)^4}{4 a^3 f}+\frac {\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}-\frac {(e+f x)^4}{4 a f}-\frac {\coth ^2(c+d x) (e+f x)^3}{2 a d}+\frac {b \text {csch}(c+d x) (e+f x)^3}{a^2 d}-\frac {\left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}-\frac {\left (a^2+b^2\right ) \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}+\frac {b^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a^3 d}+\frac {\log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a d}+\frac {(e+f x)^3}{2 a d}-\frac {3 f (e+f x)^2}{2 a d^2}+\frac {6 b f \tanh ^{-1}\left (e^{c+d x}\right ) (e+f x)^2}{a^2 d^2}-\frac {3 f \coth (c+d x) (e+f x)^2}{2 a d^2}-\frac {3 \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}-\frac {3 \left (a^2+b^2\right ) f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}+\frac {3 b^2 f \text {Li}_2\left (e^{2 (c+d x)}\right ) (e+f x)^2}{2 a^3 d^2}+\frac {3 f \text {Li}_2\left (e^{2 (c+d x)}\right ) (e+f x)^2}{2 a d^2}+\frac {3 f^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)}{a d^3}+\frac {6 b f^2 \text {Li}_2\left (-e^{c+d x}\right ) (e+f x)}{a^2 d^3}-\frac {6 b f^2 \text {Li}_2\left (e^{c+d x}\right ) (e+f x)}{a^2 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)}{a^3 d^3}+\frac {6 \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)}{a^3 d^3}-\frac {3 b^2 f^2 \text {Li}_3\left (e^{2 (c+d x)}\right ) (e+f x)}{2 a^3 d^3}-\frac {3 f^2 \text {Li}_3\left (e^{2 (c+d x)}\right ) (e+f x)}{2 a d^3}+\frac {3 f^3 \text {Li}_2\left (e^{2 (c+d x)}\right )}{2 a d^4}-\frac {6 b f^3 \text {Li}_3\left (-e^{c+d x}\right )}{a^2 d^4}+\frac {6 b f^3 \text {Li}_3\left (e^{c+d x}\right )}{a^2 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 )}{a^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 )}{a^3 d^4}+\frac {3 b^2 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a^3 d^4}+\frac {3 f^3 \text {Li}_4\left (e^{2 (c+d x)}\right )}{4 a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*f*(e + f*x)^2)/(2*a*d^2) + (e + f*x)^3/(2*a*d) - (e + f*x)^4/(4*a*f) - (b^2*(e + f*x)^4)/(4*a^3*f) + ((a^2
+ b^2)*(e + f*x)^4)/(4*a^3*f) + (6*b*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (3*f*(e + f*x)^2*Coth[c
+ d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]^2)/(2*a*d) + (b*(e + f*x)^3*Csch[c + d*x])/(a^2*d) - ((a^2 + b^
2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) - ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (3*f^2*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a*d^3) + ((e + f*x)^
3*Log[1 - E^(2*(c + d*x))])/(a*d) + (b^2*(e + f*x)^3*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (6*b*f^2*(e + f*x)*Po
lyLog[2, -E^(c + d*x)])/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (3*(a^2 + b^2)*f*(
e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) - (3*(a^2 + b^2)*f*(e + f*x)^2*Poly
Log[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^4) +
(3*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(2*(c + d*x))])/(2
*a^3*d^2) - (6*b*f^3*PolyLog[3, -E^(c + d*x)])/(a^2*d^4) + (6*b*f^3*PolyLog[3, E^(c + d*x)])/(a^2*d^4) + (6*(a
^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^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]))])/(a^3*d^3) - (3*f^2*(e + f*x)*PolyLog[3, E^(2*(c
+ d*x))])/(2*a*d^3) - (3*b^2*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*d^3) - (6*(a^2 + b^2)*f^3*Poly
Log[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^4) - (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))/
(a + Sqrt[a^2 + b^2]))])/(a^3*d^4) + (3*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a*d^4) + (3*b^2*f^3*PolyLog[4, E^(
2*(c + d*x))])/(4*a^3*d^4)

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 2317

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

Rule 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 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[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 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 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 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

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 5688

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*Cosh[c + d*x]*(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]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 55.32, size = 14876, normalized size = 15.30 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]
time = 3.12, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\coth ^{3}\left (d x +c \right )\right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^3*coth(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c)
- a^2)*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c
) + 1)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e^3 + (3*a*f^3*x^2 + 6*a*f^2*x*e + 3*a*f*e^2 + 2*(
b*d*f^3*x^3*e^(3*c) + 3*b*d*f^2*x^2*e^(3*c + 1) + 3*b*d*f*x*e^(3*c + 2))*e^(3*d*x) - (2*a*d*f^3*x^3*e^(2*c) +
3*(a*f^3*e^(2*c) + 2*a*d*f^2*e^(2*c + 1))*x^2 + 3*a*f*e^(2*c + 2) + 6*(a*d*f*e^(2*c + 2) + a*f^2*e^(2*c + 1))*
x)*e^(2*d*x) - 2*(b*d*f^3*x^3*e^c + 3*b*d*f^2*x^2*e^(c + 1) + 3*b*d*f*x*e^(c + 2))*e^(d*x))/(a^2*d^2*e^(4*d*x
+ 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) - 3*(b*d*f*e^2 + a*f^2*e)*x/(a^2*d^2) + 3*(b*d*f*e^2 - a*f^2*e)*
x/(a^2*d^2) + 3*(b*d*f*e^2 + a*f^2*e)*log(e^(d*x + c) + 1)/(a^2*d^3) - 3*(b*d*f*e^2 - a*f^2*e)*log(e^(d*x + c)
- 1)/(a^2*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*polylog(3, -e^(d*x + c
)) + 6*polylog(4, -e^(d*x + c)))*(a^2*f^3 + b^2*f^3)/(a^3*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*di
log(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*(a^2*f^3 + b^2*f^3)/(a^3*d^4) +
3*(a*b*f^3 + (a^2*d*f^2 + b^2*d*f^2)*e)*(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)))/(a^3*d^4) - 3*(a*b*f^3 - (a^2*d*f^2 + b^2*d*f^2)*e)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*d
ilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))/(a^3*d^4) + 3*(2*a*b*d*f^2*e + a^2*f^3 + (a^2*d^2*f + b^2*d^2*f
)*e^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^4) - 3*(2*a*b*d*f^2*e - a^2*f^3 - (a^2*d^2*f +
b^2*d^2*f)*e^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^4) - 1/4*((a^2*f^3 + b^2*f^3)*d^4*x^4
+ 4*(a*b*f^3 + (a^2*d*f^2 + b^2*d*f^2)*e)*d^3*x^3 + 6*(2*a*b*d*f^2*e + a^2*f^3 + (a^2*d^2*f + b^2*d^2*f)*e^2)*
d^2*x^2)/(a^3*d^4) - 1/4*((a^2*f^3 + b^2*f^3)*d^4*x^4 - 4*(a*b*f^3 - (a^2*d*f^2 + b^2*d*f^2)*e)*d^3*x^3 - 6*(2
*a*b*d*f^2*e - a^2*f^3 - (a^2*d^2*f + b^2*d^2*f)*e^2)*d^2*x^2)/(a^3*d^4) + integrate(-2*((a^2*b*f^3 + b^3*f^3)
*x^3 + 3*(a^2*b*f^2 + b^3*f^2)*x^2*e + 3*(a^2*b*f + b^3*f)*x*e^2 - ((a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3
*f^2*e^c + a*b^2*f^2*e^c)*x^2*e + 3*(a^3*f*e^c + a*b^2*f*e^c)*x*e^2)*e^(d*x))/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e
^(d*x + c) - a^3*b), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24387 vs. \(2 (932) = 1864\).
time = 0.66, size = 24387, normalized size = 25.09 \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*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(3*a^2*c^2*f^3 - 6*a^2*c*d*f^2*cosh(1) + 3*a^2*d^2*f*cosh(1)^2 + 3*a^2*d^2*f*sinh(1)^2 - 3*(a^2*d^2*f^3*x^2 -
a^2*c^2*f^3 + 2*(a^2*d^2*f^2*x + a^2*c*d*f^2)*cosh(1) + 2*(a^2*d^2*f^2*x + a^2*c*d*f^2)*sinh(1))*cosh(d*x + c)
^4 - 3*(a^2*d^2*f^3*x^2 - a^2*c^2*f^3 + 2*(a^2*d^2*f^2*x + a^2*c*d*f^2)*cosh(1) + 2*(a^2*d^2*f^2*x + a^2*c*d*f
^2)*sinh(1))*sinh(d*x + c)^4 + 2*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*f^2*x^2*cosh(1) + 3*a*b*d^3*f*x*cosh(1)^2 + a*b*
d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 3*(a*b*d^3*f*x + a*b*d^3*cosh(1))*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 + 2*a*b*d
^3*f*x*cosh(1) + a*b*d^3*cosh(1)^2)*sinh(1))*cosh(d*x + c)^3 + 2*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*f^2*x^2*cosh(1)
+ 3*a*b*d^3*f*x*cosh(1)^2 + a*b*d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 3*(a*b*d^3*f*x + a*b*d^3*cosh(1))*sinh(1)^
2 - 6*(a^2*d^2*f^3*x^2 - a^2*c^2*f^3 + 2*(a^2*d^2*f^2*x + a^2*c*d*f^2)*cosh(1) + 2*(a^2*d^2*f^2*x + a^2*c*d*f^
2)*sinh(1))*cosh(d*x + c) + 3*(a*b*d^3*f^2*x^2 + 2*a*b*d^3*f*x*cosh(1) + a*b*d^3*cosh(1)^2)*sinh(1))*sinh(d*x
+ c)^3 - (2*a^2*d^3*f^3*x^3 - 3*a^2*d^2*f^3*x^2 + 2*a^2*d^3*cosh(1)^3 + 2*a^2*d^3*sinh(1)^3 + 6*a^2*c^2*f^3 +
3*(2*a^2*d^3*f*x + a^2*d^2*f)*cosh(1)^2 + 3*(2*a^2*d^3*f*x + 2*a^2*d^3*cosh(1) + a^2*d^2*f)*sinh(1)^2 + 6*(a^2
*d^3*f^2*x^2 - a^2*d^2*f^2*x - 2*a^2*c*d*f^2)*cosh(1) + 6*(a^2*d^3*f^2*x^2 - a^2*d^2*f^2*x + a^2*d^3*cosh(1)^2
- 2*a^2*c*d*f^2 + (2*a^2*d^3*f*x + a^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 - (2*a^2*d^3*f^3*x^3 - 3*a^2*
d^2*f^3*x^2 + 2*a^2*d^3*cosh(1)^3 + 2*a^2*d^3*sinh(1)^3 + 6*a^2*c^2*f^3 + 3*(2*a^2*d^3*f*x + a^2*d^2*f)*cosh(1
)^2 + 18*(a^2*d^2*f^3*x^2 - a^2*c^2*f^3 + 2*(a^2*d^2*f^2*x + a^2*c*d*f^2)*cosh(1) + 2*(a^2*d^2*f^2*x + a^2*c*d
*f^2)*sinh(1))*cosh(d*x + c)^2 + 3*(2*a^2*d^3*f*x + 2*a^2*d^3*cosh(1) + a^2*d^2*f)*sinh(1)^2 + 6*(a^2*d^3*f^2*
x^2 - a^2*d^2*f^2*x - 2*a^2*c*d*f^2)*cosh(1) - 6*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*f^2*x^2*cosh(1) + 3*a*b*d^3*f*x*
cosh(1)^2 + a*b*d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 3*(a*b*d^3*f*x + a*b*d^3*cosh(1))*sinh(1)^2 + 3*(a*b*d^3*f
^2*x^2 + 2*a*b*d^3*f*x*cosh(1) + a*b*d^3*cosh(1)^2)*sinh(1))*cosh(d*x + c) + 6*(a^2*d^3*f^2*x^2 - a^2*d^2*f^2*
x + a^2*d^3*cosh(1)^2 - 2*a^2*c*d*f^2 + (2*a^2*d^3*f*x + a^2*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^2 - 2*(a*b
*d^3*f^3*x^3 + 3*a*b*d^3*f^2*x^2*cosh(1) + 3*a*b*d^3*f*x*cosh(1)^2 + a*b*d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 3
*(a*b*d^3*f*x + a*b*d^3*cosh(1))*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 + 2*a*b*d^3*f*x*cosh(1) + a*b*d^3*cosh(1)^2)*s
inh(1))*cosh(d*x + c) - 3*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(
1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 + ((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*
d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))
*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2
+ (a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)*s
inh(d*x + c)^3 + ((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a
^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^4 - 2
*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*
sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^
2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(1)^2 - 3*((
a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sin
h(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*f
^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh
(1))*sinh(1) + 4*(((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (
a^2 + b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^3 -
((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*s
inh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c))*sinh(d*x + c))*dilog(
(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 3
*((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*
sinh(1)^2 + ((a^2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 +
b^2)*d^2*f*sinh(1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^4 + 4*((a^
2 + b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh(1) + (a^2 + b^2)*d^2*f*cosh(1)^2 + (a^2 + b^2)*d^2*f*sinh(
1)^2 + 2*((a^2 + b^2)*d^2*f^2*x + (a^2 + b^2)*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 +
b^2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*f^2*x*cosh...

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

[Out]

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

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

[Out]

Timed out

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

[Out]

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

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