3.3.38 \(\int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [238]

Optimal. Leaf size=605 \[ -\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}-\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {6 b f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {6 f^3 \text {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \text {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {6 b f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {6 b f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.68, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5694, 4267, 2611, 6744, 2320, 6724, 3403, 2296, 2221} \begin {gather*} -\frac {6 b f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^4 \sqrt {a^2+b^2}}+\frac {6 b f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^4 \sqrt {a^2+b^2}}+\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^3 \sqrt {a^2+b^2}}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2 \sqrt {a^2+b^2}}-\frac {b (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \sqrt {a^2+b^2}}+\frac {b (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \sqrt {a^2+b^2}}-\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 5694

Int[(Csch[(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*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(Csch[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 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 \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \text {csch}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {(2 b) \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}-\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 {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 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 \sqrt {a^2+b^2}}+\frac {\left (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 \sqrt {a^2+b^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 {\left (6 f^2\right ) \int (e+f x) \text {Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 {(3 b f) \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 \sqrt {a^2+b^2} d}-\frac {(3 b f) \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 \sqrt {a^2+b^2} 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 {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b 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 \sqrt {a^2+b^2} d^2}-\frac {\left (6 b 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 \sqrt {a^2+b^2} 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 {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b 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 \sqrt {a^2+b^2} d^3}+\frac {\left (6 b 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 \sqrt {a^2+b^2} d^3}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b 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 \sqrt {a^2+b^2} d^4}+\frac {\left (6 b 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 \sqrt {a^2+b^2} d^4}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d}+\frac {b (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^3}-\frac {6 b f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} 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 b f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}+\frac {6 b f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \sqrt {a^2+b^2} d^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1264\) vs. \(2(605)=1210\).
time = 3.83, size = 1264, normalized size = 2.09 \begin {gather*} \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 )+\frac {b \left (2 d^3 e^3 \sqrt {\left (a^2+b^2\right ) e^{2 c}} \tanh ^{-1}\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 )}{\sqrt {a^2+b^2} \sqrt {\left (a^2+b^2\right ) e^{2 c}}}}{a d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-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*PolyLo
g[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*PolyLo
g[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)] + (b*(2*d^3*e^3*Sqrt[(a^2 + b^2)*E^(2*c)]*ArcTanh[(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 + 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^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*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*
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*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]))/(Sqrt[a^2 + b^2]*Sqrt[(a^2 +
b^2)*E^(2*c)]))/(a*d^4)

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \mathrm {csch}\left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(b*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a*d) +
log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^3 - 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*polylog(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*polylog(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*dilog(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*(b*f^3*x^3*e^c + 3*b*f^2*x^2*e^(c + 1) + 3*b*f*x*e^(c + 2))*e^(d*x)/(a*
b*e^(2*d*x + 2*c) + 2*a^2*e^(d*x + c) - a*b), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2509 vs. \(2 (565) = 1130\).
time = 0.40, size = 2509, normalized size = 4.15 \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*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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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} \operatorname {csch}{\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*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*csch(c + d*x)/(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*csch(d*x+c)/(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 {{\left (e+f\,x\right )}^3}{\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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