\(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [27]
Optimal result
Integrand size = 28, antiderivative size = 636 \[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=-\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3}
\]
[Out]
-1/2*b*e*f*x/a^2/d-1/4*b*f^2*x^2/a^2/d+1/3*b*(a^2+b^2)*(f*x+e)^3/a^4/f-4/3*f*(f*x+e)*cosh(d*x+c)/a/d^2-2*b^2*f
*(f*x+e)*cosh(d*x+c)/a^3/d^2-2/9*f*(f*x+e)*cosh(d*x+c)^3/a/d^2-b*(a^2+b^2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b-(a^2
+b^2)^(1/2)))/a^4/d-b*(a^2+b^2)*(f*x+e)^2*ln(1+a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d-2*b*(a^2+b^2)*f*(f*x+e)
*polylog(2,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^2-2*b*(a^2+b^2)*f*(f*x+e)*polylog(2,-a*exp(d*x+c)/(b+(a^2+
b^2)^(1/2)))/a^4/d^2+2*b*(a^2+b^2)*f^2*polylog(3,-a*exp(d*x+c)/(b-(a^2+b^2)^(1/2)))/a^4/d^3+2*b*(a^2+b^2)*f^2*
polylog(3,-a*exp(d*x+c)/(b+(a^2+b^2)^(1/2)))/a^4/d^3+14/9*f^2*sinh(d*x+c)/a/d^3+2*b^2*f^2*sinh(d*x+c)/a^3/d^3+
2/3*(f*x+e)^2*sinh(d*x+c)/a/d+b^2*(f*x+e)^2*sinh(d*x+c)/a^3/d+1/2*b*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/a^2/d^2+
1/3*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/a/d-1/4*b*f^2*sinh(d*x+c)^2/a^2/d^3-1/2*b*(f*x+e)^2*sinh(d*x+c)^2/a^2/
d+2/27*f^2*sinh(d*x+c)^3/a/d^3
Rubi [A] (verified)
Time = 0.63 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.00, number of
steps used = 24, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5698, 3392, 3377,
2717, 2713, 5684, 5554, 3391, 5680, 2221, 2611, 2320, 6724} \[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}+\frac {b f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 a^2 d^2}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}-\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {2 b f^2 \left (a^2+b^2\right ) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b f^2 \left (a^2+b^2\right ) \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}-\frac {2 b f \left (a^2+b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b f \left (a^2+b^2\right ) (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}+1\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {a e^{c+d x}}{\sqrt {a^2+b^2}+b}+1\right )}{a^4 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 a d}
\]
[In]
Int[((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]),x]
[Out]
-1/2*(b*e*f*x)/(a^2*d) - (b*f^2*x^2)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^3)/(3*a^4*f) - (4*f*(e + f*x)*Cosh[c
+ d*x])/(3*a*d^2) - (2*b^2*f*(e + f*x)*Cosh[c + d*x])/(a^3*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*a*d^2) -
(b*(a^2 + b^2)*(e + f*x)^2*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(e + f*x)
^2*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (2*b*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((a*E^(c
+ d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (2*b*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((a*E^(c + d*x))/(b + S
qrt[a^2 + b^2]))])/(a^4*d^2) + (2*b*(a^2 + b^2)*f^2*PolyLog[3, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4
*d^3) + (2*b*(a^2 + b^2)*f^2*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^3) + (14*f^2*Sinh[c
+ d*x])/(9*a*d^3) + (2*b^2*f^2*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^2*Sinh[c + d*x])/(3*a*d) + (b^2*(e + f*
x)^2*Sinh[c + d*x])/(a^3*d) + (b*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*a^2*d^2) + ((e + f*x)^2*Cosh[c +
d*x]^2*Sinh[c + d*x])/(3*a*d) - (b*f^2*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^2*Sinh[c + d*x]^2)/(2*a^2*d
) + (2*f^2*Sinh[c + d*x]^3)/(27*a*d^3)
Rule 2221
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 2320
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2611
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2713
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 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 3391
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Rule 3392
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Rule 5554
Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(c +
d*x)^m*(Sinh[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Rule 5680
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d
*x))), x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Rule 5684
Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> Dist[-a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[(e + f*x)^m*(Cosh[c + d*x]^(n - 2)/(a + b*Sinh[c + d*
x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]
Rule 5698
Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*(Sinh[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Rule 5713
Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[(e + f*x)^m*Sinh[c + d*x]*(F[c + d*x]^n/(b + a*Sinh[c + d*x])), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]
Rule 6724
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx \\ & = \frac {\int (e+f x)^2 \cosh ^3(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{b+a \sinh (c+d x)} \, dx}{a} \\ & = -\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}+\frac {2 \int (e+f x)^2 \cosh (c+d x) \, dx}{3 a}-\frac {b \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{a^2}+\frac {b^2 \int (e+f x)^2 \cosh (c+d x) \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac {\left (2 f^2\right ) \int \cosh ^3(c+d x) \, dx}{9 a d^2} \\ & = \frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^2}{b-\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^2}{b+\sqrt {a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac {(4 f) \int (e+f x) \sinh (c+d x) \, dx}{3 a d}+\frac {(b f) \int (e+f x) \sinh ^2(c+d x) \, dx}{a^2 d}-\frac {\left (2 b^2 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{a^3 d}+\frac {\left (2 i f^2\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (c+d x)\right )}{9 a d^3} \\ & = \frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}+\frac {2 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3}-\frac {(b f) \int (e+f x) \, dx}{2 a^2 d}+\frac {\left (2 b \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (2 b \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d}+\frac {\left (4 f^2\right ) \int \cosh (c+d x) \, dx}{3 a d^2}+\frac {\left (2 b^2 f^2\right ) \int \cosh (c+d x) \, dx}{a^3 d^2} \\ & = -\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3}+\frac {\left (2 b \left (a^2+b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2}+\frac {\left (2 b \left (a^2+b^2\right ) f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right ) \, dx}{a^4 d^2} \\ & = -\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3}+\frac {\left (2 b \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {a x}{-b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^3}+\frac {\left (2 b \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {a x}{b+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^3} \\ & = -\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b \left (a^2+b^2\right ) f^2 \operatorname {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3} \\
\end{align*}
Mathematica [B] (warning: unable to verify)
Leaf count is larger than twice the leaf count of optimal. \(2129\) vs. \(2(636)=1272\).
Time = 16.80 (sec) , antiderivative size = 2129, normalized size of antiderivative = 3.35
\[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Result too large to show}
\]
[In]
Integrate[((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]),x]
[Out]
(f^2*Csch[c + d*x]*((4*b*x^3)/(-1 + E^(2*c)) - 2*b*x^3*Coth[c] - (6*a^2*b*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^
2])*E^(-c - d*x))/a] - 2*d*x*PolyLog[2, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*PolyLog[3, ((-b + Sqrt[a^
2 + b^2])*E^(-c - d*x))/a]))/(Sqrt[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) - (6*a^2*b*(d^2*x^2*Log[1 + ((b + Sq
rt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x*PolyLog[2, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)] - 2*PolyLog[3, -
(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d^3) + (6*b^2*(d^2*x^2*Log[
1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*Poly
Log[3, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (6*b^2*(d^2*x^2*Log[1 + (a*E^(c + d*x
))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((a*E^(
c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) + (6*a*Cosh[d*x]*(-2*d*x*Cosh[c] + (2 + d^2*x^2)*Sin
h[c]))/d^3 + (6*a*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3)*(b + a*Sinh[c + d*x]))/(12*a^2*(a +
b*Csch[c + d*x])) - (e^2*Csch[c + d*x]*((b*Log[b + a*Sinh[c + d*x]])/a^2 - Sinh[c + d*x]/a)*(b + a*Sinh[c + d*
x]))/(2*d*(a + b*Csch[c + d*x])) + (e*f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-2*a*Cosh[c + d*x] - b*(2*c*(c +
d*x) - (c + d*x)^2 + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (a*E^(c
+ d*x))/(b + Sqrt[a^2 + b^2])] - 2*c*Log[a - 2*b*E^(c + d*x) - a*E^(2*(c + d*x))] + 2*PolyLog[2, (a*E^(c + d*x
))/(-b + Sqrt[a^2 + b^2])] + 2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]) + 2*a*d*x*Sinh[c + d*x]))
/(2*a^2*d^2*(a + b*Csch[c + d*x])) + (e^2*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-3*b*(a^2 + 2*b^2)*Log[b + a*Si
nh[c + d*x]] + 3*a*(a^2 + 2*b^2)*Sinh[c + d*x] - 3*a^2*b*Sinh[c + d*x]^2 + 2*a^3*Sinh[c + d*x]^3))/(6*a^4*d*(a
+ b*Csch[c + d*x])) + (f^2*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-36*b*(a^2 + 2*b^2)*x^3*Coth[c] + (36*b*(a^2
+ 2*b^2)*(2*x^3 - (3*a^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((b - Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*d*x*PolyL
og[2, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a] - 2*PolyLog[3, ((-b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a]))/(Sqrt
[a^2 + b^2]*(-b + Sqrt[a^2 + b^2])*d^3) - (3*a^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((b + Sqrt[a^2 + b^2])*E^(-c
- d*x))/a] - 2*d*x*PolyLog[2, -(((b + Sqrt[a^2 + b^2])*E^(-c - d*x))/a)] - 2*PolyLog[3, -(((b + Sqrt[a^2 + b^2
])*E^(-c - d*x))/a)]))/(Sqrt[a^2 + b^2]*(b + Sqrt[a^2 + b^2])*d^3) + (3*b*(-1 + E^(2*c))*(d^2*x^2*Log[1 + (a*E
^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (
a*E^(c + d*x))/(-b + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (3*b*(-1 + E^(2*c))*(d^2*x^2*Log[1 + (a*E^(c
+ d*x))/(b + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((
a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(-1 + E^(2*c)) + (a*(-27*a*b*(1 + 2*d^2*x^2)*
Cosh[2*(c + d*x)] - 12*a^2*d*x*Cosh[3*(c + d*x)] + 4*(54*b^2*(2 + d^2*x^2) + 2*a^2*(14 + 9*d^2*x^2) + a^2*(2 +
9*d^2*x^2)*Cosh[2*(c + d*x)])*Sinh[c + d*x] - 108*d*x*Cosh[c + d*x]*(a^2 + 4*b^2 - a*b*Sinh[c + d*x])))/d^3))
/(216*a^4*(a + b*Csch[c + d*x])) + (e*f*Csch[c + d*x]*(b + a*Sinh[c + d*x])*(-18*a*(a^2 + 4*b^2)*Cosh[c + d*x]
- 18*a^2*b*d*x*Cosh[2*(c + d*x)] - 2*a^3*Cosh[3*(c + d*x)] - 18*b*(a^2 + 2*b^2)*(2*c*(c + d*x) - (c + d*x)^2
+ 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a
^2 + b^2])] - 2*c*Log[a - 2*b*E^(c + d*x) - a*E^(2*(c + d*x))] + 2*PolyLog[2, (a*E^(c + d*x))/(-b + Sqrt[a^2 +
b^2])] + 2*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))]) + 18*a*(a^2 + 4*b^2)*d*x*Sinh[c + d*x] + 9*a
^2*b*Sinh[2*(c + d*x)] + 6*a^3*d*x*Sinh[3*(c + d*x)]))/(36*a^4*d^2*(a + b*Csch[c + d*x]))
Maple [F]
\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right )^{3}}{a +b \,\operatorname {csch}\left (d x +c \right )}d x\]
[In]
int((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x)
[Out]
int((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 4887 vs. \(2 (592) = 1184\).
Time = 0.34 (sec) , antiderivative size = 4887, normalized size of antiderivative = 7.68
\[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="fricas")
[Out]
-1/432*(18*a^3*d^2*f^2*x^2 + 18*a^3*d^2*e^2 - 2*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 +
6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c)^6 - 2*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3
*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*sinh(d*x + c)^6 + 12*a^3*d*e*f + 27*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2
*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b*d^2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c)^5 + 3*(18*a^2*b*d^2*f^2*
x^2 + 18*a^2*b*d^2*e^2 - 18*a^2*b*d*e*f + 9*a^2*b*f^2 + 18*(2*a^2*b*d^2*e*f - a^2*b*d*f^2)*x - 4*(9*a^3*d^2*f^
2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c
)^5 + 4*a^3*f^2 - 54*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3 + 4*a*b^2)*d^2*e^2 - 2*(3*a^3 + 4*a*b^2)*d*e*f +
2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f - (3*a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*x + c)^4 - 3*(18*(
3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + 18*(3*a^3 + 4*a*b^2)*d^2*e^2 - 36*(3*a^3 + 4*a*b^2)*d*e*f + 36*(3*a^3 + 4*a*b^2
)*f^2 + 10*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cos
h(d*x + c)^2 + 36*((3*a^3 + 4*a*b^2)*d^2*e*f - (3*a^3 + 4*a*b^2)*d*f^2)*x - 45*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*
d^2*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b*d^2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^4 - 14
4*((a^2*b + b^3)*d^3*f^2*x^3 + 3*(a^2*b + b^3)*d^3*e*f*x^2 + 3*(a^2*b + b^3)*d^3*e^2*x + 6*(a^2*b + b^3)*c*d^2
*e^2 - 6*(a^2*b + b^3)*c^2*d*e*f + 2*(a^2*b + b^3)*c^3*f^2)*cosh(d*x + c)^3 - 2*(72*(a^2*b + b^3)*d^3*f^2*x^3
+ 216*(a^2*b + b^3)*d^3*e*f*x^2 + 216*(a^2*b + b^3)*d^3*e^2*x + 432*(a^2*b + b^3)*c*d^2*e^2 - 432*(a^2*b + b^3
)*c^2*d*e*f + 144*(a^2*b + b^3)*c^3*f^2 + 20*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*
(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c)^3 - 135*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 - 2*a^2*b*d*e*f +
a^2*b*f^2 + 2*(2*a^2*b*d^2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c)^2 + 108*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3
+ 4*a*b^2)*d^2*e^2 - 2*(3*a^3 + 4*a*b^2)*d*e*f + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f - (3*
a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 54*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3 + 4*a*b^2
)*d^2*e^2 + 2*(3*a^3 + 4*a*b^2)*d*e*f + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f + (3*a^3 + 4*a*
b^2)*d*f^2)*x)*cosh(d*x + c)^2 + 6*(9*(3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + 9*(3*a^3 + 4*a*b^2)*d^2*e^2 - 5*(9*a^3*d
^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c)^4 + 18*(
3*a^3 + 4*a*b^2)*d*e*f + 45*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b*d^
2*e*f - a^2*b*d*f^2)*x)*cosh(d*x + c)^3 + 18*(3*a^3 + 4*a*b^2)*f^2 - 54*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^
3 + 4*a*b^2)*d^2*e^2 - 2*(3*a^3 + 4*a*b^2)*d*e*f + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f - (3
*a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*x + c)^2 + 18*((3*a^3 + 4*a*b^2)*d^2*e*f + (3*a^3 + 4*a*b^2)*d*f^2)*x - 72*((
a^2*b + b^3)*d^3*f^2*x^3 + 3*(a^2*b + b^3)*d^3*e*f*x^2 + 3*(a^2*b + b^3)*d^3*e^2*x + 6*(a^2*b + b^3)*c*d^2*e^2
- 6*(a^2*b + b^3)*c^2*d*e*f + 2*(a^2*b + b^3)*c^3*f^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 12*(3*a^3*d^2*e*f + a
^3*d*f^2)*x + 27*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 + 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b*d^2*e*f + a^2
*b*d*f^2)*x)*cosh(d*x + c) + 864*(((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*cosh(d*x + c)^3 + 3*((a^2*b +
b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d
*e*f)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*sinh(d*x + c)^3)*dilog((b*
cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 864*
(((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*
f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)
^2 + ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f)*sinh(d*x + c)^3)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) -
(a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1) + 432*(((a^2*b + b^3)*d^2*e^2 - 2*(a^2*
b + b^3)*c*d*e*f + (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f
+ (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f +
(a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f + (a^
2*b + b^3)*c^2*f^2)*sinh(d*x + c)^3)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2
*b) + 432*(((a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f + (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^2
*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f + (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b
+ b^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f + (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b
^3)*d^2*e^2 - 2*(a^2*b + b^3)*c*d*e*f + (a^2*b + b^3)*c^2*f^2)*sinh(d*x + c)^3)*log(2*a*cosh(d*x + c) + 2*a*si
nh(d*x + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 432*(((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x +
2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b +
b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b +
b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)*
sinh(d*x + c)^2 + ((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b +
b^3)*c^2*f^2)*sinh(d*x + c)^3)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*s
qrt((a^2 + b^2)/a^2) - a)/a) + 432*(((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c
*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2
*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d^2*f^2*x^2 +
2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 +
((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*d^2*e*f*x + 2*(a^2*b + b^3)*c*d*e*f - (a^2*b + b^3)*c^2*f^2)*sinh
(d*x + c)^3)*log(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^
2) - a)/a) - 864*((a^2*b + b^3)*f^2*cosh(d*x + c)^3 + 3*(a^2*b + b^3)*f^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a
^2*b + b^3)*f^2*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b + b^3)*f^2*sinh(d*x + c)^3)*polylog(3, (b*cosh(d*x + c)
+ b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) - 864*((a^2*b + b^3)*f^2*co
sh(d*x + c)^3 + 3*(a^2*b + b^3)*f^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b + b^3)*f^2*cosh(d*x + c)*sinh(d*x
+ c)^2 + (a^2*b + b^3)*f^2*sinh(d*x + c)^3)*polylog(3, (b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c)
+ a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 3*(18*a^2*b*d^2*f^2*x^2 + 18*a^2*b*d^2*e^2 + 18*a^2*b*d*e*f - 4
*(9*a^3*d^2*f^2*x^2 + 9*a^3*d^2*e^2 - 6*a^3*d*e*f + 2*a^3*f^2 + 6*(3*a^3*d^2*e*f - a^3*d*f^2)*x)*cosh(d*x + c)
^5 + 9*a^2*b*f^2 + 45*(2*a^2*b*d^2*f^2*x^2 + 2*a^2*b*d^2*e^2 - 2*a^2*b*d*e*f + a^2*b*f^2 + 2*(2*a^2*b*d^2*e*f
- a^2*b*d*f^2)*x)*cosh(d*x + c)^4 - 72*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3 + 4*a*b^2)*d^2*e^2 - 2*(3*a^3 +
4*a*b^2)*d*e*f + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f - (3*a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*
x + c)^3 - 144*((a^2*b + b^3)*d^3*f^2*x^3 + 3*(a^2*b + b^3)*d^3*e*f*x^2 + 3*(a^2*b + b^3)*d^3*e^2*x + 6*(a^2*b
+ b^3)*c*d^2*e^2 - 6*(a^2*b + b^3)*c^2*d*e*f + 2*(a^2*b + b^3)*c^3*f^2)*cosh(d*x + c)^2 + 18*(2*a^2*b*d^2*e*f
+ a^2*b*d*f^2)*x + 36*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + (3*a^3 + 4*a*b^2)*d^2*e^2 + 2*(3*a^3 + 4*a*b^2)*d*e*f
+ 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*e*f + (3*a^3 + 4*a*b^2)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x
+ c))/(a^4*d^3*cosh(d*x + c)^3 + 3*a^4*d^3*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^4*d^3*cosh(d*x + c)*sinh(d*x +
c)^2 + a^4*d^3*sinh(d*x + c)^3)
Sympy [F]
\[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cosh ^{3}{\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx
\]
[In]
integrate((f*x+e)**2*cosh(d*x+c)**3/(a+b*csch(d*x+c)),x)
[Out]
Integral((e + f*x)**2*cosh(c + d*x)**3/(a + b*csch(c + d*x)), x)
Maxima [F]
\[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x }
\]
[In]
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="maxima")
[Out]
-1/24*e^2*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b
+ b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(
a^3*d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4*d)) - 1/432*(144*(a^2*b*d^3*f^2
*e^(3*c) + b^3*d^3*f^2*e^(3*c))*x^3 + 432*(a^2*b*d^3*e*f*e^(3*c) + b^3*d^3*e*f*e^(3*c))*x^2 - 2*(9*a^3*d^2*f^2
*x^2*e^(6*c) + 6*(3*d^2*e*f - d*f^2)*a^3*x*e^(6*c) - 2*(3*d*e*f - f^2)*a^3*e^(6*c))*e^(3*d*x) + 27*(2*a^2*b*d^
2*f^2*x^2*e^(5*c) + 2*(2*d^2*e*f - d*f^2)*a^2*b*x*e^(5*c) - (2*d*e*f - f^2)*a^2*b*e^(5*c))*e^(2*d*x) + 54*(6*(
d*e*f - f^2)*a^3*e^(4*c) + 8*(d*e*f - f^2)*a*b^2*e^(4*c) - (3*a^3*d^2*f^2*e^(4*c) + 4*a*b^2*d^2*f^2*e^(4*c))*x
^2 - 2*(3*(d^2*e*f - d*f^2)*a^3*e^(4*c) + 4*(d^2*e*f - d*f^2)*a*b^2*e^(4*c))*x)*e^(d*x) + 54*(6*(d*e*f + f^2)*
a^3*e^(2*c) + 8*(d*e*f + f^2)*a*b^2*e^(2*c) + (3*a^3*d^2*f^2*e^(2*c) + 4*a*b^2*d^2*f^2*e^(2*c))*x^2 + 2*(3*(d^
2*e*f + d*f^2)*a^3*e^(2*c) + 4*(d^2*e*f + d*f^2)*a*b^2*e^(2*c))*x)*e^(-d*x) + 27*(2*a^2*b*d^2*f^2*x^2*e^c + 2*
(2*d^2*e*f + d*f^2)*a^2*b*x*e^c + (2*d*e*f + f^2)*a^2*b*e^c)*e^(-2*d*x) + 2*(9*a^3*d^2*f^2*x^2 + 6*(3*d^2*e*f
+ d*f^2)*a^3*x + 2*(3*d*e*f + f^2)*a^3)*e^(-3*d*x))*e^(-3*c)/(a^4*d^3) + integrate(-2*((a^3*b*f^2 + a*b^3*f^2)
*x^2 + 2*(a^3*b*e*f + a*b^3*e*f)*x - ((a^2*b^2*f^2*e^c + b^4*f^2*e^c)*x^2 + 2*(a^2*b^2*e*f*e^c + b^4*e*f*e^c)*
x)*e^(d*x))/(a^5*e^(2*d*x + 2*c) + 2*a^4*b*e^(d*x + c) - a^5), x)
Giac [F]
\[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right )^{3}}{b \operatorname {csch}\left (d x + c\right ) + a} \,d x }
\]
[In]
integrate((f*x+e)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="giac")
[Out]
integrate((f*x + e)^2*cosh(d*x + c)^3/(b*csch(d*x + c) + a), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x
\]
[In]
int((cosh(c + d*x)^3*(e + f*x)^2)/(a + b/sinh(c + d*x)),x)
[Out]
int((cosh(c + d*x)^3*(e + f*x)^2)/(a + b/sinh(c + d*x)), x)