3.1.27 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx\) [27]

Optimal. Leaf size=636 \[ -\frac {b e f x}{2 a^2 d}-\frac {b f^2 x^2}{4 a^2 d}+\frac {b \left (a^2+b^2\right ) (e+f x)^3}{3 a^4 f}-\frac {4 f (e+f x) \cosh (c+d x)}{3 a d^2}-\frac {2 b^2 f (e+f x) \cosh (c+d x)}{a^3 d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 a d^2}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {b \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^2}-\frac {2 b \left (a^2+b^2\right ) f (e+f x) \text {PolyLog}\left (2,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^2}+\frac {2 b \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{b-\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {2 b \left (a^2+b^2\right ) f^2 \text {PolyLog}\left (3,-\frac {a e^{c+d x}}{b+\sqrt {a^2+b^2}}\right )}{a^4 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 a d^3}+\frac {2 b^2 f^2 \sinh (c+d x)}{a^3 d^3}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 a d}+\frac {b^2 (e+f x)^2 \sinh (c+d x)}{a^3 d}+\frac {b f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 a^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac {b f^2 \sinh ^2(c+d x)}{4 a^2 d^3}-\frac {b (e+f x)^2 \sinh ^2(c+d x)}{2 a^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 a d^3} \]

[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]
time = 0.63, antiderivative size = 636, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5713, 5698, 3392, 3377, 2717, 2713, 5684, 5554, 3391, 5680, 2221, 2611, 2320, 6724} \begin {gather*} \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 ) \text {Li}_3\left (-\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 ) \text {Li}_3\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \text {csch}(c+d x)} \, dx &=\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 {\text {Li}_2\left (\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 {\text {Li}_2\left (-\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) \text {Li}_2\left (-\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) \text {Li}_2\left (-\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 \text {Li}_3\left (-\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 \text {Li}_3\left (-\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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.23, size = 2252, normalized size = 3.54 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^2*Cosh[c + d*x]^3)/(a + b*Csch[c + d*x]),x]

[Out]

-1/216*(-216*a^2*b*c^2*d*e*f - 216*b^3*c^2*d*e*f - (108*I)*a^2*b*c*d*e*f*Pi + 27*a^2*b*d*e*f*Pi^2 - 432*a^2*b*
c*d^2*e*f*x - 432*b^3*c*d^2*e*f*x - (108*I)*a^2*b*d^2*e*f*Pi*x - 216*a^2*b*d^3*e*f*x^2 - 216*b^3*d^3*e*f*x^2 -
72*a^2*b*d^3*f^2*x^3 - 72*b^3*d^3*f^2*x^3 - 864*a^2*b*d*e*f*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*ArcTan[((I*a +
b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[a^2 + b^2]] + 324*a^3*d*e*f*Cosh[c + d*x] + 432*a*b^2*d*e*f*Cosh[c
+ d*x] + 324*a^3*d*f^2*x*Cosh[c + d*x] + 432*a*b^2*d*f^2*x*Cosh[c + d*x] + 54*a^2*b*d^2*e^2*Cosh[2*(c + d*x)]
+ 27*a^2*b*f^2*Cosh[2*(c + d*x)] + 108*a^2*b*d^2*e*f*x*Cosh[2*(c + d*x)] + 54*a^2*b*d^2*f^2*x^2*Cosh[2*(c + d*
x)] + 12*a^3*d*e*f*Cosh[3*(c + d*x)] + 12*a^3*d*f^2*x*Cosh[3*(c + d*x)] + 216*a^2*b*c*d*e*f*Log[1 + (a*E^(c +
d*x))/(b - Sqrt[a^2 + b^2])] + 432*b^3*c*d*e*f*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 216*a^2*b*d^2*
e*f*x*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])] + 432*b^3*d^2*e*f*x*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2
+ b^2])] + 216*a^2*b*c*d*e*f*Log[1 + ((-b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] + (108*I)*a^2*b*d*e*f*Pi*Log[1 +
((-b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] + 216*a^2*b*d^2*e*f*x*Log[1 + ((-b + Sqrt[a^2 + b^2])*E^(c + d*x))/a]
+ (432*I)*a^2*b*d*e*f*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*Log[1 + ((-b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] + 216
*a^2*b*c*d*e*f*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 432*b^3*c*d*e*f*Log[1 + (a*E^(c + d*x))/(b + S
qrt[a^2 + b^2])] + 216*a^2*b*d^2*e*f*x*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 432*b^3*d^2*e*f*x*Log[
1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])] + 216*a^2*b*c*d*e*f*Log[1 - ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a]
+ (108*I)*a^2*b*d*e*f*Pi*Log[1 - ((b + Sqrt[a^2 + b^2])*E^(c + d*x))/a] + 216*a^2*b*d^2*e*f*x*Log[1 - ((b + Sq
rt[a^2 + b^2])*E^(c + d*x))/a] - (432*I)*a^2*b*d*e*f*ArcSin[Sqrt[1 + (I*b)/a]/Sqrt[2]]*Log[1 - ((b + Sqrt[a^2
+ b^2])*E^(c + d*x))/a] + 216*a^2*b*d^2*f^2*x^2*Log[1 + (a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])]
+ 216*b^3*d^2*f^2*x^2*Log[1 + (a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] + 216*a^2*b*d^2*f^2*x^2*
Log[1 + (a*E^(2*c + d*x))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 216*b^3*d^2*f^2*x^2*Log[1 + (a*E^(2*c + d*x))
/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 216*a^2*b*d^2*e^2*Log[b + a*Sinh[c + d*x]] + 216*b^3*d^2*e^2*Log[b + a
*Sinh[c + d*x]] - 216*a^2*b*c*d*e*f*Log[b + a*Sinh[c + d*x]] - 432*b^3*c*d*e*f*Log[b + a*Sinh[c + d*x]] - (108
*I)*a^2*b*d*e*f*Pi*Log[b + a*Sinh[c + d*x]] - 216*a^2*b*c*d*e*f*Log[1 + (a*Sinh[c + d*x])/b] + 216*a^2*b*d*e*f
*PolyLog[2, ((b - Sqrt[a^2 + b^2])*E^(c + d*x))/a] + 216*b*(a^2 + 2*b^2)*d*e*f*PolyLog[2, (a*E^(c + d*x))/(-b
+ Sqrt[a^2 + b^2])] + 216*a^2*b*d*e*f*PolyLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] + 432*b^3*d*e*f*Pol
yLog[2, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))] + 216*a^2*b*d*e*f*PolyLog[2, ((b + Sqrt[a^2 + b^2])*E^(c + d
*x))/a] + 432*a^2*b*d*f^2*x*PolyLog[2, -((a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*b^3*d*f
^2*x*PolyLog[2, -((a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*a^2*b*d*f^2*x*PolyLog[2, -((a*
E^(2*c + d*x))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*b^3*d*f^2*x*PolyLog[2, -((a*E^(2*c + d*x))/(b*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)]))] - 432*a^2*b*f^2*PolyLog[3, -((a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)
]))] - 432*b^3*f^2*PolyLog[3, -((a*E^(2*c + d*x))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 432*a^2*b*f^2*PolyLo
g[3, -((a*E^(2*c + d*x))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 432*b^3*f^2*PolyLog[3, -((a*E^(2*c + d*x))/(b
*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 162*a^3*d^2*e^2*Sinh[c + d*x] - 216*a*b^2*d^2*e^2*Sinh[c + d*x] - 324*a^
3*f^2*Sinh[c + d*x] - 432*a*b^2*f^2*Sinh[c + d*x] - 324*a^3*d^2*e*f*x*Sinh[c + d*x] - 432*a*b^2*d^2*e*f*x*Sinh
[c + d*x] - 162*a^3*d^2*f^2*x^2*Sinh[c + d*x] - 216*a*b^2*d^2*f^2*x^2*Sinh[c + d*x] - 54*a^2*b*d*e*f*Sinh[2*(c
+ d*x)] - 54*a^2*b*d*f^2*x*Sinh[2*(c + d*x)] - 18*a^3*d^2*e^2*Sinh[3*(c + d*x)] - 4*a^3*f^2*Sinh[3*(c + d*x)]
- 36*a^3*d^2*e*f*x*Sinh[3*(c + d*x)] - 18*a^3*d^2*f^2*x^2*Sinh[3*(c + d*x)])/(a^4*d^3)

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Maple [F]
time = 4.30, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cosh ^{3}\left (d x +c \right )\right )}{a +b \,\mathrm {csch}\left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^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)

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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)^2*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="maxima")

[Out]

-1/24*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b + b
^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(a^3*
d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4*d))*e^2 - 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*f*e^(3*c) + b^3*d^3*f*e^(3*c))*x^2*e - 2*(9*a^3*d^2*f^2*x
^2*e^(6*c) + 2*a^3*f^2*e^(6*c) - 6*a^3*d*f*e^(6*c + 1) - 6*(a^3*d*f^2*e^(6*c) - 3*a^3*d^2*f*e^(6*c + 1))*x)*e^
(3*d*x) + 27*(2*a^2*b*d^2*f^2*x^2*e^(5*c) + a^2*b*f^2*e^(5*c) - 2*a^2*b*d*f*e^(5*c + 1) - 2*(a^2*b*d*f^2*e^(5*
c) - 2*a^2*b*d^2*f*e^(5*c + 1))*x)*e^(2*d*x) - 54*(6*a^3*f^2*e^(4*c) + 8*a*b^2*f^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*a^3*d*f^2*e^(4*c) + 4*a*b^2*d*f^2*e^(4*c) - (3*a^3*d^2*f*e^(4*c) +
4*a*b^2*d^2*f*e^(4*c))*e)*x - 2*(3*a^3*d*f*e^(4*c) + 4*a*b^2*d*f*e^(4*c))*e)*e^(d*x) + 54*(6*a^3*f^2*e^(2*c)
+ 8*a*b^2*f^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*a^3*d*f^2*e^(2*c) + 4*a*b
^2*d*f^2*e^(2*c) + (3*a^3*d^2*f*e^(2*c) + 4*a*b^2*d^2*f*e^(2*c))*e)*x + 2*(3*a^3*d*f*e^(2*c) + 4*a*b^2*d*f*e^(
2*c))*e)*e^(-d*x) + 27*(2*a^2*b*d^2*f^2*x^2*e^c + 2*a^2*b*d*f*e^(c + 1) + a^2*b*f^2*e^c + 2*(2*a^2*b*d^2*f*e^(
c + 1) + a^2*b*d*f^2*e^c)*x)*e^(-2*d*x) + 2*(9*a^3*d^2*f^2*x^2 + 6*a^3*d*f*e + 2*a^3*f^2 + 6*(3*a^3*d^2*f*e +
a^3*d*f^2)*x)*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*f + a*b^3*
f)*x*e - ((a^2*b^2*f^2*e^c + b^4*f^2*e^c)*x^2 + 2*(a^2*b^2*f*e^c + b^4*f*e^c)*x*e)*e^(d*x))/(a^5*e^(2*d*x + 2*
c) + 2*a^4*b*e^(d*x + c) - a^5), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7539 vs. \(2 (606) = 1212\).
time = 0.48, size = 7539, normalized size = 11.85 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 + 12*a^3*d*f^2*x + 18*a^3*d^2*cosh(1)^2 - 2*(9*a^3*d^2*f^2*x^2 - 6*a^3*d*f^2*x + 9*
a^3*d^2*cosh(1)^2 + 9*a^3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3*d^2*f*x - a^3*d*f)*cosh(1) + 6*(3*a^3*d^2*f*x +
3*a^3*d^2*cosh(1) - a^3*d*f)*sinh(1))*cosh(d*x + c)^6 + 18*a^3*d^2*sinh(1)^2 - 2*(9*a^3*d^2*f^2*x^2 - 6*a^3*d
*f^2*x + 9*a^3*d^2*cosh(1)^2 + 9*a^3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3*d^2*f*x - a^3*d*f)*cosh(1) + 6*(3*a^
3*d^2*f*x + 3*a^3*d^2*cosh(1) - a^3*d*f)*sinh(1))*sinh(d*x + c)^6 + 27*(2*a^2*b*d^2*f^2*x^2 - 2*a^2*b*d*f^2*x
+ 2*a^2*b*d^2*cosh(1)^2 + 2*a^2*b*d^2*sinh(1)^2 + a^2*b*f^2 + 2*(2*a^2*b*d^2*f*x - a^2*b*d*f)*cosh(1) + 2*(2*a
^2*b*d^2*f*x + 2*a^2*b*d^2*cosh(1) - a^2*b*d*f)*sinh(1))*cosh(d*x + c)^5 + 3*(18*a^2*b*d^2*f^2*x^2 - 18*a^2*b*
d*f^2*x + 18*a^2*b*d^2*cosh(1)^2 + 18*a^2*b*d^2*sinh(1)^2 + 9*a^2*b*f^2 + 18*(2*a^2*b*d^2*f*x - a^2*b*d*f)*cos
h(1) - 4*(9*a^3*d^2*f^2*x^2 - 6*a^3*d*f^2*x + 9*a^3*d^2*cosh(1)^2 + 9*a^3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3
*d^2*f*x - a^3*d*f)*cosh(1) + 6*(3*a^3*d^2*f*x + 3*a^3*d^2*cosh(1) - a^3*d*f)*sinh(1))*cosh(d*x + c) + 18*(2*a
^2*b*d^2*f*x + 2*a^2*b*d^2*cosh(1) - a^2*b*d*f)*sinh(1))*sinh(d*x + c)^5 + 4*a^3*f^2 - 54*((3*a^3 + 4*a*b^2)*d
^2*f^2*x^2 - 2*(3*a^3 + 4*a*b^2)*d*f^2*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1)^2 + (3*a^3 + 4*a*b^2)*d^2*sinh(1)^2 +
2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*f*x - (3*a^3 + 4*a*b^2)*d*f)*cosh(1) + 2*((3*a^3 + 4*a*b^2
)*d^2*f*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1) - (3*a^3 + 4*a*b^2)*d*f)*sinh(1))*cosh(d*x + c)^4 - 3*(18*(3*a^3 + 4
*a*b^2)*d^2*f^2*x^2 - 36*(3*a^3 + 4*a*b^2)*d*f^2*x + 18*(3*a^3 + 4*a*b^2)*d^2*cosh(1)^2 + 18*(3*a^3 + 4*a*b^2)
*d^2*sinh(1)^2 + 36*(3*a^3 + 4*a*b^2)*f^2 + 10*(9*a^3*d^2*f^2*x^2 - 6*a^3*d*f^2*x + 9*a^3*d^2*cosh(1)^2 + 9*a^
3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3*d^2*f*x - a^3*d*f)*cosh(1) + 6*(3*a^3*d^2*f*x + 3*a^3*d^2*cosh(1) - a^3
*d*f)*sinh(1))*cosh(d*x + c)^2 + 36*((3*a^3 + 4*a*b^2)*d^2*f*x - (3*a^3 + 4*a*b^2)*d*f)*cosh(1) - 45*(2*a^2*b*
d^2*f^2*x^2 - 2*a^2*b*d*f^2*x + 2*a^2*b*d^2*cosh(1)^2 + 2*a^2*b*d^2*sinh(1)^2 + a^2*b*f^2 + 2*(2*a^2*b*d^2*f*x
- a^2*b*d*f)*cosh(1) + 2*(2*a^2*b*d^2*f*x + 2*a^2*b*d^2*cosh(1) - a^2*b*d*f)*sinh(1))*cosh(d*x + c) + 36*((3*
a^3 + 4*a*b^2)*d^2*f*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1) - (3*a^3 + 4*a*b^2)*d*f)*sinh(1))*sinh(d*x + c)^4 - 144
*((a^2*b + b^3)*d^3*f^2*x^3 + 2*(a^2*b + b^3)*c^3*f^2 + 3*((a^2*b + b^3)*d^3*x + 2*(a^2*b + b^3)*c*d^2)*cosh(1
)^2 + 3*((a^2*b + b^3)*d^3*x + 2*(a^2*b + b^3)*c*d^2)*sinh(1)^2 + 3*((a^2*b + b^3)*d^3*f*x^2 - 2*(a^2*b + b^3)
*c^2*d*f)*cosh(1) + 3*((a^2*b + b^3)*d^3*f*x^2 - 2*(a^2*b + b^3)*c^2*d*f + 2*((a^2*b + b^3)*d^3*x + 2*(a^2*b +
b^3)*c*d^2)*cosh(1))*sinh(1))*cosh(d*x + c)^3 - 2*(72*(a^2*b + b^3)*d^3*f^2*x^3 + 144*(a^2*b + b^3)*c^3*f^2 +
20*(9*a^3*d^2*f^2*x^2 - 6*a^3*d*f^2*x + 9*a^3*d^2*cosh(1)^2 + 9*a^3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3*d^2*
f*x - a^3*d*f)*cosh(1) + 6*(3*a^3*d^2*f*x + 3*a^3*d^2*cosh(1) - a^3*d*f)*sinh(1))*cosh(d*x + c)^3 + 216*((a^2*
b + b^3)*d^3*x + 2*(a^2*b + b^3)*c*d^2)*cosh(1)^2 - 135*(2*a^2*b*d^2*f^2*x^2 - 2*a^2*b*d*f^2*x + 2*a^2*b*d^2*c
osh(1)^2 + 2*a^2*b*d^2*sinh(1)^2 + a^2*b*f^2 + 2*(2*a^2*b*d^2*f*x - a^2*b*d*f)*cosh(1) + 2*(2*a^2*b*d^2*f*x +
2*a^2*b*d^2*cosh(1) - a^2*b*d*f)*sinh(1))*cosh(d*x + c)^2 + 216*((a^2*b + b^3)*d^3*x + 2*(a^2*b + b^3)*c*d^2)*
sinh(1)^2 + 216*((a^2*b + b^3)*d^3*f*x^2 - 2*(a^2*b + b^3)*c^2*d*f)*cosh(1) + 108*((3*a^3 + 4*a*b^2)*d^2*f^2*x
^2 - 2*(3*a^3 + 4*a*b^2)*d*f^2*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1)^2 + (3*a^3 + 4*a*b^2)*d^2*sinh(1)^2 + 2*(3*a^
3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*f*x - (3*a^3 + 4*a*b^2)*d*f)*cosh(1) + 2*((3*a^3 + 4*a*b^2)*d^2*f*
x + (3*a^3 + 4*a*b^2)*d^2*cosh(1) - (3*a^3 + 4*a*b^2)*d*f)*sinh(1))*cosh(d*x + c) + 216*((a^2*b + b^3)*d^3*f*x
^2 - 2*(a^2*b + b^3)*c^2*d*f + 2*((a^2*b + b^3)*d^3*x + 2*(a^2*b + b^3)*c*d^2)*cosh(1))*sinh(1))*sinh(d*x + c)
^3 + 54*((3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + 2*(3*a^3 + 4*a*b^2)*d*f^2*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1)^2 + (3*a^
3 + 4*a*b^2)*d^2*sinh(1)^2 + 2*(3*a^3 + 4*a*b^2)*f^2 + 2*((3*a^3 + 4*a*b^2)*d^2*f*x + (3*a^3 + 4*a*b^2)*d*f)*c
osh(1) + 2*((3*a^3 + 4*a*b^2)*d^2*f*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1) + (3*a^3 + 4*a*b^2)*d*f)*sinh(1))*cosh(d
*x + c)^2 + 6*(9*(3*a^3 + 4*a*b^2)*d^2*f^2*x^2 + 18*(3*a^3 + 4*a*b^2)*d*f^2*x + 9*(3*a^3 + 4*a*b^2)*d^2*cosh(1
)^2 - 5*(9*a^3*d^2*f^2*x^2 - 6*a^3*d*f^2*x + 9*a^3*d^2*cosh(1)^2 + 9*a^3*d^2*sinh(1)^2 + 2*a^3*f^2 + 6*(3*a^3*
d^2*f*x - a^3*d*f)*cosh(1) + 6*(3*a^3*d^2*f*x + 3*a^3*d^2*cosh(1) - a^3*d*f)*sinh(1))*cosh(d*x + c)^4 + 9*(3*a
^3 + 4*a*b^2)*d^2*sinh(1)^2 + 45*(2*a^2*b*d^2*f^2*x^2 - 2*a^2*b*d*f^2*x + 2*a^2*b*d^2*cosh(1)^2 + 2*a^2*b*d^2*
sinh(1)^2 + a^2*b*f^2 + 2*(2*a^2*b*d^2*f*x - a^2*b*d*f)*cosh(1) + 2*(2*a^2*b*d^2*f*x + 2*a^2*b*d^2*cosh(1) - a
^2*b*d*f)*sinh(1))*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 - 2*(3*a^3 +
4*a*b^2)*d*f^2*x + (3*a^3 + 4*a*b^2)*d^2*cosh(1)^2 + (3*a^3 + 4*a*b^2)*d^2*sinh(1)^2 + 2*(3*a^3 + 4*a*b^2)*f^
2 + 2*((3*a^3 + 4*a*b^2)*d^2*f*x - (3*a^3 + 4*a...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^3*(e + f*x)^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)

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