3.4.44 \(\int \frac {(e+f x)^2 \cosh ^3(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [344]

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

[Out]

-1/2*a*e*f*x/b^2/d-1/4*a*f^2*x^2/b^2/d+1/3*a*(a^2+b^2)*(f*x+e)^3/b^4/f-2*a^2*f*(f*x+e)*cosh(d*x+c)/b^3/d^2-4/3
*f*(f*x+e)*cosh(d*x+c)/b/d^2-2/9*f*(f*x+e)*cosh(d*x+c)^3/b/d^2-a*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/b^4/d-a*(a^2+b^2)*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d-2*a*(a^2+b^2)*f*(f*x+e)
*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^2-2*a*(a^2+b^2)*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a+(a^2+
b^2)^(1/2)))/b^4/d^2+2*a*(a^2+b^2)*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^4/d^3+2*a*(a^2+b^2)*f^2*
polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^4/d^3+2*a^2*f^2*sinh(d*x+c)/b^3/d^3+14/9*f^2*sinh(d*x+c)/b/d^3+
a^2*(f*x+e)^2*sinh(d*x+c)/b^3/d+2/3*(f*x+e)^2*sinh(d*x+c)/b/d+1/2*a*f*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b^2/d^2+
1/3*(f*x+e)^2*cosh(d*x+c)^2*sinh(d*x+c)/b/d-1/4*a*f^2*sinh(d*x+c)^2/b^2/d^3-1/2*a*(f*x+e)^2*sinh(d*x+c)^2/b^2/
d+2/27*f^2*sinh(d*x+c)^3/b/d^3

________________________________________________________________________________________

Rubi [A]
time = 0.60, antiderivative size = 636, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {5698, 3392, 3377, 2717, 2713, 5684, 5554, 3391, 5680, 2221, 2611, 2320, 6724} \begin {gather*} \frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 a f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a f^2 \left (a^2+b^2\right ) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}-\frac {2 a f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a f \left (a^2+b^2\right ) (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^4 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}+\frac {a f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b^2 d^2}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {14 f^2 \sinh (c+d x)}{9 b d^3}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {(e+f x)^2 \sinh (c+d x) \cosh ^2(c+d x)}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a*e*f*x)/(b^2*d) - (a*f^2*x^2)/(4*b^2*d) + (a*(a^2 + b^2)*(e + f*x)^3)/(3*b^4*f) - (2*a^2*f*(e + f*x)*Co
sh[c + d*x])/(b^3*d^2) - (4*f*(e + f*x)*Cosh[c + d*x])/(3*b*d^2) - (2*f*(e + f*x)*Cosh[c + d*x]^3)/(9*b*d^2) -
(a*(a^2 + b^2)*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^4*d) - (a*(a^2 + b^2)*(e + f*x)
^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^4*d) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4*d^2) - (2*a*(a^2 + b^2)*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + S
qrt[a^2 + b^2]))])/(b^4*d^2) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^4
*d^3) + (2*a*(a^2 + b^2)*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^4*d^3) + (2*a^2*f^2*Sinh
[c + d*x])/(b^3*d^3) + (14*f^2*Sinh[c + d*x])/(9*b*d^3) + (a^2*(e + f*x)^2*Sinh[c + d*x])/(b^3*d) + (2*(e + f*
x)^2*Sinh[c + d*x])/(3*b*d) + (a*f*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(2*b^2*d^2) + ((e + f*x)^2*Cosh[c +
d*x]^2*Sinh[c + d*x])/(3*b*d) - (a*f^2*Sinh[c + d*x]^2)/(4*b^2*d^3) - (a*(e + f*x)^2*Sinh[c + d*x]^2)/(2*b^2*d
) + (2*f^2*Sinh[c + d*x]^3)/(27*b*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 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) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^2 \cosh ^3(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}+\frac {a^2 \int (e+f x)^2 \cosh (c+d x) \, dx}{b^3}-\frac {a \int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b^2}+\frac {2 \int (e+f x)^2 \cosh (c+d x) \, dx}{3 b}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\left (2 f^2\right ) \int \cosh ^3(c+d x) \, dx}{9 b d^2}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^3}-\frac {\left (2 a^2 f\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^3 d}+\frac {(a f) \int (e+f x) \sinh ^2(c+d x) \, dx}{b^2 d}-\frac {(4 f) \int (e+f x) \sinh (c+d x) \, dx}{3 b 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 b d^3}\\ &=\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}+\frac {2 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}-\frac {(a f) \int (e+f x) \, dx}{2 b^2 d}+\frac {\left (2 a \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (2 a \left (a^2+b^2\right ) f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d}+\frac {\left (2 a^2 f^2\right ) \int \cosh (c+d x) \, dx}{b^3 d^2}+\frac {\left (4 f^2\right ) \int \cosh (c+d x) \, dx}{3 b d^2}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {\left (2 a \left (a^2+b^2\right ) f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}+\frac {\left (2 a \left (a^2+b^2\right ) f^2\right ) \int \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^4 d^2}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}+\frac {\left (2 a \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}+\frac {\left (2 a \left (a^2+b^2\right ) f^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^4 d^3}\\ &=-\frac {a e f x}{2 b^2 d}-\frac {a f^2 x^2}{4 b^2 d}+\frac {a \left (a^2+b^2\right ) (e+f x)^3}{3 b^4 f}-\frac {2 a^2 f (e+f x) \cosh (c+d x)}{b^3 d^2}-\frac {4 f (e+f x) \cosh (c+d x)}{3 b d^2}-\frac {2 f (e+f x) \cosh ^3(c+d x)}{9 b d^2}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {a \left (a^2+b^2\right ) (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^2}-\frac {2 a \left (a^2+b^2\right ) f (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^2}+\frac {2 a \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a \left (a^2+b^2\right ) f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^4 d^3}+\frac {2 a^2 f^2 \sinh (c+d x)}{b^3 d^3}+\frac {14 f^2 \sinh (c+d x)}{9 b d^3}+\frac {a^2 (e+f x)^2 \sinh (c+d x)}{b^3 d}+\frac {2 (e+f x)^2 \sinh (c+d x)}{3 b d}+\frac {a f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b^2 d^2}+\frac {(e+f x)^2 \cosh ^2(c+d x) \sinh (c+d x)}{3 b d}-\frac {a f^2 \sinh ^2(c+d x)}{4 b^2 d^3}-\frac {a (e+f x)^2 \sinh ^2(c+d x)}{2 b^2 d}+\frac {2 f^2 \sinh ^3(c+d x)}{27 b d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2455\) vs. \(2(636)=1272\).
time = 8.93, size = 2455, normalized size = 3.86 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(f^2*(-12*a*d*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 12*a*d*x*PolyLog[2, -((
b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + (2*a*d^3*E^c*x^3 - 6*b*Cosh[d*x] + 6*b*E^(2*c)*Cosh[d
*x] - 6*b*d*x*Cosh[d*x] - 6*b*d*E^(2*c)*x*Cosh[d*x] - 3*b*d^2*x^2*Cosh[d*x] + 3*b*d^2*E^(2*c)*x^2*Cosh[d*x] -
6*a*d^2*E^c*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 6*a*d^2*E^c*x^2*Log[1 + (b*E^
(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] + 12*a*E^c*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2
+ b^2)*E^(2*c)]))] + 12*a*E^c*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 6*b*Sinh
[d*x] + 6*b*E^(2*c)*Sinh[d*x] + 6*b*d*x*Sinh[d*x] - 6*b*d*E^(2*c)*x*Sinh[d*x] + 3*b*d^2*x^2*Sinh[d*x] + 3*b*d^
2*E^(2*c)*x^2*Sinh[d*x])/E^c))/(12*b^2*d^3) + (f^2*(144*a^3*d^3*E^(3*c)*x^3 + 72*a*b^2*d^3*E^(3*c)*x^3 - 432*a
^2*b*E^(2*c)*Cosh[d*x] - 108*b^3*E^(2*c)*Cosh[d*x] + 432*a^2*b*E^(4*c)*Cosh[d*x] + 108*b^3*E^(4*c)*Cosh[d*x] -
432*a^2*b*d*E^(2*c)*x*Cosh[d*x] - 108*b^3*d*E^(2*c)*x*Cosh[d*x] - 432*a^2*b*d*E^(4*c)*x*Cosh[d*x] - 108*b^3*d
*E^(4*c)*x*Cosh[d*x] - 216*a^2*b*d^2*E^(2*c)*x^2*Cosh[d*x] - 54*b^3*d^2*E^(2*c)*x^2*Cosh[d*x] + 216*a^2*b*d^2*
E^(4*c)*x^2*Cosh[d*x] + 54*b^3*d^2*E^(4*c)*x^2*Cosh[d*x] - 27*a*b^2*E^c*Cosh[2*d*x] - 27*a*b^2*E^(5*c)*Cosh[2*
d*x] - 54*a*b^2*d*E^c*x*Cosh[2*d*x] + 54*a*b^2*d*E^(5*c)*x*Cosh[2*d*x] - 54*a*b^2*d^2*E^c*x^2*Cosh[2*d*x] - 54
*a*b^2*d^2*E^(5*c)*x^2*Cosh[2*d*x] - 4*b^3*Cosh[3*d*x] + 4*b^3*E^(6*c)*Cosh[3*d*x] - 12*b^3*d*x*Cosh[3*d*x] -
12*b^3*d*E^(6*c)*x*Cosh[3*d*x] - 18*b^3*d^2*x^2*Cosh[3*d*x] + 18*b^3*d^2*E^(6*c)*x^2*Cosh[3*d*x] - 432*a^3*d^2
*E^(3*c)*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 216*a*b^2*d^2*E^(3*c)*x^2*Log[1
+ (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])] - 432*a^3*d^2*E^(3*c)*x^2*Log[1 + (b*E^(2*c + d*x))/(
a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 216*a*b^2*d^2*E^(3*c)*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 +
b^2)*E^(2*c)])] - 432*a*(2*a^2 + b^2)*d*E^(3*c)*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^
(2*c)]))] - 432*a*(2*a^2 + b^2)*d*E^(3*c)*x*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])
)] + 864*a^3*E^(3*c)*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*a*b^2*E^(3*c)*
PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 864*a^3*E^(3*c)*PolyLog[3, -((b*E^(2*c
+ d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 432*a*b^2*E^(3*c)*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt
[(a^2 + b^2)*E^(2*c)]))] + 432*a^2*b*E^(2*c)*Sinh[d*x] + 108*b^3*E^(2*c)*Sinh[d*x] + 432*a^2*b*E^(4*c)*Sinh[d*
x] + 108*b^3*E^(4*c)*Sinh[d*x] + 432*a^2*b*d*E^(2*c)*x*Sinh[d*x] + 108*b^3*d*E^(2*c)*x*Sinh[d*x] - 432*a^2*b*d
*E^(4*c)*x*Sinh[d*x] - 108*b^3*d*E^(4*c)*x*Sinh[d*x] + 216*a^2*b*d^2*E^(2*c)*x^2*Sinh[d*x] + 54*b^3*d^2*E^(2*c
)*x^2*Sinh[d*x] + 216*a^2*b*d^2*E^(4*c)*x^2*Sinh[d*x] + 54*b^3*d^2*E^(4*c)*x^2*Sinh[d*x] + 27*a*b^2*E^c*Sinh[2
*d*x] - 27*a*b^2*E^(5*c)*Sinh[2*d*x] + 54*a*b^2*d*E^c*x*Sinh[2*d*x] + 54*a*b^2*d*E^(5*c)*x*Sinh[2*d*x] + 54*a*
b^2*d^2*E^c*x^2*Sinh[2*d*x] - 54*a*b^2*d^2*E^(5*c)*x^2*Sinh[2*d*x] + 4*b^3*Sinh[3*d*x] + 4*b^3*E^(6*c)*Sinh[3*
d*x] + 12*b^3*d*x*Sinh[3*d*x] - 12*b^3*d*E^(6*c)*x*Sinh[3*d*x] + 18*b^3*d^2*x^2*Sinh[3*d*x] + 18*b^3*d^2*E^(6*
c)*x^2*Sinh[3*d*x]))/(432*b^4*d^3*E^(3*c)) - (e^2*((a*Log[a + b*Sinh[c + d*x]])/b^2 - Sinh[c + d*x]/b))/(2*d)
+ (e*f*(-(b*Cosh[c + d*x]) - a*(-1/2*(c + d*x)^2 + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] +
(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - c*Log[a + b*Sinh[c + d*x]] + PolyLog[2, (b*E^(c + d
*x))/(-a + Sqrt[a^2 + b^2])] + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) + b*d*x*Sinh[c + d*x]))/(
b^2*d^2) + (e^2*((-2*a*Cosh[2*(c + d*x)])/(b^2*d) - (4*(2*a^3 + a*b^2)*Log[a + b*Sinh[c + d*x]])/(b^4*d) + (2*
(4*a^2 + b^2)*Sinh[c + d*x])/(b^3*d) + (2*Sinh[3*(c + d*x)])/(3*b*d)))/8 + (e*f*(-18*b*(4*a^2 + b^2)*Cosh[c +
d*x] - 18*a*b^2*d*x*Cosh[2*(c + d*x)] - 2*b^3*Cosh[3*(c + d*x)] - 36*a*(2*a^2 + b^2)*(-1/2*(c + d*x)^2 + (c +
d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + (c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])]
- c*Log[a + b*Sinh[c + d*x]] + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + PolyLog[2, -((b*E^(c + d*
x))/(a + Sqrt[a^2 + b^2]))]) + 18*b*(4*a^2 + b^2)*d*x*Sinh[c + d*x] + 9*a*b^2*Sinh[2*(c + d*x)] + 6*b^3*d*x*Si
nh[3*(c + d*x)]))/(36*b^4*d^2)

________________________________________________________________________________________

Maple [F]
time = 1.88, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\cosh ^{3}\left (d x +c \right )\right ) \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*cosh(d*x+c)^3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/24*((3*a*b*e^(-d*x - c) - b^2 - 3*(4*a^2 + 3*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(b^3*d) + 24*(a^3 + a*b
^2)*(d*x + c)/(b^4*d) + (3*a*b*e^(-2*d*x - 2*c) + b^2*e^(-3*d*x - 3*c) + 3*(4*a^2 + 3*b^2)*e^(-d*x - c))/(b^3*
d) + 24*(a^3 + a*b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(b^4*d))*e^2 - 1/432*(144*(a^3*d^3*f^2*e
^(3*c) + a*b^2*d^3*f^2*e^(3*c))*x^3 + 432*(a^3*d^3*f*e^(3*c) + a*b^2*d^3*f*e^(3*c))*x^2*e - 2*(9*b^3*d^2*f^2*x
^2*e^(6*c) + 2*b^3*f^2*e^(6*c) - 6*b^3*d*f*e^(6*c + 1) - 6*(b^3*d*f^2*e^(6*c) - 3*b^3*d^2*f*e^(6*c + 1))*x)*e^
(3*d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^(5*c) + a*b^2*f^2*e^(5*c) - 2*a*b^2*d*f*e^(5*c + 1) - 2*(a*b^2*d*f^2*e^(5*
c) - 2*a*b^2*d^2*f*e^(5*c + 1))*x)*e^(2*d*x) - 54*(8*a^2*b*f^2*e^(4*c) + 6*b^3*f^2*e^(4*c) + (4*a^2*b*d^2*f^2*
e^(4*c) + 3*b^3*d^2*f^2*e^(4*c))*x^2 - 2*(4*a^2*b*d*f^2*e^(4*c) + 3*b^3*d*f^2*e^(4*c) - (4*a^2*b*d^2*f*e^(4*c)
+ 3*b^3*d^2*f*e^(4*c))*e)*x - 2*(4*a^2*b*d*f*e^(4*c) + 3*b^3*d*f*e^(4*c))*e)*e^(d*x) + 54*(8*a^2*b*f^2*e^(2*c
) + 6*b^3*f^2*e^(2*c) + (4*a^2*b*d^2*f^2*e^(2*c) + 3*b^3*d^2*f^2*e^(2*c))*x^2 + 2*(4*a^2*b*d*f^2*e^(2*c) + 3*b
^3*d*f^2*e^(2*c) + (4*a^2*b*d^2*f*e^(2*c) + 3*b^3*d^2*f*e^(2*c))*e)*x + 2*(4*a^2*b*d*f*e^(2*c) + 3*b^3*d*f*e^(
2*c))*e)*e^(-d*x) + 27*(2*a*b^2*d^2*f^2*x^2*e^c + 2*a*b^2*d*f*e^(c + 1) + a*b^2*f^2*e^c + 2*(2*a*b^2*d^2*f*e^(
c + 1) + a*b^2*d*f^2*e^c)*x)*e^(-2*d*x) + 2*(9*b^3*d^2*f^2*x^2 + 6*b^3*d*f*e + 2*b^3*f^2 + 6*(3*b^3*d^2*f*e +
b^3*d*f^2)*x)*e^(-3*d*x))*e^(-3*c)/(b^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^4*f^2*e^c + a^2*b^2*f^2*e^c)*x^2 + 2*(a^4*f*e^c + a^2*b^2*f*e^c)*x*e)*e^(d*x))/(b^5*e^(2*d*x + 2*
c) + 2*a*b^4*e^(d*x + c) - b^5), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 7539 vs. \(2 (606) = 1212\).
time = 0.43, 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*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/432*(18*b^3*d^2*f^2*x^2 + 12*b^3*d*f^2*x + 18*b^3*d^2*cosh(1)^2 - 2*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*
b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x +
3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^6 + 18*b^3*d^2*sinh(1)^2 - 2*(9*b^3*d^2*f^2*x^2 - 6*b^3*d
*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^
3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*sinh(d*x + c)^6 + 27*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x
+ 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a
*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c)^5 + 3*(18*a*b^2*d^2*f^2*x^2 - 18*a*b^2*
d*f^2*x + 18*a*b^2*d^2*cosh(1)^2 + 18*a*b^2*d^2*sinh(1)^2 + 9*a*b^2*f^2 + 18*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cos
h(1) - 4*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3
*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c) + 18*(2*a
*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*sinh(d*x + c)^5 + 4*b^3*f^2 - 54*((4*a^2*b + 3*b^3)*d
^2*f^2*x^2 - 2*(4*a^2*b + 3*b^3)*d*f^2*x + (4*a^2*b + 3*b^3)*d^2*cosh(1)^2 + (4*a^2*b + 3*b^3)*d^2*sinh(1)^2 +
2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*f*x - (4*a^2*b + 3*b^3)*d*f)*cosh(1) + 2*((4*a^2*b + 3*b^3
)*d^2*f*x + (4*a^2*b + 3*b^3)*d^2*cosh(1) - (4*a^2*b + 3*b^3)*d*f)*sinh(1))*cosh(d*x + c)^4 - 3*(18*(4*a^2*b +
3*b^3)*d^2*f^2*x^2 - 36*(4*a^2*b + 3*b^3)*d*f^2*x + 18*(4*a^2*b + 3*b^3)*d^2*cosh(1)^2 + 18*(4*a^2*b + 3*b^3)
*d^2*sinh(1)^2 + 36*(4*a^2*b + 3*b^3)*f^2 + 10*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^
3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3
*d*f)*sinh(1))*cosh(d*x + c)^2 + 36*((4*a^2*b + 3*b^3)*d^2*f*x - (4*a^2*b + 3*b^3)*d*f)*cosh(1) - 45*(2*a*b^2*
d^2*f^2*x^2 - 2*a*b^2*d*f^2*x + 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x
- a*b^2*d*f)*cosh(1) + 2*(2*a*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c) + 36*((4*
a^2*b + 3*b^3)*d^2*f*x + (4*a^2*b + 3*b^3)*d^2*cosh(1) - (4*a^2*b + 3*b^3)*d*f)*sinh(1))*sinh(d*x + c)^4 - 144
*((a^3 + a*b^2)*d^3*f^2*x^3 + 2*(a^3 + a*b^2)*c^3*f^2 + 3*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*cosh(1
)^2 + 3*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*sinh(1)^2 + 3*((a^3 + a*b^2)*d^3*f*x^2 - 2*(a^3 + a*b^2)
*c^2*d*f)*cosh(1) + 3*((a^3 + a*b^2)*d^3*f*x^2 - 2*(a^3 + a*b^2)*c^2*d*f + 2*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a
*b^2)*c*d^2)*cosh(1))*sinh(1))*cosh(d*x + c)^3 - 2*(72*(a^3 + a*b^2)*d^3*f^2*x^3 + 144*(a^3 + a*b^2)*c^3*f^2 +
20*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*d^2*
f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^3 + 216*((a^3
+ a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*cosh(1)^2 - 135*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x + 2*a*b^2*d^2*c
osh(1)^2 + 2*a*b^2*d^2*sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a*b^2*d^2*f*x +
2*a*b^2*d^2*cosh(1) - a*b^2*d*f)*sinh(1))*cosh(d*x + c)^2 + 216*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*
sinh(1)^2 + 216*((a^3 + a*b^2)*d^3*f*x^2 - 2*(a^3 + a*b^2)*c^2*d*f)*cosh(1) + 108*((4*a^2*b + 3*b^3)*d^2*f^2*x
^2 - 2*(4*a^2*b + 3*b^3)*d*f^2*x + (4*a^2*b + 3*b^3)*d^2*cosh(1)^2 + (4*a^2*b + 3*b^3)*d^2*sinh(1)^2 + 2*(4*a^
2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*f*x - (4*a^2*b + 3*b^3)*d*f)*cosh(1) + 2*((4*a^2*b + 3*b^3)*d^2*f*
x + (4*a^2*b + 3*b^3)*d^2*cosh(1) - (4*a^2*b + 3*b^3)*d*f)*sinh(1))*cosh(d*x + c) + 216*((a^3 + a*b^2)*d^3*f*x
^2 - 2*(a^3 + a*b^2)*c^2*d*f + 2*((a^3 + a*b^2)*d^3*x + 2*(a^3 + a*b^2)*c*d^2)*cosh(1))*sinh(1))*sinh(d*x + c)
^3 + 54*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 + 2*(4*a^2*b + 3*b^3)*d*f^2*x + (4*a^2*b + 3*b^3)*d^2*cosh(1)^2 + (4*a^
2*b + 3*b^3)*d^2*sinh(1)^2 + 2*(4*a^2*b + 3*b^3)*f^2 + 2*((4*a^2*b + 3*b^3)*d^2*f*x + (4*a^2*b + 3*b^3)*d*f)*c
osh(1) + 2*((4*a^2*b + 3*b^3)*d^2*f*x + (4*a^2*b + 3*b^3)*d^2*cosh(1) + (4*a^2*b + 3*b^3)*d*f)*sinh(1))*cosh(d
*x + c)^2 + 6*(9*(4*a^2*b + 3*b^3)*d^2*f^2*x^2 + 18*(4*a^2*b + 3*b^3)*d*f^2*x + 9*(4*a^2*b + 3*b^3)*d^2*cosh(1
)^2 - 5*(9*b^3*d^2*f^2*x^2 - 6*b^3*d*f^2*x + 9*b^3*d^2*cosh(1)^2 + 9*b^3*d^2*sinh(1)^2 + 2*b^3*f^2 + 6*(3*b^3*
d^2*f*x - b^3*d*f)*cosh(1) + 6*(3*b^3*d^2*f*x + 3*b^3*d^2*cosh(1) - b^3*d*f)*sinh(1))*cosh(d*x + c)^4 + 9*(4*a
^2*b + 3*b^3)*d^2*sinh(1)^2 + 45*(2*a*b^2*d^2*f^2*x^2 - 2*a*b^2*d*f^2*x + 2*a*b^2*d^2*cosh(1)^2 + 2*a*b^2*d^2*
sinh(1)^2 + a*b^2*f^2 + 2*(2*a*b^2*d^2*f*x - a*b^2*d*f)*cosh(1) + 2*(2*a*b^2*d^2*f*x + 2*a*b^2*d^2*cosh(1) - a
*b^2*d*f)*sinh(1))*cosh(d*x + c)^3 + 18*(4*a^2*b + 3*b^3)*f^2 - 54*((4*a^2*b + 3*b^3)*d^2*f^2*x^2 - 2*(4*a^2*b
+ 3*b^3)*d*f^2*x + (4*a^2*b + 3*b^3)*d^2*cosh(1)^2 + (4*a^2*b + 3*b^3)*d^2*sinh(1)^2 + 2*(4*a^2*b + 3*b^3)*f^
2 + 2*((4*a^2*b + 3*b^3)*d^2*f*x - (4*a^2*b + 3...

________________________________________________________________________________________

Sympy [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)**2*cosh(d*x+c)**3*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

integrate((f*x + e)^2*cosh(d*x + c)^3*sinh(d*x + c)/(b*sinh(d*x + c) + a), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(c + d*x)^3*sinh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)),x)

[Out]

int((cosh(c + d*x)^3*sinh(c + d*x)*(e + f*x)^2)/(a + b*sinh(c + d*x)), x)

________________________________________________________________________________________