\(\int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [363]

Optimal result

Integrand size = 34, antiderivative size = 449 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5698, 5554, 3391, 3377, 2717, 5680, 2221, 2611, 2320, 6724} \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}-\frac {f (e+f x) \sinh (c+d x) \cosh (c+d x)}{2 b d^2}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d} \]

[In]

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

[Out]

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

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 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 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 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*} \text {integral}& = \frac {\int (e+f x)^2 \cosh (c+d x) \sinh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b} \\ & = \frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {a \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}-\frac {f \int (e+f x) \sinh ^2(c+d x) \, dx}{b d} \\ & = -\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)^2}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {a^2 \int \frac {e^{c+d x} (e+f x)^2}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b^2}+\frac {(2 a f) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d}+\frac {f \int (e+f x) \, dx}{2 b d} \\ & = \frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (2 a^2 f\right ) \int (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (2 a f^2\right ) \int \cosh (c+d x) \, dx}{b^2 d^2} \\ & = \frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (2 a^2 f^2\right ) \int \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2} \\ & = \frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d}-\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3}-\frac {\left (2 a^2 f^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^3} \\ & = \frac {e f x}{2 b d}+\frac {f^2 x^2}{4 b d}-\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 a f (e+f x) \cosh (c+d x)}{b^2 d^2}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {2 a^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {2 a f^2 \sinh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^2 \sinh (c+d x)}{b^2 d}-\frac {f (e+f x) \cosh (c+d x) \sinh (c+d x)}{2 b d^2}+\frac {f^2 \sinh ^2(c+d x)}{4 b d^3}+\frac {(e+f x)^2 \sinh ^2(c+d x)}{2 b d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1453\) vs. \(2(449)=898\).

Time = 9.37 (sec) , antiderivative size = 1453, normalized size of antiderivative = 3.24 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {e^2 \log (a+b \sinh (c+d x))}{4 b d}-\frac {1}{2} e f \left (-\frac {x^2}{2 b}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d}+\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d}+\frac {\operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}\right )-\frac {1}{4} f^2 \left (-\frac {x^3}{3 b}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d}+\frac {x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2}-\frac {2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )}{b d^3}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^3}\right )+\frac {f^2 \left (2 \left (4 a^2+b^2\right ) x^3 \coth (c)-\frac {2 \left (4 a^2+b^2\right ) \left (2 x^3-\frac {3 b^2 \left (-1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {\left (a-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-2 d x \operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-2 \operatorname {PolyLog}\left (3,\frac {\left (-a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2+b^2} \left (-a+\sqrt {a^2+b^2}\right ) d^3}-\frac {3 b^2 \left (-1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-2 d x \operatorname {PolyLog}\left (2,-\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-2 \operatorname {PolyLog}\left (3,-\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )\right )}{\sqrt {a^2+b^2} \left (a+\sqrt {a^2+b^2}\right ) d^3}+\frac {3 a \left (-1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}-\frac {3 a \left (-1+e^{2 c}\right ) \left (d^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d x \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{\sqrt {a^2+b^2} d^3}\right )}{-1+e^{2 c}}-\frac {24 a b \cosh (d x) \left (-2 d x \cosh (c)+\left (2+d^2 x^2\right ) \sinh (c)\right )}{d^3}+\frac {3 b^2 \cosh (2 d x) \left (\left (1+2 d^2 x^2\right ) \cosh (2 c)-2 d x \sinh (2 c)\right )}{d^3}-\frac {24 a b \left (\left (2+d^2 x^2\right ) \cosh (c)-2 d x \sinh (c)\right ) \sinh (d x)}{d^3}+\frac {3 b^2 \left (-2 d x \cosh (2 c)+\left (1+2 d^2 x^2\right ) \sinh (2 c)\right ) \sinh (2 d x)}{d^3}\right )}{24 b^3}+\frac {e^2 \left (\left (4 a^2+b^2\right ) \log (a+b \sinh (c+d x))-4 a b \sinh (c+d x)+2 b^2 \sinh ^2(c+d x)\right )}{4 b^3 d}+\frac {e f \left (8 a b \cosh (c+d x)+2 b^2 d x \cosh (2 (c+d x))+\left (4 a^2+b^2\right ) \left (2 c (c+d x)-(c+d x)^2+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )-8 a b d x \sinh (c+d x)-b^2 \sinh (2 (c+d x))\right )}{4 b^3 d^2} \]

[In]

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

[Out]

-1/4*(e^2*Log[a + b*Sinh[c + d*x]])/(b*d) - (e*f*(-1/2*x^2/b + (x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)])/(b*d) + (x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d) + PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a
^2 + b^2])]/(b*d^2) + PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]/(b*d^2)))/2 - (f^2*(-1/3*x^3/b + (x
^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d) + (x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])]
)/(b*d) + (2*x*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b*d^2) + (2*x*PolyLog[2, -((b*E^(c + d*x
))/(a + Sqrt[a^2 + b^2]))])/(b*d^2) - (2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])])/(b*d^3) - (2*Poly
Log[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b*d^3)))/4 + (f^2*(2*(4*a^2 + b^2)*x^3*Coth[c] - (2*(4*a^2
+ b^2)*(2*x^3 - (3*b^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*d*x*PolyLog
[2, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - 2*PolyLog[3, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b]))/(Sqrt[a
^2 + b^2]*(-a + Sqrt[a^2 + b^2])*d^3) - (3*b^2*(-1 + E^(2*c))*(d^2*x^2*Log[1 + ((a + Sqrt[a^2 + b^2])*E^(-c -
d*x))/b] - 2*d*x*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] - 2*PolyLog[3, -(((a + Sqrt[a^2 + b^2])
*E^(-c - d*x))/b)]))/(Sqrt[a^2 + b^2]*(a + Sqrt[a^2 + b^2])*d^3) + (3*a*(-1 + E^(2*c))*(d^2*x^2*Log[1 + (b*E^(
c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*PolyLog[3, (b*
E^(c + d*x))/(-a + Sqrt[a^2 + b^2])]))/(Sqrt[a^2 + b^2]*d^3) - (3*a*(-1 + E^(2*c))*(d^2*x^2*Log[1 + (b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(-1 + E^(2*c)) - (24*a*b*Cosh[d*x]*(-2*d*x*Cosh
[c] + (2 + d^2*x^2)*Sinh[c]))/d^3 + (3*b^2*Cosh[2*d*x]*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c]))/d^3 - (2
4*a*b*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh
[2*c])*Sinh[2*d*x])/d^3))/(24*b^3) + (e^2*((4*a^2 + b^2)*Log[a + b*Sinh[c + d*x]] - 4*a*b*Sinh[c + d*x] + 2*b^
2*Sinh[c + d*x]^2))/(4*b^3*d) + (e*f*(8*a*b*Cosh[c + d*x] + 2*b^2*d*x*Cosh[2*(c + d*x)] + (4*a^2 + b^2)*(2*c*(
c + d*x) - (c + d*x)^2 + 2*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 2*(c + d*x)*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*Log[b - 2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*PolyLog[2, (b*E^(c +
d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]) - 8*a*b*d*x*Sinh[c +
d*x] - b^2*Sinh[2*(c + d*x)]))/(4*b^3*d^2)

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \cosh \left (d x +c \right ) \sinh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2414 vs. \(2 (417) = 834\).

Time = 0.29 (sec) , antiderivative size = 2414, normalized size of antiderivative = 5.38 \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/48*(6*b^2*d^2*f^2*x^2 + 6*b^2*d^2*e^2 + 6*b^2*d*e*f + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b
^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^4 + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f
+ b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*sinh(d*x + c)^4 + 3*b^2*f^2 - 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 - 12*(2*a*b*d^2*f^2*x^2 + 2*a*b*d^2
*e^2 - 4*a*b*d*e*f + 4*a*b*f^2 + 4*(a*b*d^2*e*f - a*b*d*f^2)*x - (2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*
e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c)^3 - 16*(a^2*d^3*f^2*x^3 + 3*a^2*
d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2)*cosh(d*x + c)^2 - 2*(8*a^2*
d^3*f^2*x^3 + 24*a^2*d^3*e*f*x^2 + 24*a^2*d^3*e^2*x + 48*a^2*c*d^2*e^2 - 48*a^2*c^2*d*e*f + 16*a^2*c^3*f^2 - 9
*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2 - 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^2
+ 36*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 - 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c))
*sinh(d*x + c)^2 + 6*(2*b^2*d^2*e*f + b^2*d*f^2)*x + 24*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2 + 2*a*b*d*e*f + 2*a*b*f
^2 + 2*(a*b*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x + c) + 96*((a^2*d*f^2*x + a^2*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*d*f
^2*x + a^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d*f^2*x + a^2*d*e*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x +
c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 96*((a^2*d*f^2
*x + a^2*d*e*f)*cosh(d*x + c)^2 + 2*(a^2*d*f^2*x + a^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d*f^2*x + a^2
*d*e*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((
a^2 + b^2)/b^2) - b)/b + 1) + 48*((a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*e^2
- 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*sinh
(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*((a^2*d^2*e^2 -
2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c) + (a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*
x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 48*((a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f
^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)*sinh(d
*x + c) + (a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*sinh(d*x + c)^2)*log(-(a*cosh(d*x
+ c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 48*((a^2*d^2*f^2*
x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x +
2*a^2*c*d*e*f - a^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) + (a^2*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*a^2*c*d*e*f
- a^2*c^2*f^2)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*
sqrt((a^2 + b^2)/b^2) - b)/b) - 96*(a^2*f^2*cosh(d*x + c)^2 + 2*a^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*f^2*
sinh(d*x + c)^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2
+ b^2)/b^2))/b) - 96*(a^2*f^2*cosh(d*x + c)^2 + 2*a^2*f^2*cosh(d*x + c)*sinh(d*x + c) + a^2*f^2*sinh(d*x + c)
^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))
/b) + 4*(6*a*b*d^2*f^2*x^2 + 6*a*b*d^2*e^2 + 12*a*b*d*e*f + 12*a*b*f^2 + 3*(2*b^2*d^2*f^2*x^2 + 2*b^2*d^2*e^2
- 2*b^2*d*e*f + b^2*f^2 + 2*(2*b^2*d^2*e*f - b^2*d*f^2)*x)*cosh(d*x + c)^3 - 18*(a*b*d^2*f^2*x^2 + a*b*d^2*e^2
- 2*a*b*d*e*f + 2*a*b*f^2 + 2*(a*b*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 + 12*(a*b*d^2*e*f + a*b*d*f^2)*x -
8*(a^2*d^3*f^2*x^3 + 3*a^2*d^3*e*f*x^2 + 3*a^2*d^3*e^2*x + 6*a^2*c*d^2*e^2 - 6*a^2*c^2*d*e*f + 2*a^2*c^3*f^2)
*cosh(d*x + c))*sinh(d*x + c))/(b^3*d^3*cosh(d*x + c)^2 + 2*b^3*d^3*cosh(d*x + c)*sinh(d*x + c) + b^3*d^3*sinh
(d*x + c)^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

1/8*e^2*(8*(d*x + c)*a^2/(b^3*d) - (4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*a^2*log(-2*a*e^(-d*x - c
) + b*e^(-2*d*x - 2*c) - b)/(b^3*d) + (4*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c))/(b^2*d)) + 1/48*(16*a^2*d^3*f^2*
x^3*e^(2*c) + 48*a^2*d^3*e*f*x^2*e^(2*c) + 3*(2*b^2*d^2*f^2*x^2*e^(4*c) + 2*(2*d^2*e*f - d*f^2)*b^2*x*e^(4*c)
- (2*d*e*f - f^2)*b^2*e^(4*c))*e^(2*d*x) - 24*(a*b*d^2*f^2*x^2*e^(3*c) + 2*(d^2*e*f - d*f^2)*a*b*x*e^(3*c) - 2
*(d*e*f - f^2)*a*b*e^(3*c))*e^(d*x) + 24*(a*b*d^2*f^2*x^2*e^c + 2*(d^2*e*f + d*f^2)*a*b*x*e^c + 2*(d*e*f + f^2
)*a*b*e^c)*e^(-d*x) + 3*(2*b^2*d^2*f^2*x^2 + 2*(2*d^2*e*f + d*f^2)*b^2*x + (2*d*e*f + f^2)*b^2)*e^(-2*d*x))*e^
(-2*c)/(b^3*d^3) - integrate(-2*(a^2*b*f^2*x^2 + 2*a^2*b*e*f*x - (a^3*f^2*x^2*e^c + 2*a^3*e*f*x*e^c)*e^(d*x))/
(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

Giac [F]

\[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \cosh (c+d x) \sinh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^2}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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