### 3.723 $$\int \frac{\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx$$

Optimal. Leaf size=314 $-\frac{a^3 b x}{\left (a^2-b^2\right )^3}-\frac{6 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac{a b x}{4 \left (a^2-b^2\right )^2}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac{2 a^2 b^2 \sinh ^2(x)}{\left (a^2-b^2\right )^3}+\frac{b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac{a b \sinh (x) \cosh ^3(x)}{2 \left (a^2-b^2\right )^2}+\frac{a^3 b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{a b \sinh (x) \cosh (x)}{4 \left (a^2-b^2\right )^2}+\frac{a b^3 \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{3 a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac{3 a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}$

[Out]

(-6*a^3*b^3*x)/(a^2 - b^2)^4 - (a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 + (a*b*x)/(4*(a^2 - b^2)^2) +
(b^2*Cosh[x]^4)/(4*(a^2 - b^2)^2) + (3*a^4*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (3*a^2*b^4*Log[a*C
osh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (a^3*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^3 + (a*b^3*Cosh[x]*Sinh[x])/(a^2 - b^
2)^3 + (a*b*Cosh[x]*Sinh[x])/(4*(a^2 - b^2)^2) - (a*b*Cosh[x]^3*Sinh[x])/(2*(a^2 - b^2)^2) - (2*a^2*b^2*Sinh[x
]^2)/(a^2 - b^2)^3 + (a^2*Sinh[x]^4)/(4*(a^2 - b^2)^2) + (a^2*b^3*Sinh[x])/((a^2 - b^2)^3*(a*Cosh[x] + b*Sinh[
x]))

________________________________________________________________________________________

Rubi [A]  time = 1.73207, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 12, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.6, Rules used = {3111, 3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133, 3097, 3075} $-\frac{a^3 b x}{\left (a^2-b^2\right )^3}-\frac{6 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac{a b x}{4 \left (a^2-b^2\right )^2}+\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac{2 a^2 b^2 \sinh ^2(x)}{\left (a^2-b^2\right )^3}+\frac{b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}-\frac{a b \sinh (x) \cosh ^3(x)}{2 \left (a^2-b^2\right )^2}+\frac{a^3 b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{a b \sinh (x) \cosh (x)}{4 \left (a^2-b^2\right )^2}+\frac{a b^3 \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{3 a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac{3 a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-6*a^3*b^3*x)/(a^2 - b^2)^4 - (a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 + (a*b*x)/(4*(a^2 - b^2)^2) +
(b^2*Cosh[x]^4)/(4*(a^2 - b^2)^2) + (3*a^4*b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (3*a^2*b^4*Log[a*C
osh[x] + b*Sinh[x]])/(a^2 - b^2)^4 + (a^3*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^3 + (a*b^3*Cosh[x]*Sinh[x])/(a^2 - b^
2)^3 + (a*b*Cosh[x]*Sinh[x])/(4*(a^2 - b^2)^2) - (a*b*Cosh[x]^3*Sinh[x])/(2*(a^2 - b^2)^2) - (2*a^2*b^2*Sinh[x
]^2)/(a^2 - b^2)^3 + (a^2*Sinh[x]^4)/(4*(a^2 - b^2)^2) + (a^2*b^3*Sinh[x])/((a^2 - b^2)^3*(a*Cosh[x] + b*Sinh[
x]))

Rule 3111

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*
x] + b*Sin[c + d*x])^(p + 1), x], x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n*(a*Cos[c +
d*x] + b*Sin[c + d*x])^(p + 1), x], x] - Dist[(a*b)/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1
)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] &&
IGtQ[n, 0] && ILtQ[p, 0]

Rule 3109

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m
- 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 3098

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(a*x)/(a^2 + b^2), x] + Dist[b/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rule 3097

Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(b*x)/(a^2 + b^2), x] - Dist[a/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3075

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-2), x_Symbol] :> Simp[Sin[c + d*x]/(a*d*
(a*Cos[c + d*x] + b*Sin[c + d*x])), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin{align*} \int \frac{\cosh ^3(x) \sinh ^3(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{a \int \frac{\cosh ^2(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \frac{\cosh ^3(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac{(a b) \int \frac{\cosh ^2(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac{a^2 \int \cosh (x) \sinh ^3(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{(a b) \int \cosh ^2(x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac{\left (a^2 b\right ) \int \frac{\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{b^2 \int \cosh ^3(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{\left (a b^2\right ) \int \frac{\cosh ^2(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{\cosh (x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{\left (a^3 b^2\right ) \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}+2 \left (\frac{\left (a^3 b\right ) \int \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^3}-\frac{\left (a^2 b^2\right ) \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac{\left (a^3 b^2\right ) \int \frac{\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-\frac{\left (a^2 b^3\right ) \int \frac{\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}-2 \left (\frac{\left (a^2 b^2\right ) \int \cosh (x) \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac{\left (a b^3\right ) \int \cosh ^2(x) \, dx}{\left (a^2-b^2\right )^3}+\frac{\left (a^2 b^3\right ) \int \frac{\cosh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )+\frac{\left (a^3 b^3\right ) \int \frac{1}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^3}+\frac{a^2 \operatorname{Subst}\left (\int x^3 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}-2 \left (\frac{a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}-\frac{(a b) \int \cosh ^2(x) \, dx}{4 \left (a^2-b^2\right )^2}\right )+\frac{b^2 \operatorname{Subst}\left (\int x^3 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac{2 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac{b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac{a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{\left (i a^4 b^2\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^4}+\frac{\left (i a^2 b^4\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^4}+2 \left (-\frac{a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac{a^3 b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}+\frac{\left (i a^4 b^2\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^4}-\frac{\left (a^3 b\right ) \int 1 \, dx}{2 \left (a^2-b^2\right )^3}+\frac{\left (a^2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^3}\right )-2 \left (\frac{a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac{a b^3 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}-\frac{\left (i a^2 b^4\right ) \int \frac{-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^4}-\frac{\left (a^2 b^2\right ) \operatorname{Subst}(\int x \, dx,x,i \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{\left (a b^3\right ) \int 1 \, dx}{2 \left (a^2-b^2\right )^3}\right )-2 \left (-\frac{a b \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )^2}+\frac{a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}-\frac{(a b) \int 1 \, dx}{8 \left (a^2-b^2\right )^2}\right )\\ &=-\frac{2 a^3 b^3 x}{\left (a^2-b^2\right )^4}+\frac{b^2 \cosh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac{a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac{a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac{a^2 \sinh ^4(x)}{4 \left (a^2-b^2\right )^2}+\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}-2 \left (-\frac{a b x}{8 \left (a^2-b^2\right )^2}-\frac{a b \cosh (x) \sinh (x)}{8 \left (a^2-b^2\right )^2}+\frac{a b \cosh ^3(x) \sinh (x)}{4 \left (a^2-b^2\right )^2}\right )+2 \left (-\frac{a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac{a^3 b x}{2 \left (a^2-b^2\right )^3}+\frac{a^4 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}+\frac{a^3 b \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}-\frac{a^2 b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^3}\right )-2 \left (\frac{a^3 b^3 x}{\left (a^2-b^2\right )^4}-\frac{a b^3 x}{2 \left (a^2-b^2\right )^3}-\frac{a^2 b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^4}-\frac{a b^3 \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )^3}+\frac{a^2 b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^3}\right )\\ \end{align*}

Mathematica [A]  time = 1.37939, size = 366, normalized size = 1.17 $\frac{-48 a^5 b^2 x \sinh (x)+84 a^4 b^3 \sinh (x)-15 a^4 b^3 \sinh (3 x)+3 a^4 b^3 \sinh (5 x)-288 a^3 b^4 x \sinh (x)-100 a^2 b^5 \sinh (x)+3 a^2 b^5 \sinh (3 x)-3 a^2 b^5 \sinh (5 x)-3 a^5 b^2 \cosh (5 x)+3 a^3 b^4 \cosh (5 x)-3 a \left (a^2-b^2\right )^2 \left (a^2+3 b^2\right ) \cosh (3 x)+192 a^4 b^3 \sinh (x) \log (a \cosh (x)+b \sinh (x))+192 a^2 b^5 \sinh (x) \log (a \cosh (x)+b \sinh (x))-4 a \cosh (x) \left (72 a^3 b^3 x-48 a^2 b^2 \left (a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))+9 a^4 b^2-5 a^2 b^4+12 a^5 b x+a^6+12 a b^5 x-5 b^6\right )+20 a^6 b \sinh (x)+9 a^6 b \sinh (3 x)-a^6 b \sinh (5 x)+a^7 \cosh (5 x)-48 a b^6 x \sinh (x)-a b^6 \cosh (5 x)-4 b^7 \sinh (x)+3 b^7 \sinh (3 x)+b^7 \sinh (5 x)}{64 (a-b)^4 (a+b)^4 (a \cosh (x)+b \sinh (x))}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-3*a*(a^2 - b^2)^2*(a^2 + 3*b^2)*Cosh[3*x] + a^7*Cosh[5*x] - 3*a^5*b^2*Cosh[5*x] + 3*a^3*b^4*Cosh[5*x] - a*b^
6*Cosh[5*x] - 4*a*Cosh[x]*(a^6 + 9*a^4*b^2 - 5*a^2*b^4 - 5*b^6 + 12*a^5*b*x + 72*a^3*b^3*x + 12*a*b^5*x - 48*a
^2*b^2*(a^2 + b^2)*Log[a*Cosh[x] + b*Sinh[x]]) + 20*a^6*b*Sinh[x] + 84*a^4*b^3*Sinh[x] - 100*a^2*b^5*Sinh[x] -
4*b^7*Sinh[x] - 48*a^5*b^2*x*Sinh[x] - 288*a^3*b^4*x*Sinh[x] - 48*a*b^6*x*Sinh[x] + 192*a^4*b^3*Log[a*Cosh[x]
+ b*Sinh[x]]*Sinh[x] + 192*a^2*b^5*Log[a*Cosh[x] + b*Sinh[x]]*Sinh[x] + 9*a^6*b*Sinh[3*x] - 15*a^4*b^3*Sinh[3
*x] + 3*a^2*b^5*Sinh[3*x] + 3*b^7*Sinh[3*x] - a^6*b*Sinh[5*x] + 3*a^4*b^3*Sinh[5*x] - 3*a^2*b^5*Sinh[5*x] + b^
7*Sinh[5*x])/(64*(a - b)^4*(a + b)^4*(a*Cosh[x] + b*Sinh[x]))

________________________________________________________________________________________

Maple [A]  time = 0.089, size = 398, normalized size = 1.3 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^3*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x)

[Out]

1/4/(a-b)^2/(tanh(1/2*x)+1)^4-1/2/(a-b)^2/(tanh(1/2*x)+1)^3+1/8/(a-b)^3/(tanh(1/2*x)+1)*a+3/8/(a-b)^3/(tanh(1/
2*x)+1)*b+1/8/(a-b)^3/(tanh(1/2*x)+1)^2*a-5/8/(a-b)^3/(tanh(1/2*x)+1)^2*b-3/4*a*b/(a-b)^4*ln(tanh(1/2*x)+1)+2*
a^4*b^3/(a-b)^4/(a+b)^4*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)-2*a^2*b^5/(a-b)^4/(a+b)^4*tanh(1/2*x)/
(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+3*a^4*b^2/(a-b)^4/(a+b)^4*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+3*a^2*b^4/
(a-b)^4/(a+b)^4*ln(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+1/4/(a+b)^2/(tanh(1/2*x)-1)^4+1/2/(a+b)^2/(tanh(1/2*x)-1
)^3+1/8/(a+b)^3/(tanh(1/2*x)-1)^2*a+5/8/(a+b)^3/(tanh(1/2*x)-1)^2*b-1/8/(a+b)^3/(tanh(1/2*x)-1)*a+3/8/(a+b)^3/
(tanh(1/2*x)-1)*b+3/4*a*b/(a+b)^4*ln(tanh(1/2*x)-1)

________________________________________________________________________________________

Maxima [A]  time = 1.15133, size = 518, normalized size = 1.65 \begin{align*} -\frac{3 \, a b x}{4 \,{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} + \frac{3 \,{\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac{4 \,{\left (a + b\right )} e^{\left (-2 \, x\right )} -{\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - 3 \,{\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} e^{\left (-2 \, x\right )} - 4 \,{\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 52 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}}{64 \,{\left ({\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} e^{\left (-4 \, x\right )} +{\left (a^{8} - 2 \, a^{7} b - 2 \, a^{6} b^{2} + 6 \, a^{5} b^{3} - 6 \, a^{3} b^{5} + 2 \, a^{2} b^{6} + 2 \, a b^{7} - b^{8}\right )} e^{\left (-6 \, x\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*a*b*x/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 3*(a^4*b^2 + a^2*b^4)*log(-(a - b)*e^(-2*x) - a - b)/
(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) - 1/64*(4*(a + b)*e^(-2*x) - (a - b)*e^(-4*x))/(a^3 - 3*a^2*b
+ 3*a*b^2 - b^3) + 1/64*(a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 3*(a^6 - 4*a^5*b + 5*
a^4*b^2 - 5*a^2*b^4 + 4*a*b^5 - b^6)*e^(-2*x) - 4*(a^6 - 6*a^5*b + 15*a^4*b^2 - 52*a^3*b^3 + 15*a^2*b^4 - 6*a*
b^5 + b^6)*e^(-4*x))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*e^(-4*x) + (a^8 - 2*a^7*b - 2*a^6*b^2 +
6*a^5*b^3 - 6*a^3*b^5 + 2*a^2*b^6 + 2*a*b^7 - b^8)*e^(-6*x))

________________________________________________________________________________________

Fricas [B]  time = 3.16378, size = 8631, normalized size = 27.49 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x, algorithm="fricas")

[Out]

1/64*((a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^10 + 10*(a^7 - a^6*b
- 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)*sinh(x)^9 + (a^7 - a^6*b - 3*a^5*b^2 +
3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*sinh(x)^10 - 3*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*
b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^8 - 3*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*
b^6 - b^7 - 15*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^2)*sinh(x)^
8 + 24*(5*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^3 - (a^7 - 3*a^6
*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x))*sinh(x)^7 + a^7 + a^6*b - 3*a^5*b^2 -
3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7 - 4*(a^7 - 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 9*a^
2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6)*x)*cosh(x)^6 - 2*
(2*a^7 - 10*a^6*b + 18*a^5*b^2 - 10*a^4*b^3 - 10*a^3*b^4 + 18*a^2*b^5 - 10*a*b^6 + 2*b^7 - 105*(a^7 - a^6*b -
3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^4 + 42*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b
^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^2 + 24*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*
b^5 + a*b^6)*x)*sinh(x)^6 + 12*(21*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)
*cosh(x)^5 - 14*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^3 - 2*(a^7
- 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^
3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6)*x)*cosh(x))*sinh(x)^5 - 4*(a^7 + 5*a^6*b + 9*a^5*b^2 + 37*a^4*b^3 - 37*a^3
*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7 + 12*(a^6*b + 3*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - 3*a^2*b^5 - a*b^6)*x)*cosh(
x)^4 - 2*(2*a^7 + 10*a^6*b + 18*a^5*b^2 + 74*a^4*b^3 - 74*a^3*b^4 - 18*a^2*b^5 - 10*a*b^6 - 2*b^7 - 105*(a^7 -
a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^6 + 105*(a^7 - 3*a^6*b + a^5*b^2
+ 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^4 + 30*(a^7 - 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*
a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6)*x)*
cosh(x)^2 + 24*(a^6*b + 3*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - 3*a^2*b^5 - a*b^6)*x)*sinh(x)^4 + 8*(15*(a^7 - a^6
*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^7 - 21*(a^7 - 3*a^6*b + a^5*b^2 + 5*
a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^5 - 10*(a^7 - 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b
^4 + 9*a^2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6)*x)*cosh(
x)^3 - 2*(a^7 + 5*a^6*b + 9*a^5*b^2 + 37*a^4*b^3 - 37*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7 + 12*(a^6*b + 3*a^5*
b^2 + 2*a^4*b^3 - 2*a^3*b^4 - 3*a^2*b^5 - a*b^6)*x)*cosh(x))*sinh(x)^3 - 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^
3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*cosh(x)^2 + 3*(15*(a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 -
3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^8 - a^7 - 3*a^6*b - a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 - a^2*b^5 - 3*a*b^6 - b^7
- 28*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^6 - 20*(a^7 - 5*a^6*
b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 9*a^2*b^5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^
3*b^4 + 5*a^2*b^5 + a*b^6)*x)*cosh(x)^4 - 8*(a^7 + 5*a^6*b + 9*a^5*b^2 + 37*a^4*b^3 - 37*a^3*b^4 - 9*a^2*b^5 -
5*a*b^6 - b^7 + 12*(a^6*b + 3*a^5*b^2 + 2*a^4*b^3 - 2*a^3*b^4 - 3*a^2*b^5 - a*b^6)*x)*cosh(x)^2)*sinh(x)^2 +
192*((a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)^6 + 6*(a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)*sin
h(x)^5 + (a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*sinh(x)^6 + (a^5*b^2 - a^4*b^3 + a^3*b^4 - a^2*b^5)*cosh(x)^4
+ (a^5*b^2 - a^4*b^3 + a^3*b^4 - a^2*b^5 + 15*(a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)^2)*sinh(x)^4 +
4*(5*(a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)^3 + (a^5*b^2 - a^4*b^3 + a^3*b^4 - a^2*b^5)*cosh(x))*sinh
(x)^3 + 3*(5*(a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)^4 + 2*(a^5*b^2 - a^4*b^3 + a^3*b^4 - a^2*b^5)*cos
h(x)^2)*sinh(x)^2 + 2*(3*(a^5*b^2 + a^4*b^3 + a^3*b^4 + a^2*b^5)*cosh(x)^5 + 2*(a^5*b^2 - a^4*b^3 + a^3*b^4 -
a^2*b^5)*cosh(x)^3)*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) + 2*(5*(a^7 - a^6*b - 3*a^5*b^
2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x)^9 - 12*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a
^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^7 - 12*(a^7 - 5*a^6*b + 9*a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + 9*a^2*b^
5 - 5*a*b^6 + b^7 + 12*(a^6*b + 5*a^5*b^2 + 10*a^4*b^3 + 10*a^3*b^4 + 5*a^2*b^5 + a*b^6)*x)*cosh(x)^5 - 8*(a^7
+ 5*a^6*b + 9*a^5*b^2 + 37*a^4*b^3 - 37*a^3*b^4 - 9*a^2*b^5 - 5*a*b^6 - b^7 + 12*(a^6*b + 3*a^5*b^2 + 2*a^4*b
^3 - 2*a^3*b^4 - 3*a^2*b^5 - a*b^6)*x)*cosh(x)^3 - 3*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^
5 + 3*a*b^6 + b^7)*cosh(x))*sinh(x))/((a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6
- 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^6 + 6*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3
*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)*sinh(x)^5 + (a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b
^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*sinh(x)^6 + (a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4
*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^4 + (a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a
^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9 + 15*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5
- 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a
^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^3 + (a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6
*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x))*sinh(x)^3 + 3*(5*(a^9 + a^8*b - 4*a^7*b^2
- 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^4 + 2*(a^9 - a^8*b - 4*a^7
*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^2)*sinh(x)^2 + 2*(3*(a
^9 + a^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b^7 + a*b^8 + b^9)*cosh(x)^5 +
2*(a^9 - a^8*b - 4*a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b^8 - b^9)*cosh(x)^
3)*sinh(x))

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**3*sinh(x)**3/(a*cosh(x)+b*sinh(x))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18333, size = 518, normalized size = 1.65 \begin{align*} -\frac{3 \, a b x}{4 \,{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac{{\left (36 \, a b e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \,{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )}} + \frac{3 \,{\left (a^{4} b^{2} + a^{2} b^{4}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac{a^{2} e^{\left (4 \, x\right )} + 2 \, a b e^{\left (4 \, x\right )} + b^{2} e^{\left (4 \, x\right )} - 4 \, a^{2} e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )}}{64 \,{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )}} - \frac{3 \, a^{5} b^{2} e^{\left (2 \, x\right )} + 3 \, a^{4} b^{3} e^{\left (2 \, x\right )} + 3 \, a^{3} b^{4} e^{\left (2 \, x\right )} + 3 \, a^{2} b^{5} e^{\left (2 \, x\right )} + 3 \, a^{5} b^{2} - a^{4} b^{3} + a^{3} b^{4} - 3 \, a^{2} b^{5}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3*sinh(x)^3/(a*cosh(x)+b*sinh(x))^2,x, algorithm="giac")

[Out]

-3/4*a*b*x/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 1/64*(36*a*b*e^(4*x) - 4*a^2*e^(2*x) + 4*b^2*e^(2*x)
+ a^2 - 2*a*b + b^2)*e^(-4*x)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 3*(a^4*b^2 + a^2*b^4)*log(abs(a*e^
(2*x) + b*e^(2*x) + a - b))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + 1/64*(a^2*e^(4*x) + 2*a*b*e^(4*x
) + b^2*e^(4*x) - 4*a^2*e^(2*x) + 4*b^2*e^(2*x))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - (3*a^5*b^2*e^(2
*x) + 3*a^4*b^3*e^(2*x) + 3*a^3*b^4*e^(2*x) + 3*a^2*b^5*e^(2*x) + 3*a^5*b^2 - a^4*b^3 + a^3*b^4 - 3*a^2*b^5)/(
(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(a*e^(2*x) + b*e^(2*x) + a - b))