### 3.722 $$\int \frac{\cosh ^3(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx$$

Optimal. Leaf size=259 $\frac{b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh (x)}{\left (a^2-b^2\right )^2}-\frac{4 a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac{2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{2 a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{2 a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{2 a b^4 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac{3 a^3 b^2 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}$

[Out]

(3*a^3*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a*b^4*ArcTan[(b*Cosh[x] + a
*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a^3*b*Cosh[x])/(a^2 - b^2)^3 + (2*a*b^3*Cosh[x])/(a^2 - b^2
)^3 - (2*a*b*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (4*a^2*b^2*Sinh[x])/(a^2 - b^2)^3 + (b^2*Sinh[x])/(a^2 - b^2)^2 +
(a^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (b^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (a^2*b^3)/((a^2 - b^2)^3*(a*Cosh[x] +
b*Sinh[x]))

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Rubi [A]  time = 0.907976, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 12, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.6, Rules used = {3111, 3109, 2633, 2565, 30, 3100, 2637, 3074, 206, 2564, 2638, 3155} $\frac{b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh (x)}{\left (a^2-b^2\right )^2}-\frac{4 a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac{2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{2 a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{2 a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{2 a b^4 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac{3 a^3 b^2 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*a^3*b^2*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a*b^4*ArcTan[(b*Cosh[x] + a
*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*a^3*b*Cosh[x])/(a^2 - b^2)^3 + (2*a*b^3*Cosh[x])/(a^2 - b^2
)^3 - (2*a*b*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (4*a^2*b^2*Sinh[x])/(a^2 - b^2)^3 + (b^2*Sinh[x])/(a^2 - b^2)^2 +
(a^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (b^2*Sinh[x]^3)/(3*(a^2 - b^2)^2) + (a^2*b^3)/((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 2633

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 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 3100

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3074

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3155

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.))/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(
x_)])^2, x_Symbol] :> Simp[(c*B + c*A*Cos[d + e*x] + (a*B - b*A)*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos
[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B)/(a^2 - b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e
*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B, 0]

Rubi steps

\begin{align*} \int \frac{\cosh ^3(x) \sinh ^2(x)}{(a \cosh (x)+b \sinh (x))^2} \, dx &=\frac{a \int \frac{\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}-\frac{b \int \frac{\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}+\frac{(a b) \int \frac{\cosh ^2(x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{a^2-b^2}\\ &=\frac{a^2 \int \cosh (x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{(a b) \int \cosh ^2(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}+2 \frac{\left (a^2 b\right ) \int \frac{\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}+\frac{b^2 \int \cosh ^3(x) \, dx}{\left (a^2-b^2\right )^2}-2 \frac{\left (a b^2\right ) \int \frac{\cosh ^2(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)}{(a \cosh (x)+b \sinh (x))^2} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{\left (a^3 b^2\right ) \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}+2 \left (\frac{\left (a^3 b\right ) \int \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac{\left (a^2 b^2\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac{\left (a^3 b^2\right ) \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac{a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{\left (a^2 b^2\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac{\left (a b^4\right ) \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}\right )+\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}-2 \frac{(a b) \operatorname{Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac{2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac{a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+\frac{\left (i a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}+2 \left (\frac{a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac{\left (i a^3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac{a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}-\frac{\left (i a b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\right )\\ &=\frac{a^3 b^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac{2 a b \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac{a^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{b^2 \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^2 b^3}{\left (a^2-b^2\right )^3 (a \cosh (x)+b \sinh (x))}+2 \left (\frac{a^3 b^2 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac{a^3 b \cosh (x)}{\left (a^2-b^2\right )^3}-\frac{a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}\right )-2 \left (-\frac{a b^4 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}-\frac{a b^3 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b^2 \sinh (x)}{\left (a^2-b^2\right )^3}\right )\\ \end{align*}

Mathematica [A]  time = 2.0564, size = 481, normalized size = 1.86 $\frac{1}{16} \left (\frac{4 \left (a^2+b^2\right ) \sinh (x)}{(a-b)^2 (a+b)^2}-\frac{6 a \left (a^2+3 b^2\right ) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}-\frac{b \left (3 a^2+b^2\right )}{(a-b)^2 (a+b)^2 (a \cosh (x)+b \sinh (x))}-\frac{8 a b \cosh (x)}{(a-b)^2 (a+b)^2}\right )+\frac{1}{16} \left (-\frac{8 \left (6 a^2 b^2+a^4+b^4\right ) \sinh (x)}{(a-b)^3 (a+b)^3}+\frac{4 \left (a^2+b^2\right ) \sinh (3 x)}{3 (a-b)^2 (a+b)^2}+\frac{32 a b \left (a^2+b^2\right ) \cosh (x)}{(a-b)^3 (a+b)^3}+\frac{10 a \left (10 a^2 b^2+a^4+5 b^4\right ) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac{b \left (10 a^2 b^2+5 a^4+b^4\right )}{(a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))}-\frac{8 a b \cosh (3 x)}{3 (a-b)^2 (a+b)^2}\right )-\frac{2 a^2 \sqrt{a+b} \cosh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+2 a b \sqrt{a+b} \sinh (x) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )+b \sqrt{a-b} (a+b)}{8 (a-b)^{3/2} (a+b)^2 (a \cosh (x)+b \sinh (x))}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Sqrt[a - b]*b*(a + b) + 2*a^2*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Cosh[x] + 2*a*
b*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]*Sinh[x])/(8*(a - b)^(3/2)*(a + b)^2*(a*Cosh[
x] + b*Sinh[x])) + ((-6*a*(a^2 + 3*b^2)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(5/2)*(a
+ b)^(5/2)) - (8*a*b*Cosh[x])/((a - b)^2*(a + b)^2) + (4*(a^2 + b^2)*Sinh[x])/((a - b)^2*(a + b)^2) - (b*(3*a
^2 + b^2))/((a - b)^2*(a + b)^2*(a*Cosh[x] + b*Sinh[x])))/16 + ((10*a*(a^4 + 10*a^2*b^2 + 5*b^4)*ArcTan[(b + a
*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(7/2)*(a + b)^(7/2)) + (32*a*b*(a^2 + b^2)*Cosh[x])/((a - b)^
3*(a + b)^3) - (8*a*b*Cosh[3*x])/(3*(a - b)^2*(a + b)^2) - (8*(a^4 + 6*a^2*b^2 + b^4)*Sinh[x])/((a - b)^3*(a +
b)^3) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/((a - b)^3*(a + b)^3*(a*Cosh[x] + b*Sinh[x])) + (4*(a^2 + b^2)*Sinh[3*
x])/(3*(a - b)^2*(a + b)^2))/16

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Maple [A]  time = 0.08, size = 289, normalized size = 1.1 \begin{align*} -{\frac{1}{3\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{2\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{b}{ \left ( a-b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+2\,{\frac{{b}^{4}a\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{{a}^{2}{b}^{3}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+6\,{\frac{{a}^{3}{b}^{2}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }+4\,{\frac{{b}^{4}a}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{3\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{2\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{b}{ \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/(a-b)^2/(tanh(1/2*x)+1)^3+1/2/(a-b)^2/(tanh(1/2*x)+1)^2+1/(a-b)^3/(tanh(1/2*x)+1)*b+2*a*b^4/(a-b)^3/(a+b)
^3*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+2*a^2*b^3/(a-b)^3/(a+b)^3/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^
2)+6*a^3*b^2/(a-b)^3/(a+b)^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))+4*a*b^4/(a-b)^3
/(a+b)^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))-1/3/(a+b)^2/(tanh(1/2*x)-1)^3-1/2/(
a+b)^2/(tanh(1/2*x)-1)^2-1/(a+b)^3/(tanh(1/2*x)-1)*b

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.4068, size = 11158, normalized size = 43.08 \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)^2/(a*cosh(x)+b*sinh(x))^2,x, algorithm="fricas")

[Out]

[1/24*((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 + 8*(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)^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)*sinh(x)^8 - 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 - 2*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7
)*cosh(x)^6 - 2*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7 - 14*(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)^6 + 4*(14*(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 - 3*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a
^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x))*sinh(x)^5 + 6*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 -
3*b^7)*cosh(x)^4 + 2*(21*a^6*b + 69*a^4*b^3 - 81*a^2*b^5 - 9*b^7 + 35*(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 - 15*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^
2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x)^2)*sinh(x)^4 + 8*(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)*cosh(x)^5 - 5*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4
*b^7)*cosh(x)^3 + 3*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x))*sinh(x)^3 + 2*(a^7 + 6*a^6*b + 7*a^5*
b^2 - 8*a^4*b^3 - 17*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*b^7)*cosh(x)^2 + 2*(a^7 + 6*a^6*b + 7*a^5*b^2 - 8*a^4*b
^3 - 17*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*b^7 + 14*(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 - 15*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*
b^7)*cosh(x)^4 + 18*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x)^2)*sinh(x)^2 + 24*((3*a^4*b^2 + 3*a^3*
b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^5 + 5*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)*sinh(x)^4 + (3*
a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*sinh(x)^5 + (3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5)*cosh(x)^3
+ (3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 + 10*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^2)*
sinh(x)^3 + (10*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^3 + 3*(3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4
- 2*a*b^5)*cosh(x))*sinh(x)^2 + (5*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^4 + 3*(3*a^4*b^2 - 3
*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5)*cosh(x)^2)*sinh(x))*sqrt(-a^2 + b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x
)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)
*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 4*(2*(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 - 3*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 -
4*b^7)*cosh(x)^5 + 6*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x)^3 + (a^7 + 6*a^6*b + 7*a^5*b^2 - 8*a^
4*b^3 - 17*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*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)^5 + 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)*sinh(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)*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)*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 + 10*(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)^3 + (10*(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 + 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)^2
+ (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 + 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)^2)*sinh(x)), 1/24*((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 + 8*(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)^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)*sinh(x)^8 - 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 - 2*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b
^5 + 9*a*b^6 - 4*b^7)*cosh(x)^6 - 2*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6
- 4*b^7 - 14*(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)^6
+ 4*(14*(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 - 3*(a^7 - 6*a^6
*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x))*sinh(x)^5 + 6*(7*a^6*b + 23*a^
4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x)^4 + 2*(21*a^6*b + 69*a^4*b^3 - 81*a^2*b^5 - 9*b^7 + 35*(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 - 15*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3
- 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x)^2)*sinh(x)^4 + 8*(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)*cosh(x)^5 - 5*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a
^2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x)^3 + 3*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x))*sinh(x)^3 + 2*(a^
7 + 6*a^6*b + 7*a^5*b^2 - 8*a^4*b^3 - 17*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*b^7)*cosh(x)^2 + 2*(a^7 + 6*a^6*b +
7*a^5*b^2 - 8*a^4*b^3 - 17*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*b^7 + 14*(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 - 15*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^
2*b^5 + 9*a*b^6 - 4*b^7)*cosh(x)^4 + 18*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x)^2)*sinh(x)^2 - 48*
((3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^5 + 5*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cos
h(x)*sinh(x)^4 + (3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*sinh(x)^5 + (3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4
- 2*a*b^5)*cosh(x)^3 + (3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5 + 10*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 +
2*a*b^5)*cosh(x)^2)*sinh(x)^3 + (10*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^3 + 3*(3*a^4*b^2 - 3
*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5)*cosh(x))*sinh(x)^2 + (5*(3*a^4*b^2 + 3*a^3*b^3 + 2*a^2*b^4 + 2*a*b^5)*cosh(x)^
4 + 3*(3*a^4*b^2 - 3*a^3*b^3 + 2*a^2*b^4 - 2*a*b^5)*cosh(x)^2)*sinh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)
/((a + b)*cosh(x) + (a + b)*sinh(x))) + 4*(2*(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 - 3*(a^7 - 6*a^6*b + 7*a^5*b^2 + 8*a^4*b^3 - 17*a^3*b^4 + 2*a^2*b^5 + 9*a*b^6 - 4*b^7)*co
sh(x)^5 + 6*(7*a^6*b + 23*a^4*b^3 - 27*a^2*b^5 - 3*b^7)*cosh(x)^3 + (a^7 + 6*a^6*b + 7*a^5*b^2 - 8*a^4*b^3 - 1
7*a^3*b^4 - 2*a^2*b^5 + 9*a*b^6 + 4*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)^5 + 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)*sinh(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)*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)*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 + 10*(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)^3 + (10*(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 + 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)^2 + (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 + 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)^2
)*sinh(x))]

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

[Out]

Timed out

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Giac [A]  time = 1.15524, size = 419, normalized size = 1.62 \begin{align*} \frac{2 \, a^{2} b^{3} e^{x}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left (a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b\right )}} + \frac{{\left (3 \, a e^{\left (2 \, x\right )} + 9 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{2 \,{\left (3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt{a^{2} - b^{2}}} + \frac{a^{4} e^{\left (3 \, x\right )} + 4 \, a^{3} b e^{\left (3 \, x\right )} + 6 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 4 \, a b^{3} e^{\left (3 \, x\right )} + b^{4} e^{\left (3 \, x\right )} - 3 \, a^{4} e^{x} + 18 \, a^{2} b^{2} e^{x} + 24 \, a b^{3} e^{x} + 9 \, b^{4} e^{x}}{24 \,{\left (a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^2*b^3*e^x/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(a*e^(2*x) + b*e^(2*x) + a - b)) + 1/24*(3*a*e^(2*x) + 9*b*
e^(2*x) - a + b)*e^(-3*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 2*(3*a^3*b^2 + 2*a*b^4)*arctan((a*e^x + b*e^x)/sqr
t(a^2 - b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(a^2 - b^2)) + 1/24*(a^4*e^(3*x) + 4*a^3*b*e^(3*x) + 6*
a^2*b^2*e^(3*x) + 4*a*b^3*e^(3*x) + b^4*e^(3*x) - 3*a^4*e^x + 18*a^2*b^2*e^x + 24*a*b^3*e^x + 9*b^4*e^x)/(a^6
+ 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)