Integrand size = 18, antiderivative size = 137 \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=-\frac {a b^3 \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {a b^2 \cosh (x)}{\left (a^2-b^2\right )^2}+\frac {a \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {a^2 b \sinh (x)}{\left (a^2-b^2\right )^2}-\frac {b \sinh (x)}{a^2-b^2}-\frac {b \sinh ^3(x)}{3 \left (a^2-b^2\right )} \]
-a*b^3*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)-a*b^2 *cosh(x)/(a^2-b^2)^2+1/3*a*cosh(x)^3/(a^2-b^2)+a^2*b*sinh(x)/(a^2-b^2)^2-b *sinh(x)/(a^2-b^2)-1/3*b*sinh(x)^3/(a^2-b^2)
Time = 0.87 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22 \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {1}{12} \left (-\frac {24 a b^3 \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}+\frac {3 a \left (a^2-5 b^2\right ) \cosh (x)}{(a-b)^2 (a+b)^2}+\frac {a \cosh (3 x)}{(a-b) (a+b)}+\frac {3 b \left (a^2+3 b^2\right ) \sinh (x)}{(a-b)^2 (a+b)^2}-\frac {a^2 b \sinh (3 x)}{(a-b)^2 (a+b)^2}+\frac {b^3 \sinh (3 x)}{(a-b)^2 (a+b)^2}\right ) \]
((-24*a*b^3*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^ (5/2)*(a + b)^(5/2)) + (3*a*(a^2 - 5*b^2)*Cosh[x])/((a - b)^2*(a + b)^2) + (a*Cosh[3*x])/((a - b)*(a + b)) + (3*b*(a^2 + 3*b^2)*Sinh[x])/((a - b)^2* (a + b)^2) - (a^2*b*Sinh[3*x])/((a - b)^2*(a + b)^2) + (b^3*Sinh[3*x])/((a - b)^2*(a + b)^2))/12
Result contains complex when optimal does not.
Time = 0.75 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.14, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {3042, 26, 3588, 26, 3042, 26, 3045, 15, 3113, 2009, 3579, 3042, 3117, 3553, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sinh (x) \cosh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x) \cos (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cos (i x)^3 \sin (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -i \left (-\frac {i b \int \cosh ^3(x)dx}{a^2-b^2}+\frac {a \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-\frac {i b \int \cosh ^3(x)dx}{a^2-b^2}+\frac {i a \int \cosh ^2(x) \sinh (x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cosh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}+\frac {i a \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}+\frac {a \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}+\frac {i a \int \cosh ^2(x)d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -i \left (-\frac {i b \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -i \left (\frac {b \int \left (\sinh ^2(x)+1\right )d(-i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a b \int \frac {\cos (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle -i \left (\frac {i a b \left (\frac {a \int \cosh (x)dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a b \left (\frac {a \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}-\frac {b^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -i \left (\frac {i a b \left (-\frac {b^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -i \left (\frac {i a b \left (-\frac {i b^2 \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}+\frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -i \left (\frac {i a b \left (-\frac {i b^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}+\frac {a \sinh (x)}{a^2-b^2}-\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {b \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}+\frac {i a \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
(-I)*(((I/3)*a*Cosh[x]^3)/(a^2 - b^2) + (I*a*b*(((-I)*b^2*ArcTanh[((-I)*b* Cosh[x] - I*a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (b*Cosh[x])/( a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)))/(a^2 - b^2) + (b*((-I)*Sinh[x] - (I /3)*Sinh[x]^3))/(a^2 - b^2))
3.8.12.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(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])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-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]
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] + (Simp[a/(a^2 + b^2) Int[Cos[c + d*x]^(m - 1), x], x] + Simp[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]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[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]
Time = 1.92 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.27
| method | result | size |
| risch | \(\frac {{\mathrm e}^{3 x}}{24 a +24 b}+\frac {{\mathrm e}^{x} a}{8 \left (a +b \right )^{2}}+\frac {3 \,{\mathrm e}^{x} b}{8 \left (a +b \right )^{2}}+\frac {{\mathrm e}^{-x} a}{8 \left (a -b \right )^{2}}-\frac {3 \,{\mathrm e}^{-x} b}{8 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a -24 b}-\frac {b^{3} a \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {b^{3} a \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(174\) |
| default | \(-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3} \left (4 a +4 b \right )}-\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {a +2 b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {2}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3} \left (4 a -4 b \right )}-\frac {-a +2 b}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 a \,b^{3} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a +b \right )^{2} \left (a -b \right )^{2} \sqrt {a^{2}-b^{2}}}\) | \(176\) |
1/24/(a+b)*exp(x)^3+1/8/(a+b)^2*exp(x)*a+3/8/(a+b)^2*exp(x)*b+1/8/(a-b)^2/ exp(x)*a-3/8/(a-b)^2/exp(x)*b+1/24/(a-b)/exp(x)^3-1/(-a^2+b^2)^(1/2)*b^3*a /(a+b)^2/(a-b)^2*ln(exp(x)+(a-b)/(-a^2+b^2)^(1/2))+1/(-a^2+b^2)^(1/2)*b^3* a/(a+b)^2/(a-b)^2*ln(exp(x)-(a-b)/(-a^2+b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 887 vs. \(2 (129) = 258\).
Time = 0.30 (sec) , antiderivative size = 1829, normalized size of antiderivative = 13.35 \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Too large to display} \]
[1/24*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 + 6*( a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)*sinh(x)^5 + (a^ 5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^6 + a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + 3*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2* b^3 + 5*a*b^4 - 3*b^5)*cosh(x)^4 + 3*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5 + 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)* cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 + 3*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5 )*cosh(x))*sinh(x)^3 + 3*(a^5 - a^4*b - 6*a^3*b^2 - 2*a^2*b^3 + 5*a*b^4 + 3*b^5)*cosh(x)^2 + 3*(a^5 - a^4*b - 6*a^3*b^2 - 2*a^2*b^3 + 5*a*b^4 + 3*b^ 5 + 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^4 + 6*(a ^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5)*cosh(x)^2)*sinh(x)^2 - 24*(a*b^3*cosh(x)^3 + 3*a*b^3*cosh(x)^2*sinh(x) + 3*a*b^3*cosh(x)*sinh( x)^2 + a*b^3*sinh(x)^3)*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)*sin h(x)^2 + a - b)) + 6*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)* cosh(x)^5 + 2*(a^5 + a^4*b - 6*a^3*b^2 + 2*a^2*b^3 + 5*a*b^4 - 3*b^5)*cosh (x)^3 + (a^5 - a^4*b - 6*a^3*b^2 - 2*a^2*b^3 + 5*a*b^4 + 3*b^5)*cosh(x))*s inh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4...
Timed out. \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.26 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=-\frac {2 \, a b^{3} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {{\left (3 \, a e^{\left (2 \, x\right )} - 9 \, b e^{\left (2 \, x\right )} + a - b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} + b^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{x} + 12 \, a b e^{x} + 9 \, b^{2} e^{x}}{24 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} \]
-2*a*b^3*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/((a^4 - 2*a^2*b^2 + b^4)* sqrt(a^2 - b^2)) + 1/24*(3*a*e^(2*x) - 9*b*e^(2*x) + a - b)*e^(-3*x)/(a^2 - 2*a*b + b^2) + 1/24*(a^2*e^(3*x) + 2*a*b*e^(3*x) + b^2*e^(3*x) + 3*a^2*e ^x + 12*a*b*e^x + 9*b^2*e^x)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
Time = 2.61 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.89 \[ \int \frac {\cosh ^3(x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a-24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,a+24\,b}+\frac {{\mathrm {e}}^x\,\left (a+3\,b\right )}{8\,{\left (a+b\right )}^2}-\frac {2\,\mathrm {atan}\left (\frac {a\,b^3\,{\mathrm {e}}^x\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}{a^5\,\sqrt {a^2\,b^6}-b^5\,\sqrt {a^2\,b^6}+2\,a^2\,b^3\,\sqrt {a^2\,b^6}-2\,a^3\,b^2\,\sqrt {a^2\,b^6}+a\,b^4\,\sqrt {a^2\,b^6}-a^4\,b\,\sqrt {a^2\,b^6}}\right )\,\sqrt {a^2\,b^6}}{\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}+\frac {{\mathrm {e}}^{-x}\,\left (a-3\,b\right )}{8\,{\left (a-b\right )}^2} \]
exp(-3*x)/(24*a - 24*b) + exp(3*x)/(24*a + 24*b) + (exp(x)*(a + 3*b))/(8*( a + b)^2) - (2*atan((a*b^3*exp(x)*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)^(1/2))/(a^5*(a^2*b^6)^(1/2) - b^5*(a^2*b^6)^(1/2) + 2*a^2*b^3*(a^2*b^6)^(1/2) - 2*a^3*b^2*(a^2*b^6)^(1/2) + a*b^4*(a^2*b^6)^ (1/2) - a^4*b*(a^2*b^6)^(1/2)))*(a^2*b^6)^(1/2))/(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)^(1/2) + (exp(-x)*(a - 3*b))/(8*(a - b)^2)