Integrand size = 13, antiderivative size = 135 \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\frac {a^3 b \text {arctanh}\left (\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {a^2 b \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {b \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {a b^2 \sinh (x)}{\left (a^2-b^2\right )^2}+\frac {a \sinh (x)}{a^2-b^2}+\frac {a \sinh ^3(x)}{3 \left (a^2-b^2\right )} \]
a^3*b*arctanh((a*cosh(x)+b*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)-a^2*b *cosh(x)/(a^2-b^2)^2-1/3*b*cosh(x)^3/(a^2-b^2)+a*b^2*sinh(x)/(a^2-b^2)^2+a *sinh(x)/(a^2-b^2)+1/3*a*sinh(x)^3/(a^2-b^2)
Time = 1.71 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.24 \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\frac {1}{12} \left (-\frac {24 a^3 b \arctan \left (\frac {a+b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a+b} \sqrt {a+b}}\right )}{(-a+b)^{5/2} (a+b)^{5/2}}+\frac {3 b \left (-5 a^2+b^2\right ) \cosh (x)}{(a-b)^2 (a+b)^2}+\frac {b \cosh (3 x)}{(-a+b) (a+b)}+\frac {3 a \left (3 a^2+b^2\right ) \sinh (x)}{(a-b)^2 (a+b)^2}+\frac {a^3 \sinh (3 x)}{(a-b)^2 (a+b)^2}-\frac {a b^2 \sinh (3 x)}{(a-b)^2 (a+b)^2}\right ) \]
((-24*a^3*b*ArcTan[(a + b*Tanh[x/2])/(Sqrt[-a + b]*Sqrt[a + b])])/((-a + b )^(5/2)*(a + b)^(5/2)) + (3*b*(-5*a^2 + b^2)*Cosh[x])/((a - b)^2*(a + b)^2 ) + (b*Cosh[3*x])/((-a + b)*(a + b)) + (3*a*(3*a^2 + b^2)*Sinh[x])/((a - b )^2*(a + b)^2) + (a^3*Sinh[3*x])/((a - b)^2*(a + b)^2) - (a*b^2*Sinh[3*x]) /((a - b)^2*(a + b)^2))/12
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.16, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.462, Rules used = {3042, 4001, 26, 26, 3042, 26, 3588, 26, 3042, 26, 3045, 15, 3113, 2009, 3579, 3042, 3117, 3553, 217}
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.
| \(\displaystyle \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^3}{a-i b \tan \left (\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 4001 |
| \(\displaystyle \int -\frac {i \sinh (x) \cosh ^3(x)}{-i a \sinh (x)-i b \cosh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {i \cosh ^3(x) \sinh (x)}{b \cosh (x)+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh (x) \cosh ^3(x)}{a \sinh (x)+b \cosh (x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \sin (i x) \cos (i x)^3}{b \cos (i x)-i a \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\cos (i x)^3 \sin (i x)}{b \cos (i x)-i a \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle -i \left (\frac {i a \int \cosh ^3(x)dx}{a^2-b^2}-\frac {b \int i \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x)}{b \cosh (x)+a \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \cosh ^3(x)dx}{a^2-b^2}-\frac {i b \int \cosh ^2(x) \sinh (x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cosh ^2(x)}{b \cosh (x)+a \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i a \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}-\frac {i b \int -i \cos (i x)^2 \sin (i x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i a \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}-\frac {b \int \cos (i x)^2 \sin (i x)dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle -i \left (\frac {i a \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}-\frac {i b \int \cosh ^2(x)d\cosh (x)}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -i \left (\frac {i a \int \sin \left (i x+\frac {\pi }{2}\right )^3dx}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle -i \left (-\frac {a \int \left (\sinh ^2(x)+1\right )d(-i \sinh (x))}{a^2-b^2}-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (-\frac {i a b \int \frac {\cos (i x)^2}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle -i \left (-\frac {i a b \left (-\frac {b \int \cosh (x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{b \cosh (x)+a \sinh (x)}dx}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (-\frac {i a b \left (-\frac {b \int \sin \left (i x+\frac {\pi }{2}\right )dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle -i \left (-\frac {i a b \left (\frac {a^2 \int \frac {1}{b \cos (i x)-i a \sin (i x)}dx}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle -i \left (-\frac {i a b \left (\frac {i a^2 \int \frac {1}{-a^2+b^2-(-i a \cosh (x)-i b \sinh (x))^2}d(-i a \cosh (x)-i b \sinh (x))}{a^2-b^2}-\frac {b \sinh (x)}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -i \left (-\frac {i a b \left (-\frac {i a^2 \arctan \left (\frac {-i a \cosh (x)-i b \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {b \sinh (x)}{a^2-b^2}+\frac {a \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}-\frac {a \left (-\frac {1}{3} i \sinh ^3(x)-i \sinh (x)\right )}{a^2-b^2}-\frac {i b \cosh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
(-I)*(((-1/3*I)*b*Cosh[x]^3)/(a^2 - b^2) - (I*a*b*(((-I)*a^2*ArcTan[((-I)* a*Cosh[x] - I*b*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Cosh[x]) /(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2)))/(a^2 - b^2) - (a*((-I)*Sinh[x] - (I/3)*Sinh[x]^3))/(a^2 - b^2))
3.2.15.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 2.00 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(\frac {{\mathrm e}^{3 x}}{24 a +24 b}+\frac {3 \,{\mathrm e}^{x} a}{8 \left (a +b \right )^{2}}+\frac {{\mathrm e}^{x} b}{8 \left (a +b \right )^{2}}-\frac {3 \,{\mathrm e}^{-x} a}{8 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-x} b}{8 \left (a -b \right )^{2}}-\frac {{\mathrm e}^{-3 x}}{24 \left (a -b \right )}+\frac {b \,a^{3} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {b \,a^{3} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(174\) |
| default | \(-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3} \left (4 a -4 b \right )}+\frac {2}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 a -b}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 a^{3} b \arctan \left (\frac {2 b \tanh \left (\frac {x}{2}\right )+2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {-a^{2}+b^{2}}}-\frac {4}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3} \left (4 a +4 b \right )}-\frac {2}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a +b}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(176\) |
1/24/(a+b)*exp(x)^3+3/8/(a+b)^2*exp(x)*a+1/8/(a+b)^2*exp(x)*b-3/8/(a-b)^2/ exp(x)*a+1/8/(a-b)^2/exp(x)*b-1/24/(a-b)/exp(x)^3+1/(a^2-b^2)^(1/2)*b*a^3/ (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*a^3/( 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. 909 vs. \(2 (127) = 254\).
Time = 0.29 (sec) , antiderivative size = 1873, normalized size of antiderivative = 13.87 \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \]
[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*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6* a^2*b^3 - a*b^4 - b^5)*cosh(x)^4 + 3*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2* b^3 - a*b^4 - 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*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5 )*cosh(x))*sinh(x)^3 - 3*(3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + b^5)*cosh(x)^2 - 3*(3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + 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*(3 *a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5)*cosh(x)^2)*sinh(x)^2 + 24*(a^3*b*cosh(x)^3 + 3*a^3*b*cosh(x)^2*sinh(x) + 3*a^3*b*cosh(x)*sinh( x)^2 + a^3*b*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)*sinh( 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)*co sh(x)^5 + 2*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5)*cosh(x )^3 - (3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + b^5)*cosh(x))*sin h(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b...
\[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\text {Exception raised: ValueError} \]
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*a^2-4*b^2>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.21 \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\frac {2 \, a^{3} b \arctan \left (-\frac {a e^{x} + b e^{x}}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {{\left (9 \, a e^{\left (2 \, x\right )} - 3 \, b e^{\left (2 \, x\right )} + a - b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} + b^{2} e^{\left (3 \, x\right )} + 9 \, a^{2} e^{x} + 12 \, a b e^{x} + 3 \, b^{2} e^{x}}{24 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} \]
2*a^3*b*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*(9*a*e^(2*x) - 3*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) + 9*a^2 *e^x + 12*a*b*e^x + 3*b^2*e^x)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
Time = 2.70 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.94 \[ \int \frac {\cosh ^3(x)}{a+b \coth (x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,a+24\,b}-\frac {{\mathrm {e}}^{-3\,x}}{24\,a-24\,b}+\frac {{\mathrm {e}}^x\,\left (3\,a+b\right )}{8\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a-b\right )}{8\,{\left (a-b\right )}^2}+\frac {2\,\mathrm {atan}\left (\frac {a^3\,b\,{\mathrm {e}}^x\,\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}}{a^5\,\sqrt {a^6\,b^2}-b^5\,\sqrt {a^6\,b^2}+2\,a^2\,b^3\,\sqrt {a^6\,b^2}-2\,a^3\,b^2\,\sqrt {a^6\,b^2}+a\,b^4\,\sqrt {a^6\,b^2}-a^4\,b\,\sqrt {a^6\,b^2}}\right )\,\sqrt {a^6\,b^2}}{\sqrt {-a^{10}+5\,a^8\,b^2-10\,a^6\,b^4+10\,a^4\,b^6-5\,a^2\,b^8+b^{10}}} \]
exp(3*x)/(24*a + 24*b) - exp(-3*x)/(24*a - 24*b) + (exp(x)*(3*a + b))/(8*( a + b)^2) - (exp(-x)*(3*a - b))/(8*(a - b)^2) + (2*atan((a^3*b*exp(x)*(b^1 0 - a^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 ^6*b^2)^(1/2) - b^5*(a^6*b^2)^(1/2) + 2*a^2*b^3*(a^6*b^2)^(1/2) - 2*a^3*b^ 2*(a^6*b^2)^(1/2) + a*b^4*(a^6*b^2)^(1/2) - a^4*b*(a^6*b^2)^(1/2)))*(a^6*b ^2)^(1/2))/(b^10 - a^10 - 5*a^2*b^8 + 10*a^4*b^6 - 10*a^6*b^4 + 5*a^8*b^2) ^(1/2)