Integrand size = 13, antiderivative size = 159 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\frac {2 b^6 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}+\frac {\left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right ) \text {csch}(x)}{15 \left (a^2-b^2\right )^3}+\frac {\left (5 b^3+a \left (4 a^2-9 b^2\right ) \cosh (x)\right ) \text {csch}^3(x)}{15 \left (a^2-b^2\right )^2}+\frac {(b-a \cosh (x)) \text {csch}^5(x)}{5 \left (a^2-b^2\right )} \] Output:
2*b^6*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b)^(7/2)/(a+b)^(7/2) +1/15*(15*b^5-a*(8*a^4-26*a^2*b^2+33*b^4)*cosh(x))*csch(x)/(a^2-b^2)^3+1/1 5*(5*b^3+a*(4*a^2-9*b^2)*cosh(x))*csch(x)^3/(a^2-b^2)^2+(b-a*cosh(x))*csch (x)^5/(5*a^2-5*b^2)
Time = 1.21 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.26 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\frac {1}{480} \left (\frac {960 b^6 \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}-\frac {2 \left (64 a^2+183 a b+149 b^2\right ) \coth \left (\frac {x}{2}\right )}{(a+b)^3}-\frac {8 (19 a-29 b) \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )}{(a-b)^2}-\frac {96 \text {csch}^5(x) \sinh ^6\left (\frac {x}{2}\right )}{a-b}+\frac {(19 a+29 b) \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)^2}-\frac {3 \text {csch}^6\left (\frac {x}{2}\right ) \sinh (x)}{2 (a+b)}-\frac {2 \left (64 a^2-183 a b+149 b^2\right ) \tanh \left (\frac {x}{2}\right )}{(a-b)^3}\right ) \] Input:
Integrate[Csch[x]^6/(a + b*Cosh[x]),x]
Output:
((960*b^6*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) - (2*(64*a^2 + 183*a*b + 149*b^2)*Coth[x/2])/(a + b)^3 - (8*(19*a - 29*b) *Csch[x]^3*Sinh[x/2]^4)/(a - b)^2 - (96*Csch[x]^5*Sinh[x/2]^6)/(a - b) + ( (19*a + 29*b)*Csch[x/2]^4*Sinh[x])/(2*(a + b)^2) - (3*Csch[x/2]^6*Sinh[x]) /(2*(a + b)) - (2*(64*a^2 - 183*a*b + 149*b^2)*Tanh[x/2])/(a - b)^3)/480
Time = 0.97 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.26, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 25, 3175, 25, 3042, 3345, 3042, 25, 3345, 27, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\cos \left (-\frac {\pi }{2}+i x\right )^6 \left (a-b \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right )^6 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3175 |
| \(\displaystyle \frac {\int -\frac {\left (4 a^2+4 b \cosh (x) a-5 b^2\right ) \text {csch}^4(x)}{a+b \cosh (x)}dx}{5 \left (a^2-b^2\right )}+\frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\int \frac {\left (4 a^2+4 b \cosh (x) a-5 b^2\right ) \text {csch}^4(x)}{a+b \cosh (x)}dx}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\int \frac {4 a^2-4 b \sin \left (i x-\frac {\pi }{2}\right ) a-5 b^2}{\cos \left (i x-\frac {\pi }{2}\right )^4 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {-\frac {\int \frac {\left (8 a^4-18 b^2 a^2+2 b \left (4 a^2-9 b^2\right ) \cosh (x) a+15 b^4\right ) \text {csch}^2(x)}{a+b \cosh (x)}dx}{3 \left (a^2-b^2\right )}-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}-\frac {\int -\frac {8 a^4-18 b^2 a^2-2 b \left (4 a^2-9 b^2\right ) \sin \left (i x-\frac {\pi }{2}\right ) a+15 b^4}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}+\frac {\int \frac {8 a^4-18 b^2 a^2-2 b \left (4 a^2-9 b^2\right ) \sin \left (i x-\frac {\pi }{2}\right ) a+15 b^4}{\cos \left (i x-\frac {\pi }{2}\right )^2 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\frac {-\frac {\int \frac {15 b^6}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\frac {-\frac {15 b^6 \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}+\frac {-\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{a^2-b^2}-\frac {15 b^6 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\frac {-\frac {30 b^6 \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\text {csch}^5(x) (b-a \cosh (x))}{5 \left (a^2-b^2\right )}-\frac {\frac {-\frac {30 b^6 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\text {csch}(x) \left (15 b^5-a \left (8 a^4-26 a^2 b^2+33 b^4\right ) \cosh (x)\right )}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\text {csch}^3(x) \left (a \left (4 a^2-9 b^2\right ) \cosh (x)+5 b^3\right )}{3 \left (a^2-b^2\right )}}{5 \left (a^2-b^2\right )}\) |
Input:
Int[Csch[x]^6/(a + b*Cosh[x]),x]
Output:
((b - a*Cosh[x])*Csch[x]^5)/(5*(a^2 - b^2)) - (-1/3*((5*b^3 + a*(4*a^2 - 9 *b^2)*Cosh[x])*Csch[x]^3)/(a^2 - b^2) + ((-30*b^6*ArcTanh[(Sqrt[a - b]*Tan h[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) - ((15*b^5 - a *(8*a^4 - 26*a^2*b^2 + 33*b^4)*Cosh[x])*Csch[x])/(a^2 - b^2))/(3*(a^2 - b^ 2)))/(5*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^ (m + 1)*((b - a*Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2* (a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m* (a^2*(p + 2) - b^2*(m + p + 2) + a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; F reeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegersQ [2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Time = 29.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.34
| method | result | size |
| default | \(-\frac {\frac {\tanh \left (\frac {x}{2}\right )^{5} a^{2}}{5}-\frac {2 \tanh \left (\frac {x}{2}\right )^{5} a b}{5}+\frac {b^{2} \tanh \left (\frac {x}{2}\right )^{5}}{5}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3} a^{2}}{3}+4 \tanh \left (\frac {x}{2}\right )^{3} a b -\frac {7 b^{2} \tanh \left (\frac {x}{2}\right )^{3}}{3}+10 a^{2} \tanh \left (\frac {x}{2}\right )-28 a b \tanh \left (\frac {x}{2}\right )+22 b^{2} \tanh \left (\frac {x}{2}\right )}{32 \left (a -b \right )^{3}}-\frac {1}{160 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{5}}-\frac {-5 a -7 b}{96 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {10 a^{2}+28 a b +22 b^{2}}{32 \left (a +b \right )^{3} \tanh \left (\frac {x}{2}\right )}+\frac {2 b^{6} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(213\) |
| risch | \(-\frac {2 \left (-15 b^{5} {\mathrm e}^{9 x}+15 a \,b^{4} {\mathrm e}^{8 x}-20 a^{2} b^{3} {\mathrm e}^{7 x}+80 b^{5} {\mathrm e}^{7 x}+30 a^{3} b^{2} {\mathrm e}^{6 x}-90 a \,b^{4} {\mathrm e}^{6 x}-48 a^{4} b \,{\mathrm e}^{5 x}+136 a^{2} b^{3} {\mathrm e}^{5 x}-178 b^{5} {\mathrm e}^{5 x}+80 a^{5} {\mathrm e}^{4 x}-230 a^{3} b^{2} {\mathrm e}^{4 x}+240 a \,b^{4} {\mathrm e}^{4 x}-20 a^{2} b^{3} {\mathrm e}^{3 x}+80 b^{5} {\mathrm e}^{3 x}-40 a^{5} {\mathrm e}^{2 x}+130 a^{3} b^{2} {\mathrm e}^{2 x}-150 a \,b^{4} {\mathrm e}^{2 x}-15 b^{5} {\mathrm e}^{x}+8 a^{5}-26 a^{3} b^{2}+33 a \,b^{4}\right )}{15 \left (a^{2}-b^{2}\right )^{3} \left ({\mathrm e}^{2 x}-1\right )^{5}}+\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(363\) |
Input:
int(csch(x)^6/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
Output:
-1/32/(a-b)^3*(1/5*tanh(1/2*x)^5*a^2-2/5*tanh(1/2*x)^5*a*b+1/5*b^2*tanh(1/ 2*x)^5-5/3*tanh(1/2*x)^3*a^2+4*tanh(1/2*x)^3*a*b-7/3*b^2*tanh(1/2*x)^3+10* a^2*tanh(1/2*x)-28*a*b*tanh(1/2*x)+22*b^2*tanh(1/2*x))-1/160/(a+b)/tanh(1/ 2*x)^5-1/96*(-5*a-7*b)/(a+b)^2/tanh(1/2*x)^3-1/32/(a+b)^3*(10*a^2+28*a*b+2 2*b^2)/tanh(1/2*x)+2/(a-b)^3/(a+b)^3*b^6/((a+b)*(a-b))^(1/2)*arctanh((a-b) *tanh(1/2*x)/((a+b)*(a-b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 3156 vs. \(2 (144) = 288\).
Time = 0.16 (sec) , antiderivative size = 6381, normalized size of antiderivative = 40.13 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^6/(a+b*cosh(x)),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\text {Timed out} \] Input:
integrate(csch(x)**6/(a+b*cosh(x)),x)
Output:
Timed out
Exception generated. \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(csch(x)^6/(a+b*cosh(x)),x, algorithm="maxima")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (144) = 288\).
Time = 0.12 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.91 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\frac {2 \, b^{6} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, {\left (15 \, b^{5} e^{\left (9 \, x\right )} - 15 \, a b^{4} e^{\left (8 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (7 \, x\right )} - 80 \, b^{5} e^{\left (7 \, x\right )} - 30 \, a^{3} b^{2} e^{\left (6 \, x\right )} + 90 \, a b^{4} e^{\left (6 \, x\right )} + 48 \, a^{4} b e^{\left (5 \, x\right )} - 136 \, a^{2} b^{3} e^{\left (5 \, x\right )} + 178 \, b^{5} e^{\left (5 \, x\right )} - 80 \, a^{5} e^{\left (4 \, x\right )} + 230 \, a^{3} b^{2} e^{\left (4 \, x\right )} - 240 \, a b^{4} e^{\left (4 \, x\right )} + 20 \, a^{2} b^{3} e^{\left (3 \, x\right )} - 80 \, b^{5} e^{\left (3 \, x\right )} + 40 \, a^{5} e^{\left (2 \, x\right )} - 130 \, a^{3} b^{2} e^{\left (2 \, x\right )} + 150 \, a b^{4} e^{\left (2 \, x\right )} + 15 \, b^{5} e^{x} - 8 \, a^{5} + 26 \, a^{3} b^{2} - 33 \, a b^{4}\right )}}{15 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \] Input:
integrate(csch(x)^6/(a+b*cosh(x)),x, algorithm="giac")
Output:
2*b^6*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(-a^2 + b^2)) + 2/15*(15*b^5*e^(9*x) - 15*a*b^4*e^(8*x) + 20*a^2 *b^3*e^(7*x) - 80*b^5*e^(7*x) - 30*a^3*b^2*e^(6*x) + 90*a*b^4*e^(6*x) + 48 *a^4*b*e^(5*x) - 136*a^2*b^3*e^(5*x) + 178*b^5*e^(5*x) - 80*a^5*e^(4*x) + 230*a^3*b^2*e^(4*x) - 240*a*b^4*e^(4*x) + 20*a^2*b^3*e^(3*x) - 80*b^5*e^(3 *x) + 40*a^5*e^(2*x) - 130*a^3*b^2*e^(2*x) + 150*a*b^4*e^(2*x) + 15*b^5*e^ x - 8*a^5 + 26*a^3*b^2 - 33*a*b^4)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*(e ^(2*x) - 1)^5)
Time = 3.64 (sec) , antiderivative size = 1031, normalized size of antiderivative = 6.48 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx =\text {Too large to display} \] Input:
int(1/(sinh(x)^6*(a + b*cosh(x))),x)
Output:
((16*(a*b^2 - a^3))/(a^2 - b^2)^2 + (64*exp(x)*(a^2*b - b^3))/(5*(a^2 - b^ 2)^2))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - ((2*a*b^4)/ (a^2 - b^2)^3 - (2*b^5*exp(x))/(a^2 - b^2)^3)/(exp(2*x) - 1) - ((32*a)/(5* (a^2 - b^2)) - (32*b*exp(x))/(5*(a^2 - b^2)))/(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1) + ((8*(3*a*b^2 - 4*a^3))/(3*(a^2 - b^2)^2) + (8*exp(x)*(12*a^2*b - 7*b^3))/(15*(a^2 - b^2)^2))/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) + ((4*(a*b^4 - a^3*b^2))/(a^2 - b^2)^3 - (8* exp(x)*(b^5 - a^2*b^3))/(3*(a^2 - b^2)^3))/(exp(4*x) - 2*exp(2*x) + 1) - ( 2*atan((exp(x)*((2*b^4)/((a^2 - b^2)^3*(b^12)^(1/2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (2*a*(a^7*(b^12)^(1/2) + 3*a^3*b^4*(b^12)^(1/2) - 3*a^5*b ^2*(b^12)^(1/2) - a*b^6*(b^12)^(1/2)))/(b^8*(-(a^2 - b^2)^7)^(1/2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)*(b^14 - a^14 - 7*a^2*b^12 + 21*a^4*b^10 - 35* a^6*b^8 + 35*a^8*b^6 - 21*a^10*b^4 + 7*a^12*b^2)^(1/2))) - (2*a*(b^7*(b^12 )^(1/2) - 3*a^2*b^5*(b^12)^(1/2) + 3*a^4*b^3*(b^12)^(1/2) - a^6*b*(b^12)^( 1/2)))/(b^8*(-(a^2 - b^2)^7)^(1/2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)*(b^ 14 - a^14 - 7*a^2*b^12 + 21*a^4*b^10 - 35*a^6*b^8 + 35*a^8*b^6 - 21*a^10*b ^4 + 7*a^12*b^2)^(1/2)))*((b^7*(b^14 - a^14 - 7*a^2*b^12 + 21*a^4*b^10 - 3 5*a^6*b^8 + 35*a^8*b^6 - 21*a^10*b^4 + 7*a^12*b^2)^(1/2))/2 - (a^6*b*(b^14 - a^14 - 7*a^2*b^12 + 21*a^4*b^10 - 35*a^6*b^8 + 35*a^8*b^6 - 21*a^10*b^4 + 7*a^12*b^2)^(1/2))/2 - (3*a^2*b^5*(b^14 - a^14 - 7*a^2*b^12 + 21*a^4...
Time = 0.25 (sec) , antiderivative size = 919, normalized size of antiderivative = 5.78 \[ \int \frac {\text {csch}^6(x)}{a+b \cosh (x)} \, dx=\frac {30 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}-6 e^{10 x} a^{3} b^{4}+6 e^{10 x} a \,b^{6}+30 e^{9 x} a^{2} b^{5}+40 e^{7 x} a^{4} b^{3}-200 e^{7 x} a^{2} b^{5}-60 e^{6 x} a^{5} b^{2}+180 e^{6 x} a^{3} b^{4}-120 e^{6 x} a \,b^{6}+96 e^{5 x} a^{6} b -368 e^{5 x} a^{4} b^{3}+628 e^{5 x} a^{2} b^{5}+620 e^{4 x} a^{5} b^{2}-880 e^{4 x} a^{3} b^{4}+420 e^{4 x} a \,b^{6}+40 e^{3 x} a^{4} b^{3}-200 e^{3 x} a^{2} b^{5}-340 e^{2 x} a^{5} b^{2}+530 e^{2 x} a^{3} b^{4}-270 e^{2 x} a \,b^{6}+30 e^{x} a^{2} b^{5}-300 e^{6 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}+300 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}-150 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}-30 e^{10 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}+150 e^{8 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) b^{6}-16 a^{7}-30 e^{9 x} b^{7}+160 e^{7 x} b^{7}-356 e^{5 x} b^{7}-160 e^{4 x} a^{7}+160 e^{3 x} b^{7}+80 e^{2 x} a^{7}-30 e^{x} b^{7}+68 a^{5} b^{2}-112 a^{3} b^{4}+60 a \,b^{6}}{-600 e^{6 x} a^{2} b^{6}-60 e^{10 x} a^{6} b^{2}+90 e^{10 x} a^{4} b^{4}-60 e^{10 x} a^{2} b^{6}+300 e^{8 x} a^{6} b^{2}-450 e^{8 x} a^{4} b^{4}+300 e^{8 x} a^{2} b^{6}-600 e^{6 x} a^{6} b^{2}+900 e^{6 x} a^{4} b^{4}-15 a^{8}-150 e^{4 x} b^{8}+75 e^{2 x} a^{8}+75 e^{2 x} b^{8}-15 b^{8}-900 e^{4 x} a^{4} b^{4}+600 e^{4 x} a^{2} b^{6}-300 e^{2 x} a^{6} b^{2}+450 e^{2 x} a^{4} b^{4}-300 e^{2 x} a^{2} b^{6}+15 e^{10 x} a^{8}+15 e^{10 x} b^{8}-75 e^{8 x} a^{8}-75 e^{8 x} b^{8}+150 e^{6 x} a^{8}+150 e^{6 x} b^{8}-150 e^{4 x} a^{8}+600 e^{4 x} a^{6} b^{2}+60 a^{6} b^{2}-90 a^{4} b^{4}+60 a^{2} b^{6}} \] Input:
int(csch(x)^6/(a+b*cosh(x)),x)
Output:
(2*( - 15*e**(10*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*b**6 + 75*e**(8*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a **2 + b**2))*b**6 - 150*e**(6*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sq rt( - a**2 + b**2))*b**6 + 150*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*b**6 - 75*e**(2*x)*sqrt( - a**2 + b**2)*atan((e **x*b + a)/sqrt( - a**2 + b**2))*b**6 + 15*sqrt( - a**2 + b**2)*atan((e**x *b + a)/sqrt( - a**2 + b**2))*b**6 - 3*e**(10*x)*a**3*b**4 + 3*e**(10*x)*a *b**6 + 15*e**(9*x)*a**2*b**5 - 15*e**(9*x)*b**7 + 20*e**(7*x)*a**4*b**3 - 100*e**(7*x)*a**2*b**5 + 80*e**(7*x)*b**7 - 30*e**(6*x)*a**5*b**2 + 90*e* *(6*x)*a**3*b**4 - 60*e**(6*x)*a*b**6 + 48*e**(5*x)*a**6*b - 184*e**(5*x)* a**4*b**3 + 314*e**(5*x)*a**2*b**5 - 178*e**(5*x)*b**7 - 80*e**(4*x)*a**7 + 310*e**(4*x)*a**5*b**2 - 440*e**(4*x)*a**3*b**4 + 210*e**(4*x)*a*b**6 + 20*e**(3*x)*a**4*b**3 - 100*e**(3*x)*a**2*b**5 + 80*e**(3*x)*b**7 + 40*e** (2*x)*a**7 - 170*e**(2*x)*a**5*b**2 + 265*e**(2*x)*a**3*b**4 - 135*e**(2*x )*a*b**6 + 15*e**x*a**2*b**5 - 15*e**x*b**7 - 8*a**7 + 34*a**5*b**2 - 56*a **3*b**4 + 30*a*b**6))/(15*(e**(10*x)*a**8 - 4*e**(10*x)*a**6*b**2 + 6*e** (10*x)*a**4*b**4 - 4*e**(10*x)*a**2*b**6 + e**(10*x)*b**8 - 5*e**(8*x)*a** 8 + 20*e**(8*x)*a**6*b**2 - 30*e**(8*x)*a**4*b**4 + 20*e**(8*x)*a**2*b**6 - 5*e**(8*x)*b**8 + 10*e**(6*x)*a**8 - 40*e**(6*x)*a**6*b**2 + 60*e**(6*x) *a**4*b**4 - 40*e**(6*x)*a**2*b**6 + 10*e**(6*x)*b**8 - 10*e**(4*x)*a**...