Integrand size = 13, antiderivative size = 167 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=-\frac {1}{16 (a+b) (1-\cosh (x))^2}-\frac {3 a+5 b}{16 (a+b)^2 (1-\cosh (x))}+\frac {1}{16 (a-b) (1+\cosh (x))^2}+\frac {3 a-5 b}{16 (a-b)^2 (1+\cosh (x))}+\frac {\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac {\left (3 a^2-9 a b+8 b^2\right ) \log (1+\cosh (x))}{16 (a-b)^3}+\frac {b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3} \] Output:
-1/16/(a+b)/(1-cosh(x))^2-1/16*(3*a+5*b)/(a+b)^2/(1-cosh(x))+1/16/(a-b)/(1 +cosh(x))^2+1/16*(3*a-5*b)/(a-b)^2/(1+cosh(x))+1/16*(3*a^2+9*a*b+8*b^2)*ln (1-cosh(x))/(a+b)^3-1/16*(3*a^2-9*a*b+8*b^2)*ln(1+cosh(x))/(a-b)^3+b^5*ln( a+b*cosh(x))/(a^2-b^2)^3
Time = 0.47 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\frac {1}{64} \left (\frac {2 (3 a+5 b) \text {csch}^2\left (\frac {x}{2}\right )}{(a+b)^2}-\frac {\text {csch}^4\left (\frac {x}{2}\right )}{a+b}-\frac {8 \left (3 a^2-9 a b+8 b^2\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )}{(a-b)^3}+\frac {64 b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac {8 \left (3 a^2+9 a b+8 b^2\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )}{(a+b)^3}+\frac {2 (3 a-5 b) \text {sech}^2\left (\frac {x}{2}\right )}{(a-b)^2}+\frac {\text {sech}^4\left (\frac {x}{2}\right )}{a-b}\right ) \] Input:
Integrate[Csch[x]^5/(a + b*Cosh[x]),x]
Output:
((2*(3*a + 5*b)*Csch[x/2]^2)/(a + b)^2 - Csch[x/2]^4/(a + b) - (8*(3*a^2 - 9*a*b + 8*b^2)*Log[Cosh[x/2]])/(a - b)^3 + (64*b^5*Log[a + b*Cosh[x]])/(a ^2 - b^2)^3 + (8*(3*a^2 + 9*a*b + 8*b^2)*Log[Sinh[x/2]])/(a + b)^3 + (2*(3 *a - 5*b)*Sech[x/2]^2)/(a - b)^2 + Sech[x/2]^4/(a - b))/64
Time = 0.53 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 3147, 477, 2009}
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.
| \(\displaystyle \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\cos \left (-\frac {\pi }{2}+i x\right )^5 \left (a-b \sin \left (-\frac {\pi }{2}+i x\right )\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\cos \left (i x-\frac {\pi }{2}\right )^5 \left (a-b \sin \left (i x-\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle -b^5 \int \frac {1}{(a+b \cosh (x)) \left (b^2-b^2 \cosh ^2(x)\right )^3}d(b \cosh (x))\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle -\frac {\int \left (-\frac {b^6}{\left (a^2-b^2\right )^3 (a+b \cosh (x))}+\frac {b^3}{8 (a+b) (b-b \cosh (x))^3}+\frac {b^3}{8 (a-b) (\cosh (x) b+b)^3}+\frac {(3 a+5 b) b^2}{16 (a+b)^2 (b-b \cosh (x))^2}+\frac {(3 a-5 b) b^2}{16 (a-b)^2 (\cosh (x) b+b)^2}+\frac {\left (3 a^2+9 b a+8 b^2\right ) b}{16 (a+b)^3 (b-b \cosh (x))}+\frac {\left (3 a^2-9 b a+8 b^2\right ) b}{16 (a-b)^3 (\cosh (x) b+b)}\right )d(b \cosh (x))}{b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {b \left (3 a^2+9 a b+8 b^2\right ) \log (b-b \cosh (x))}{16 (a+b)^3}+\frac {b \left (3 a^2-9 a b+8 b^2\right ) \log (b \cosh (x)+b)}{16 (a-b)^3}-\frac {b^6 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac {b^3}{16 (a+b) (b-b \cosh (x))^2}-\frac {b^3}{16 (a-b) (b \cosh (x)+b)^2}+\frac {b^2 (3 a+5 b)}{16 (a+b)^2 (b-b \cosh (x))}-\frac {b^2 (3 a-5 b)}{16 (a-b)^2 (b \cosh (x)+b)}}{b}\) |
Input:
Int[Csch[x]^5/(a + b*Cosh[x]),x]
Output:
-((b^3/(16*(a + b)*(b - b*Cosh[x])^2) + (b^2*(3*a + 5*b))/(16*(a + b)^2*(b - b*Cosh[x])) - b^3/(16*(a - b)*(b + b*Cosh[x])^2) - ((3*a - 5*b)*b^2)/(1 6*(a - b)^2*(b + b*Cosh[x])) - (b*(3*a^2 + 9*a*b + 8*b^2)*Log[b - b*Cosh[x ]])/(16*(a + b)^3) - (b^6*Log[a + b*Cosh[x]])/(a^2 - b^2)^3 + (b*(3*a^2 - 9*a*b + 8*b^2)*Log[b + b*Cosh[x]])/(16*(a - b)^3))/b)
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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 14.53 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {\left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-4 a +6 b \right )^{2}}{64 \left (a -b \right )^{3}}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}-\frac {-4 a -6 b}{32 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (6 a^{2}+18 a b +16 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 \left (a +b \right )^{3}}+\frac {b^{5} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-a -b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}\) | \(138\) |
| risch | \(-\frac {3 x \,a^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {9 x a b}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {x \,b^{2}}{a^{3}+3 a^{2} b +3 b^{2} a +b^{3}}+\frac {3 x \,a^{2}}{8 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {9 x a b}{8 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}+\frac {x \,b^{2}}{a^{3}-3 a^{2} b +3 b^{2} a -b^{3}}-\frac {2 x \,b^{5}}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {\left (3 a^{3} {\mathrm e}^{6 x}-7 a \,b^{2} {\mathrm e}^{6 x}+8 b^{3} {\mathrm e}^{5 x}-11 a^{3} {\mathrm e}^{4 x}+15 a \,b^{2} {\mathrm e}^{4 x}+16 a^{2} b \,{\mathrm e}^{3 x}-32 b^{3} {\mathrm e}^{3 x}-11 a^{3} {\mathrm e}^{2 x}+15 a \,b^{2} {\mathrm e}^{2 x}+8 b^{3} {\mathrm e}^{x}+3 a^{3}-7 b^{2} a \right ) {\mathrm e}^{x}}{4 \left (a^{2}-b^{2}\right )^{2} \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {9 \ln \left ({\mathrm e}^{x}-1\right ) a b}{8 \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{a^{3}+3 a^{2} b +3 b^{2} a +b^{3}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right ) a^{2}}{8 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}+\frac {9 \ln \left ({\mathrm e}^{x}+1\right ) a b}{8 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {\ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{a^{3}-3 a^{2} b +3 b^{2} a -b^{3}}+\frac {b^{5} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}\) | \(562\) |
Input:
int(csch(x)^5/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
Output:
1/64*(tanh(1/2*x)^2*a-b*tanh(1/2*x)^2-4*a+6*b)^2/(a-b)^3-1/64/(a+b)/tanh(1 /2*x)^4-1/32*(-4*a-6*b)/(a+b)^2/tanh(1/2*x)^2+1/16/(a+b)^3*(6*a^2+18*a*b+1 6*b^2)*ln(tanh(1/2*x))+1/(a-b)^3*b^5/(a+b)^3*ln(tanh(1/2*x)^2*a-b*tanh(1/2 *x)^2-a-b)
Leaf count of result is larger than twice the leaf count of optimal. 3450 vs. \(2 (151) = 302\).
Time = 0.19 (sec) , antiderivative size = 3450, normalized size of antiderivative = 20.66 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\int \frac {\operatorname {csch}^{5}{\left (x \right )}}{a + b \cosh {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**5/(a+b*cosh(x)),x)
Output:
Integral(csch(x)**5/(a + b*cosh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (151) = 302\).
Time = 0.05 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.08 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\frac {b^{5} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {8 \, b^{3} e^{\left (-2 \, x\right )} + 8 \, b^{3} e^{\left (-6 \, x\right )} + {\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-x\right )} - {\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-4 \, x\right )} - {\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-5 \, x\right )} + {\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \] Input:
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="maxima")
Output:
b^5*log(2*a*e^(-x) + b*e^(-2*x) + b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(3*a^2 - 9*a*b + 8*b^2)*log(e^(-x) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^ 3) + 1/8*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) - 1)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) + 1/4*(8*b^3*e^(-2*x) + 8*b^3*e^(-6*x) + (3*a^3 - 7*a*b^2)*e^(-x) - (11*a^3 - 15*a*b^2)*e^(-3*x) + 16*(a^2*b - 2*b^3)*e^(-4*x) - (11*a^3 - 15 *a*b^2)*e^(-5*x) + (3*a^3 - 7*a*b^2)*e^(-7*x))/(a^4 - 2*a^2*b^2 + b^4 - 4* (a^4 - 2*a^2*b^2 + b^4)*e^(-2*x) + 6*(a^4 - 2*a^2*b^2 + b^4)*e^(-4*x) - 4* (a^4 - 2*a^2*b^2 + b^4)*e^(-6*x) + (a^4 - 2*a^2*b^2 + b^4)*e^(-8*x))
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (151) = 302\).
Time = 0.12 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.02 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\frac {b^{6} \log \left ({\left | b {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac {{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {3 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 3 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 10 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 7 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 8 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 32 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 20 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )} + 56 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )} - 36 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )} + 16 \, a^{4} b - 64 \, a^{2} b^{3} + 96 \, b^{5}}{4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \] Input:
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="giac")
Output:
b^6*log(abs(b*(e^(-x) + e^x) + 2*a))/(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7) - 1/16*(3*a^2 - 9*a*b + 8*b^2)*log(e^(-x) + e^x + 2)/(a^3 - 3*a^2*b + 3*a *b^2 - b^3) + 1/16*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) + e^x - 2)/(a^3 + 3* a^2*b + 3*a*b^2 + b^3) + 1/4*(3*b^5*(e^(-x) + e^x)^4 + 3*a^5*(e^(-x) + e^x )^3 - 10*a^3*b^2*(e^(-x) + e^x)^3 + 7*a*b^4*(e^(-x) + e^x)^3 + 8*a^2*b^3*( e^(-x) + e^x)^2 - 32*b^5*(e^(-x) + e^x)^2 - 20*a^5*(e^(-x) + e^x) + 56*a^3 *b^2*(e^(-x) + e^x) - 36*a*b^4*(e^(-x) + e^x) + 16*a^4*b - 64*a^2*b^3 + 96 *b^5)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((e^(-x) + e^x)^2 - 4)^2)
Time = 2.89 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.35 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx=\frac {\frac {4\,b}{a^2-b^2}-\frac {4\,a\,{\mathrm {e}}^x}{a^2-b^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {\frac {2\,\left (b^5-a^2\,b^3\right )}{{\left (a^2-b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (3\,a^5-10\,a^3\,b^2+7\,a\,b^4\right )}{4\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {8\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}+\frac {6\,{\mathrm {e}}^x\,\left (a\,b^2-a^3\right )}{{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}+\frac {\frac {2\,\left (2\,a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^3+3\,a\,b^2\right )}{2\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {b^5\,\ln \left (256\,b^{11}\,{\mathrm {e}}^{2\,x}-9\,a^{10}\,b+256\,b^{11}-225\,a^2\,b^9+300\,a^4\,b^7-190\,a^6\,b^5+60\,a^8\,b^3-18\,a^{11}\,{\mathrm {e}}^x-225\,a^2\,b^9\,{\mathrm {e}}^{2\,x}+300\,a^4\,b^7\,{\mathrm {e}}^{2\,x}-190\,a^6\,b^5\,{\mathrm {e}}^{2\,x}+60\,a^8\,b^3\,{\mathrm {e}}^{2\,x}+512\,a\,b^{10}\,{\mathrm {e}}^x-9\,a^{10}\,b\,{\mathrm {e}}^{2\,x}-450\,a^3\,b^8\,{\mathrm {e}}^x+600\,a^5\,b^6\,{\mathrm {e}}^x-380\,a^7\,b^4\,{\mathrm {e}}^x+120\,a^9\,b^2\,{\mathrm {e}}^x\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (3\,a^2+9\,a\,b+8\,b^2\right )}{8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (3\,a^2-9\,a\,b+8\,b^2\right )}{8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3} \] Input:
int(1/(sinh(x)^5*(a + b*cosh(x))),x)
Output:
((4*b)/(a^2 - b^2) - (4*a*exp(x))/(a^2 - b^2))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) - ((2*(b^5 - a^2*b^3))/(a^2 - b^2)^3 - (exp(x)* (7*a*b^4 + 3*a^5 - 10*a^3*b^2))/(4*(a^2 - b^2)^3))/(exp(2*x) - 1) + ((8*(a ^2*b - b^3))/(a^2 - b^2)^2 + (6*exp(x)*(a*b^2 - a^3))/(a^2 - b^2)^2)/(3*ex p(2*x) - 3*exp(4*x) + exp(6*x) - 1) + ((2*(2*a^2*b - b^3))/(a^2 - b^2)^2 - (exp(x)*(3*a*b^2 + a^3))/(2*(a^2 - b^2)^2))/(exp(4*x) - 2*exp(2*x) + 1) + (b^5*log(256*b^11*exp(2*x) - 9*a^10*b + 256*b^11 - 225*a^2*b^9 + 300*a^4* b^7 - 190*a^6*b^5 + 60*a^8*b^3 - 18*a^11*exp(x) - 225*a^2*b^9*exp(2*x) + 3 00*a^4*b^7*exp(2*x) - 190*a^6*b^5*exp(2*x) + 60*a^8*b^3*exp(2*x) + 512*a*b ^10*exp(x) - 9*a^10*b*exp(2*x) - 450*a^3*b^8*exp(x) + 600*a^5*b^6*exp(x) - 380*a^7*b^4*exp(x) + 120*a^9*b^2*exp(x)))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^4* b^2) + (log(exp(x) - 1)*(9*a*b + 3*a^2 + 8*b^2))/(24*a*b^2 + 24*a^2*b + 8* a^3 + 8*b^3) - (log(exp(x) + 1)*(3*a^2 - 9*a*b + 8*b^2))/(24*a*b^2 - 24*a^ 2*b + 8*a^3 - 8*b^3)
Time = 0.24 (sec) , antiderivative size = 1180, normalized size of antiderivative = 7.07 \[ \int \frac {\text {csch}^5(x)}{a+b \cosh (x)} \, dx =\text {Too large to display} \] Input:
int(csch(x)^5/(a+b*cosh(x)),x)
Output:
(3*e**(8*x)*log(e**x - 1)*a**5 - 10*e**(8*x)*log(e**x - 1)*a**3*b**2 + 15* e**(8*x)*log(e**x - 1)*a*b**4 - 8*e**(8*x)*log(e**x - 1)*b**5 - 3*e**(8*x) *log(e**x + 1)*a**5 + 10*e**(8*x)*log(e**x + 1)*a**3*b**2 - 15*e**(8*x)*lo g(e**x + 1)*a*b**4 - 8*e**(8*x)*log(e**x + 1)*b**5 + 8*e**(8*x)*log(e**(2* x)*b + 2*e**x*a + b)*b**5 + 4*e**(8*x)*a**2*b**3 - 4*e**(8*x)*b**5 + 6*e** (7*x)*a**5 - 20*e**(7*x)*a**3*b**2 + 14*e**(7*x)*a*b**4 - 12*e**(6*x)*log( e**x - 1)*a**5 + 40*e**(6*x)*log(e**x - 1)*a**3*b**2 - 60*e**(6*x)*log(e** x - 1)*a*b**4 + 32*e**(6*x)*log(e**x - 1)*b**5 + 12*e**(6*x)*log(e**x + 1) *a**5 - 40*e**(6*x)*log(e**x + 1)*a**3*b**2 + 60*e**(6*x)*log(e**x + 1)*a* b**4 + 32*e**(6*x)*log(e**x + 1)*b**5 - 32*e**(6*x)*log(e**(2*x)*b + 2*e** x*a + b)*b**5 - 22*e**(5*x)*a**5 + 52*e**(5*x)*a**3*b**2 - 30*e**(5*x)*a*b **4 + 18*e**(4*x)*log(e**x - 1)*a**5 - 60*e**(4*x)*log(e**x - 1)*a**3*b**2 + 90*e**(4*x)*log(e**x - 1)*a*b**4 - 48*e**(4*x)*log(e**x - 1)*b**5 - 18* e**(4*x)*log(e**x + 1)*a**5 + 60*e**(4*x)*log(e**x + 1)*a**3*b**2 - 90*e** (4*x)*log(e**x + 1)*a*b**4 - 48*e**(4*x)*log(e**x + 1)*b**5 + 48*e**(4*x)* log(e**(2*x)*b + 2*e**x*a + b)*b**5 + 32*e**(4*x)*a**4*b - 72*e**(4*x)*a** 2*b**3 + 40*e**(4*x)*b**5 - 22*e**(3*x)*a**5 + 52*e**(3*x)*a**3*b**2 - 30* e**(3*x)*a*b**4 - 12*e**(2*x)*log(e**x - 1)*a**5 + 40*e**(2*x)*log(e**x - 1)*a**3*b**2 - 60*e**(2*x)*log(e**x - 1)*a*b**4 + 32*e**(2*x)*log(e**x - 1 )*b**5 + 12*e**(2*x)*log(e**x + 1)*a**5 - 40*e**(2*x)*log(e**x + 1)*a**...