Integrand size = 13, antiderivative size = 140 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=\frac {a \left (a^4-3 a^2 b^2+3 b^4\right ) \coth (x)}{b^6}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \coth ^2(x)}{2 b^5}+\frac {a \left (a^2-3 b^2\right ) \coth ^3(x)}{3 b^4}-\frac {\left (a^2-3 b^2\right ) \coth ^4(x)}{4 b^3}+\frac {a \coth ^5(x)}{5 b^2}-\frac {\coth ^6(x)}{6 b}-\frac {\left (a^2-b^2\right )^3 \log (a+b \coth (x))}{b^7} \] Output:
a*(a^4-3*a^2*b^2+3*b^4)*coth(x)/b^6-1/2*(a^4-3*a^2*b^2+3*b^4)*coth(x)^2/b^ 5+1/3*a*(a^2-3*b^2)*coth(x)^3/b^4-1/4*(a^2-3*b^2)*coth(x)^4/b^3+1/5*a*coth (x)^5/b^2-1/6*coth(x)^6/b-(a^2-b^2)^3*ln(a+b*coth(x))/b^7
Time = 3.41 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=\frac {4 a b \coth (x) \left (15 a^4-40 a^2 b^2+33 b^4+b^2 \left (5 a^2-9 b^2\right ) \text {csch}^2(x)+3 b^4 \text {csch}^4(x)\right )+5 \left (-6 b^2 \left (a^2-b^2\right )^2 \text {csch}^2(x)+3 b^4 \left (-a^2+b^2\right ) \text {csch}^4(x)-2 b^6 \text {csch}^6(x)+12 \left (a^2-b^2\right )^3 (\log (\sinh (x))-\log (b \cosh (x)+a \sinh (x)))\right )}{60 b^7} \] Input:
Integrate[Csch[x]^8/(a + b*Coth[x]),x]
Output:
(4*a*b*Coth[x]*(15*a^4 - 40*a^2*b^2 + 33*b^4 + b^2*(5*a^2 - 9*b^2)*Csch[x] ^2 + 3*b^4*Csch[x]^4) + 5*(-6*b^2*(a^2 - b^2)^2*Csch[x]^2 + 3*b^4*(-a^2 + b^2)*Csch[x]^4 - 2*b^6*Csch[x]^6 + 12*(a^2 - b^2)^3*(Log[Sinh[x]] - Log[b* Cosh[x] + a*Sinh[x]])))/(60*b^7)
Time = 0.64 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3987, 27, 476, 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.
| \(\displaystyle \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec \left (-\frac {\pi }{2}+i x\right )^8}{a-i b \tan \left (-\frac {\pi }{2}+i x\right )}dx\) |
| \(\Big \downarrow \) 3987 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \coth ^2(x)\right )^3}{b^6 (a+b \coth (x))}d(b \coth (x))}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \coth ^2(x)\right )^3}{a+b \coth (x)}d(b \coth (x))}{b^7}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (\left (\frac {3 \left (b^2-a^2\right ) b^2}{a^4}+1\right ) a^5+b^4 \coth ^4(x) a+b^2 \left (a^2-3 b^2\right ) \coth ^2(x) a-b^5 \coth ^5(x)-b^3 \left (a^2-3 b^2\right ) \coth ^3(x)-b \left (a^4-3 b^2 a^2+3 b^4\right ) \coth (x)-\frac {\left (a^2-b^2\right )^3}{a+b \coth (x)}\right )d(b \coth (x))}{b^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\left (a^2-b^2\right )^3 \log (a+b \coth (x))-\frac {1}{4} b^4 \left (a^2-3 b^2\right ) \coth ^4(x)+\frac {1}{3} a b^3 \left (a^2-3 b^2\right ) \coth ^3(x)-\frac {1}{2} b^2 \left (a^4-3 a^2 b^2+3 b^4\right ) \coth ^2(x)+a b \left (a^4-3 a^2 b^2+3 b^4\right ) \coth (x)+\frac {1}{5} a b^5 \coth ^5(x)-\frac {1}{6} b^6 \coth ^6(x)}{b^7}\) |
Input:
Int[Csch[x]^8/(a + b*Coth[x]),x]
Output:
(a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Coth[x] - (b^2*(a^4 - 3*a^2*b^2 + 3*b^4)*Co th[x]^2)/2 + (a*b^3*(a^2 - 3*b^2)*Coth[x]^3)/3 - (b^4*(a^2 - 3*b^2)*Coth[x ]^4)/4 + (a*b^5*Coth[x]^5)/5 - (b^6*Coth[x]^6)/6 - (a^2 - b^2)^3*Log[a + b *Coth[x]])/b^7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(b*f) Subst[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b^2, 0] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(365\) vs. \(2(130)=260\).
Time = 25.21 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.61
| method | result | size |
| default | \(\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{6} b^{5}}{6}+\frac {2 a \tanh \left (\frac {x}{2}\right )^{5} b^{4}}{5}-a^{2} b^{3} \tanh \left (\frac {x}{2}\right )^{4}+2 \tanh \left (\frac {x}{2}\right )^{4} b^{5}+\frac {8 a^{3} b^{2} \tanh \left (\frac {x}{2}\right )^{3}}{3}-6 a \,b^{4} \tanh \left (\frac {x}{2}\right )^{3}-8 a^{4} b \tanh \left (\frac {x}{2}\right )^{2}+20 a^{2} b^{3} \tanh \left (\frac {x}{2}\right )^{2}-\frac {29 b^{5} \tanh \left (\frac {x}{2}\right )^{2}}{2}+32 \tanh \left (\frac {x}{2}\right ) a^{5}-88 \tanh \left (\frac {x}{2}\right ) a^{3} b^{2}+76 b^{4} a \tanh \left (\frac {x}{2}\right )}{64 b^{6}}+\frac {\left (-64 a^{6}+192 a^{4} b^{2}-192 a^{2} b^{4}+64 b^{6}\right ) \ln \left (b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{64 b^{7}}-\frac {1}{384 b \tanh \left (\frac {x}{2}\right )^{6}}-\frac {4 a^{2}-8 b^{2}}{256 b^{3} \tanh \left (\frac {x}{2}\right )^{4}}-\frac {16 a^{4}-40 a^{2} b^{2}+29 b^{4}}{128 b^{5} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (64 a^{6}-192 a^{4} b^{2}+192 a^{2} b^{4}-64 b^{6}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{64 b^{7}}+\frac {a}{160 b^{2} \tanh \left (\frac {x}{2}\right )^{5}}+\frac {a \left (4 a^{2}-9 b^{2}\right )}{96 b^{4} \tanh \left (\frac {x}{2}\right )^{3}}+\frac {a \left (8 a^{4}-22 a^{2} b^{2}+19 b^{4}\right )}{16 b^{6} \tanh \left (\frac {x}{2}\right )}\) | \(366\) |
| risch | \(\frac {32 a^{2} b^{3} {\mathrm e}^{6 x}+\frac {16 a^{3} b^{2}}{3}-\frac {160 a^{3} b^{2} {\mathrm e}^{6 x}}{3}+24 a^{3} b^{2} {\mathrm e}^{8 x}-20 a^{2} b^{3} {\mathrm e}^{8 x}-14 a \,b^{4} {\mathrm e}^{8 x}-12 a^{4} b \,{\mathrm e}^{6 x}-4 a^{3} b^{2} {\mathrm e}^{10 x}+4 a^{2} b^{3} {\mathrm e}^{10 x}-20 a^{2} b^{3} {\mathrm e}^{4 x}-52 a \,b^{4} {\mathrm e}^{4 x}-2 a^{5}-\frac {22 b^{4} a}{5}+2 a^{5} {\mathrm e}^{10 x}-2 b^{5} {\mathrm e}^{10 x}-2 a^{4} b \,{\mathrm e}^{2 x}-28 a^{3} b^{2} {\mathrm e}^{2 x}+\frac {122 a \,b^{4} {\mathrm e}^{2 x}}{5}+2 a \,b^{4} {\mathrm e}^{10 x}+8 a^{4} b \,{\mathrm e}^{8 x}-10 a^{5} {\mathrm e}^{8 x}+12 b^{5} {\mathrm e}^{8 x}+20 a^{5} {\mathrm e}^{6 x}-\frac {92 b^{5} {\mathrm e}^{6 x}}{3}-20 a^{5} {\mathrm e}^{4 x}+12 b^{5} {\mathrm e}^{4 x}+10 a^{5} {\mathrm e}^{2 x}-2 b^{5} {\mathrm e}^{2 x}+44 a \,b^{4} {\mathrm e}^{6 x}+8 a^{4} b \,{\mathrm e}^{4 x}+56 a^{3} b^{2} {\mathrm e}^{4 x}+4 a^{2} b^{3} {\mathrm e}^{2 x}-2 a^{4} b \,{\mathrm e}^{10 x}}{b^{6} \left ({\mathrm e}^{2 x}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{2 x}-1\right ) a^{6}}{b^{7}}-\frac {3 \ln \left ({\mathrm e}^{2 x}-1\right ) a^{4}}{b^{5}}+\frac {3 \ln \left ({\mathrm e}^{2 x}-1\right ) a^{2}}{b^{3}}-\frac {\ln \left ({\mathrm e}^{2 x}-1\right )}{b}-\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{6}}{b^{7}}+\frac {3 \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{4}}{b^{5}}-\frac {3 \ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}-\frac {a -b}{a +b}\right )}{b}\) | \(501\) |
Input:
int(csch(x)^8/(a+b*coth(x)),x,method=_RETURNVERBOSE)
Output:
1/64/b^6*(-1/6*tanh(1/2*x)^6*b^5+2/5*a*tanh(1/2*x)^5*b^4-a^2*b^3*tanh(1/2* x)^4+2*tanh(1/2*x)^4*b^5+8/3*a^3*b^2*tanh(1/2*x)^3-6*a*b^4*tanh(1/2*x)^3-8 *a^4*b*tanh(1/2*x)^2+20*a^2*b^3*tanh(1/2*x)^2-29/2*b^5*tanh(1/2*x)^2+32*ta nh(1/2*x)*a^5-88*tanh(1/2*x)*a^3*b^2+76*b^4*a*tanh(1/2*x))+1/64/b^7*(-64*a ^6+192*a^4*b^2-192*a^2*b^4+64*b^6)*ln(b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b)-1 /384/b/tanh(1/2*x)^6-1/256*(4*a^2-8*b^2)/b^3/tanh(1/2*x)^4-1/128/b^5*(16*a ^4-40*a^2*b^2+29*b^4)/tanh(1/2*x)^2+1/64/b^7*(64*a^6-192*a^4*b^2+192*a^2*b ^4-64*b^6)*ln(tanh(1/2*x))+1/160/b^2*a/tanh(1/2*x)^5+1/96*a/b^4*(4*a^2-9*b ^2)/tanh(1/2*x)^3+1/16*a*(8*a^4-22*a^2*b^2+19*b^4)/b^6/tanh(1/2*x)
Leaf count of result is larger than twice the leaf count of optimal. 5283 vs. \(2 (130) = 260\).
Time = 0.15 (sec) , antiderivative size = 5283, normalized size of antiderivative = 37.74 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^8/(a+b*coth(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=\int \frac {\operatorname {csch}^{8}{\left (x \right )}}{a + b \coth {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**8/(a+b*coth(x)),x)
Output:
Integral(csch(x)**8/(a + b*coth(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (130) = 260\).
Time = 0.06 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.01 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=-\frac {2 \, {\left (15 \, a^{5} - 40 \, a^{3} b^{2} + 33 \, a b^{4} - 3 \, {\left (25 \, a^{5} + 5 \, a^{4} b - 70 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 61 \, a b^{4} + 5 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 30 \, {\left (5 \, a^{5} + 2 \, a^{4} b - 14 \, a^{3} b^{2} - 5 \, a^{2} b^{3} + 13 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-4 \, x\right )} - 10 \, {\left (15 \, a^{5} + 9 \, a^{4} b - 40 \, a^{3} b^{2} - 24 \, a^{2} b^{3} + 33 \, a b^{4} + 23 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 15 \, {\left (5 \, a^{5} + 4 \, a^{4} b - 12 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 7 \, a b^{4} + 6 \, b^{5}\right )} e^{\left (-8 \, x\right )} - 15 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} e^{\left (-10 \, x\right )}\right )}}{15 \, {\left (6 \, b^{6} e^{\left (-2 \, x\right )} - 15 \, b^{6} e^{\left (-4 \, x\right )} + 20 \, b^{6} e^{\left (-6 \, x\right )} - 15 \, b^{6} e^{\left (-8 \, x\right )} + 6 \, b^{6} e^{\left (-10 \, x\right )} - b^{6} e^{\left (-12 \, x\right )} - b^{6}\right )}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{b^{7}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{7}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{7}} \] Input:
integrate(csch(x)^8/(a+b*coth(x)),x, algorithm="maxima")
Output:
-2/15*(15*a^5 - 40*a^3*b^2 + 33*a*b^4 - 3*(25*a^5 + 5*a^4*b - 70*a^3*b^2 - 10*a^2*b^3 + 61*a*b^4 + 5*b^5)*e^(-2*x) + 30*(5*a^5 + 2*a^4*b - 14*a^3*b^ 2 - 5*a^2*b^3 + 13*a*b^4 + 3*b^5)*e^(-4*x) - 10*(15*a^5 + 9*a^4*b - 40*a^3 *b^2 - 24*a^2*b^3 + 33*a*b^4 + 23*b^5)*e^(-6*x) + 15*(5*a^5 + 4*a^4*b - 12 *a^3*b^2 - 10*a^2*b^3 + 7*a*b^4 + 6*b^5)*e^(-8*x) - 15*(a^5 + a^4*b - 2*a^ 3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*e^(-10*x))/(6*b^6*e^(-2*x) - 15*b^6*e^(-4 *x) + 20*b^6*e^(-6*x) - 15*b^6*e^(-8*x) + 6*b^6*e^(-10*x) - b^6*e^(-12*x) - b^6) - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(-(a - b)*e^(-2*x) + a + b )/b^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(e^(-x) + 1)/b^7 + (a^6 - 3 *a^4*b^2 + 3*a^2*b^4 - b^6)*log(e^(-x) - 1)/b^7
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (130) = 260\).
Time = 0.14 (sec) , antiderivative size = 594, normalized size of antiderivative = 4.24 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx =\text {Too large to display} \] Input:
integrate(csch(x)^8/(a+b*coth(x)),x, algorithm="giac")
Output:
-(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)*log(abs(a*e^(2*x) + b*e^(2*x) - a + b))/(a*b^7 + b^8) + (a^6 - 3*a^4*b^ 2 + 3*a^2*b^4 - b^6)*log(abs(e^(2*x) - 1))/b^7 - 1/60*(147*a^6*e^(12*x) - 441*a^4*b^2*e^(12*x) + 441*a^2*b^4*e^(12*x) - 147*b^6*e^(12*x) - 882*a^6*e ^(10*x) - 120*a^5*b*e^(10*x) + 2766*a^4*b^2*e^(10*x) + 240*a^3*b^3*e^(10*x ) - 2886*a^2*b^4*e^(10*x) - 120*a*b^5*e^(10*x) + 1002*b^6*e^(10*x) + 2205* a^6*e^(8*x) + 600*a^5*b*e^(8*x) - 7095*a^4*b^2*e^(8*x) - 1440*a^3*b^3*e^(8 *x) + 7815*a^2*b^4*e^(8*x) + 840*a*b^5*e^(8*x) - 2925*b^6*e^(8*x) - 2940*a ^6*e^(6*x) - 1200*a^5*b*e^(6*x) + 9540*a^4*b^2*e^(6*x) + 3200*a^3*b^3*e^(6 *x) - 10740*a^2*b^4*e^(6*x) - 2640*a*b^5*e^(6*x) + 4780*b^6*e^(6*x) + 2205 *a^6*e^(4*x) + 1200*a^5*b*e^(4*x) - 7095*a^4*b^2*e^(4*x) - 3360*a^3*b^3*e^ (4*x) + 7815*a^2*b^4*e^(4*x) + 3120*a*b^5*e^(4*x) - 2925*b^6*e^(4*x) - 882 *a^6*e^(2*x) - 600*a^5*b*e^(2*x) + 2766*a^4*b^2*e^(2*x) + 1680*a^3*b^3*e^( 2*x) - 2886*a^2*b^4*e^(2*x) - 1464*a*b^5*e^(2*x) + 1002*b^6*e^(2*x) + 147* a^6 + 120*a^5*b - 441*a^4*b^2 - 320*a^3*b^3 + 441*a^2*b^4 + 264*a*b^5 - 14 7*b^6)/(b^7*(e^(2*x) - 1)^6)
Time = 2.70 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.15 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx=\frac {32\,\left (a-5\,b\right )}{5\,b^2\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {32}{3\,b\,\left (15\,{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}-20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}-6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1\right )}-\frac {4\,\left (a^2-4\,a\,b+7\,b^2\right )}{b^3\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}+\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}+\frac {8\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{3\,b^4\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}+\frac {2\,{\left (a+b\right )}^2\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^6\,\left ({\mathrm {e}}^{2\,x}-1\right )}-\frac {2\,\left (a+b\right )\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^5\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )} \] Input:
int(1/(sinh(x)^8*(a + b*coth(x))),x)
Output:
(32*(a - 5*b))/(5*b^2*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1)) - 32/(3*b*(15*exp(4*x) - 6*exp(2*x) - 20*exp(6*x) + 15* exp(8*x) - 6*exp(10*x) + exp(12*x) + 1)) - (4*(a^2 - 4*a*b + 7*b^2))/(b^3* (6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - (log(b - a + a*ex p(2*x) + b*exp(2*x))*(a + b)^3*(a - b)^3)/b^7 + (log(exp(2*x) - 1)*(a + b) ^3*(a - b)^3)/b^7 + (8*(a - b)*(a^2 - 2*a*b + b^2))/(3*b^4*(3*exp(2*x) - 3 *exp(4*x) + exp(6*x) - 1)) + (2*(a + b)^2*(a - b)*(a^2 - 2*a*b + b^2))/(b^ 6*(exp(2*x) - 1)) - (2*(a + b)*(a - b)*(a^2 - 2*a*b + b^2))/(b^5*(exp(4*x) - 2*exp(2*x) + 1))
Time = 0.22 (sec) , antiderivative size = 2212, normalized size of antiderivative = 15.80 \[ \int \frac {\text {csch}^8(x)}{a+b \coth (x)} \, dx =\text {Too large to display} \] Input:
int(csch(x)^8/(a+b*coth(x)),x)
Output:
(15*e**(12*x)*log(e**x - 1)*a**6 - 45*e**(12*x)*log(e**x - 1)*a**4*b**2 + 45*e**(12*x)*log(e**x - 1)*a**2*b**4 - 15*e**(12*x)*log(e**x - 1)*b**6 + 1 5*e**(12*x)*log(e**x + 1)*a**6 - 45*e**(12*x)*log(e**x + 1)*a**4*b**2 + 45 *e**(12*x)*log(e**x + 1)*a**2*b**4 - 15*e**(12*x)*log(e**x + 1)*b**6 - 15* e**(12*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**6 + 45*e**(12*x)*log(e** (2*x)*a + e**(2*x)*b - a + b)*a**4*b**2 - 45*e**(12*x)*log(e**(2*x)*a + e* *(2*x)*b - a + b)*a**2*b**4 + 15*e**(12*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*b**6 + 5*e**(12*x)*a**5*b - 5*e**(12*x)*a**4*b**2 - 10*e**(12*x)*a** 3*b**3 + 10*e**(12*x)*a**2*b**4 + 5*e**(12*x)*a*b**5 - 5*e**(12*x)*b**6 - 90*e**(10*x)*log(e**x - 1)*a**6 + 270*e**(10*x)*log(e**x - 1)*a**4*b**2 - 270*e**(10*x)*log(e**x - 1)*a**2*b**4 + 90*e**(10*x)*log(e**x - 1)*b**6 - 90*e**(10*x)*log(e**x + 1)*a**6 + 270*e**(10*x)*log(e**x + 1)*a**4*b**2 - 270*e**(10*x)*log(e**x + 1)*a**2*b**4 + 90*e**(10*x)*log(e**x + 1)*b**6 + 90*e**(10*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**6 - 270*e**(10*x)*log (e**(2*x)*a + e**(2*x)*b - a + b)*a**4*b**2 + 270*e**(10*x)*log(e**(2*x)*a + e**(2*x)*b - a + b)*a**2*b**4 - 90*e**(10*x)*log(e**(2*x)*a + e**(2*x)* b - a + b)*b**6 + 225*e**(8*x)*log(e**x - 1)*a**6 - 675*e**(8*x)*log(e**x - 1)*a**4*b**2 + 675*e**(8*x)*log(e**x - 1)*a**2*b**4 - 225*e**(8*x)*log(e **x - 1)*b**6 + 225*e**(8*x)*log(e**x + 1)*a**6 - 675*e**(8*x)*log(e**x + 1)*a**4*b**2 + 675*e**(8*x)*log(e**x + 1)*a**2*b**4 - 225*e**(8*x)*log(...