Integrand size = 13, antiderivative size = 130 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=-\frac {\left (a^2-b^2\right )^2 \coth (x)}{a^5}-\frac {b \left (2 a^2-b^2\right ) \coth ^2(x)}{2 a^4}+\frac {\left (2 a^2-b^2\right ) \coth ^3(x)}{3 a^3}+\frac {b \coth ^4(x)}{4 a^2}-\frac {\coth ^5(x)}{5 a}-\frac {b \left (a^2-b^2\right )^2 \log (\tanh (x))}{a^6}+\frac {b \left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{a^6} \] Output:
-(a^2-b^2)^2*coth(x)/a^5-1/2*b*(2*a^2-b^2)*coth(x)^2/a^4+1/3*(2*a^2-b^2)*c oth(x)^3/a^3+1/4*b*coth(x)^4/a^2-1/5*coth(x)^5/a-b*(a^2-b^2)^2*ln(tanh(x)) /a^6+b*(a^2-b^2)^2*ln(a+b*tanh(x))/a^6
Time = 5.44 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\frac {-4 \coth (x) \left (8 a^5-25 a^3 b^2+15 a b^4+\left (-4 a^5+5 a^3 b^2\right ) \text {csch}^2(x)+3 a^5 \text {csch}^4(x)\right )+15 b \left (-2 a^2 \left (a^2-b^2\right ) \text {csch}^2(x)+a^4 \text {csch}^4(x)-4 \left (a^2-b^2\right )^2 (\log (\sinh (x))-\log (a \cosh (x)+b \sinh (x)))\right )}{60 a^6} \] Input:
Integrate[Csch[x]^6/(a + b*Tanh[x]),x]
Output:
(-4*Coth[x]*(8*a^5 - 25*a^3*b^2 + 15*a*b^4 + (-4*a^5 + 5*a^3*b^2)*Csch[x]^ 2 + 3*a^5*Csch[x]^4) + 15*b*(-2*a^2*(a^2 - b^2)*Csch[x]^2 + a^4*Csch[x]^4 - 4*(a^2 - b^2)^2*(Log[Sinh[x]] - Log[a*Cosh[x] + b*Sinh[x]])))/(60*a^6)
Time = 0.39 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 25, 3999, 522, 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.
| \(\displaystyle \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i x)^6 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\sin (i x)^6 (a-i b \tan (i x))}dx\) |
| \(\Big \downarrow \) 3999 |
| \(\displaystyle b \int \frac {\coth ^6(x) \left (b^2-b^2 \tanh ^2(x)\right )^2}{b^6 (a+b \tanh (x))}d(b \tanh (x))\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle b \int \left (\frac {\coth ^6(x)}{a b^2}-\frac {\coth ^5(x)}{a^2 b}+\frac {\left (b^4-2 a^2 b^2\right ) \coth ^4(x)}{a^3 b^4}+\frac {\left (2 a^2 b^2-b^4\right ) \coth ^3(x)}{a^4 b^3}+\frac {\left (a^2-b^2\right )^2 \coth ^2(x)}{a^5 b^2}-\frac {\left (a^2-b^2\right )^2 \coth (x)}{a^6 b}+\frac {\left (a^2-b^2\right )^2}{a^6 (a+b \tanh (x))}\right )d(b \tanh (x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle b \left (\frac {\coth ^4(x)}{4 a^2}-\frac {\left (a^2-b^2\right )^2 \log (b \tanh (x))}{a^6}+\frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{a^6}-\frac {\left (a^2-b^2\right )^2 \coth (x)}{a^5 b}-\frac {\left (2 a^2-b^2\right ) \coth ^2(x)}{2 a^4}+\frac {\left (2 a^2-b^2\right ) \coth ^3(x)}{3 a^3 b}-\frac {\coth ^5(x)}{5 a b}\right )\) |
Input:
Int[Csch[x]^6/(a + b*Tanh[x]),x]
Output:
b*(-(((a^2 - b^2)^2*Coth[x])/(a^5*b)) - ((2*a^2 - b^2)*Coth[x]^2)/(2*a^4) + ((2*a^2 - b^2)*Coth[x]^3)/(3*a^3*b) + Coth[x]^4/(4*a^2) - Coth[x]^5/(5*a *b) - ((a^2 - b^2)^2*Log[b*Tanh[x]])/a^6 + ((a^2 - b^2)^2*Log[a + b*Tanh[x ]])/a^6)
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[b/f Subst[Int[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/2]
Leaf count of result is larger than twice the leaf count of optimal. \(276\) vs. \(2(122)=244\).
Time = 12.85 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(-\frac {\frac {a^{4} \tanh \left (\frac {x}{2}\right )^{5}}{5}-\frac {b \tanh \left (\frac {x}{2}\right )^{4} a^{3}}{2}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3} a^{4}}{3}+\frac {4 a^{2} b^{2} \tanh \left (\frac {x}{2}\right )^{3}}{3}+6 a^{3} b \tanh \left (\frac {x}{2}\right )^{2}-4 b^{3} \tanh \left (\frac {x}{2}\right )^{2} a +10 a^{4} \tanh \left (\frac {x}{2}\right )-28 a^{2} b^{2} \tanh \left (\frac {x}{2}\right )+16 b^{4} \tanh \left (\frac {x}{2}\right )}{32 a^{5}}-\frac {1}{160 a \tanh \left (\frac {x}{2}\right )^{5}}-\frac {-5 a^{2}+4 b^{2}}{96 a^{3} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {10 a^{4}-28 a^{2} b^{2}+16 b^{4}}{32 a^{5} \tanh \left (\frac {x}{2}\right )}+\frac {b}{64 a^{2} \tanh \left (\frac {x}{2}\right )^{4}}-\frac {b \left (3 a^{2}-2 b^{2}\right )}{16 a^{4} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{6}}+\frac {2 b \left (\frac {1}{2} a^{4}-a^{2} b^{2}+\frac {1}{2} b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{a^{6}}\) | \(277\) |
| risch | \(-\frac {2 \left (15 a^{3} b \,{\mathrm e}^{8 x}-15 a^{2} b^{2} {\mathrm e}^{8 x}-15 a \,b^{3} {\mathrm e}^{8 x}+15 b^{4} {\mathrm e}^{8 x}-75 b \,a^{3} {\mathrm e}^{6 x}+90 a^{2} b^{2} {\mathrm e}^{6 x}+45 a \,b^{3} {\mathrm e}^{6 x}-60 b^{4} {\mathrm e}^{6 x}+80 \,{\mathrm e}^{4 x} a^{4}+75 \,{\mathrm e}^{4 x} a^{3} b -160 \,{\mathrm e}^{4 x} a^{2} b^{2}-45 \,{\mathrm e}^{4 x} a \,b^{3}+90 b^{4} {\mathrm e}^{4 x}-40 \,{\mathrm e}^{2 x} a^{4}-15 b \,{\mathrm e}^{2 x} a^{3}+110 \,{\mathrm e}^{2 x} a^{2} b^{2}+15 b^{3} {\mathrm e}^{2 x} a -60 b^{4} {\mathrm e}^{2 x}+8 a^{4}-25 a^{2} b^{2}+15 b^{4}\right )}{15 a^{5} \left ({\mathrm e}^{2 x}-1\right )^{5}}-\frac {b \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{2}}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{4}}-\frac {b^{5} \ln \left ({\mathrm e}^{2 x}-1\right )}{a^{6}}+\frac {b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{2}}-\frac {2 b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4}}+\frac {b^{5} \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{6}}\) | \(330\) |
Input:
int(csch(x)^6/(a+b*tanh(x)),x,method=_RETURNVERBOSE)
Output:
-1/32/a^5*(1/5*a^4*tanh(1/2*x)^5-1/2*b*tanh(1/2*x)^4*a^3-5/3*tanh(1/2*x)^3 *a^4+4/3*a^2*b^2*tanh(1/2*x)^3+6*a^3*b*tanh(1/2*x)^2-4*b^3*tanh(1/2*x)^2*a +10*a^4*tanh(1/2*x)-28*a^2*b^2*tanh(1/2*x)+16*b^4*tanh(1/2*x))-1/160/a/tan h(1/2*x)^5-1/96*(-5*a^2+4*b^2)/a^3/tanh(1/2*x)^3-1/32/a^5*(10*a^4-28*a^2*b ^2+16*b^4)/tanh(1/2*x)+1/64/a^2*b/tanh(1/2*x)^4-1/16/a^4*b*(3*a^2-2*b^2)/t anh(1/2*x)^2-1/a^6*b*(a^4-2*a^2*b^2+b^4)*ln(tanh(1/2*x))+2/a^6*b*(1/2*a^4- a^2*b^2+1/2*b^4)*ln(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)
Leaf count of result is larger than twice the leaf count of optimal. 2972 vs. \(2 (122) = 244\).
Time = 0.14 (sec) , antiderivative size = 2972, normalized size of antiderivative = 22.86 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \] Input:
integrate(csch(x)^6/(a+b*tanh(x)),x, algorithm="fricas")
Output:
-1/15*(30*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^8 + 240*(a^4*b - a^3 *b^2 - a^2*b^3 + a*b^4)*cosh(x)*sinh(x)^7 + 30*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*sinh(x)^8 - 30*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x )^6 - 30*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4 - 28*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^2)*sinh(x)^6 + 60*(28*(a^4*b - a^3*b^2 - a^2*b^ 3 + a*b^4)*cosh(x)^3 - 3*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh( x))*sinh(x)^5 + 16*a^5 - 50*a^3*b^2 + 30*a*b^4 + 10*(16*a^5 + 15*a^4*b - 3 2*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4)*cosh(x)^4 + 10*(16*a^5 + 15*a^4*b - 32*a ^3*b^2 - 9*a^2*b^3 + 18*a*b^4 + 210*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*co sh(x)^4 - 45*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^2)*sinh(x )^4 + 40*(42*(a^4*b - a^3*b^2 - a^2*b^3 + a*b^4)*cosh(x)^5 - 15*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^3 + (16*a^5 + 15*a^4*b - 32*a^3* b^2 - 9*a^2*b^3 + 18*a*b^4)*cosh(x))*sinh(x)^3 - 10*(8*a^5 + 3*a^4*b - 22* a^3*b^2 - 3*a^2*b^3 + 12*a*b^4)*cosh(x)^2 + 10*(84*(a^4*b - a^3*b^2 - a^2* b^3 + a*b^4)*cosh(x)^6 - 8*a^5 - 3*a^4*b + 22*a^3*b^2 + 3*a^2*b^3 - 12*a*b ^4 - 45*(5*a^4*b - 6*a^3*b^2 - 3*a^2*b^3 + 4*a*b^4)*cosh(x)^4 + 6*(16*a^5 + 15*a^4*b - 32*a^3*b^2 - 9*a^2*b^3 + 18*a*b^4)*cosh(x)^2)*sinh(x)^2 - 15* ((a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^10 + 10*(a^4*b - 2*a^2*b^3 + b^5)*cosh( x)*sinh(x)^9 + (a^4*b - 2*a^2*b^3 + b^5)*sinh(x)^10 - 5*(a^4*b - 2*a^2*b^3 + b^5)*cosh(x)^8 - 5*(a^4*b - 2*a^2*b^3 + b^5 - 9*(a^4*b - 2*a^2*b^3 +...
\[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {csch}^{6}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \] Input:
integrate(csch(x)**6/(a+b*tanh(x)),x)
Output:
Integral(csch(x)**6/(a + b*tanh(x)), x)
Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (122) = 244\).
Time = 0.05 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.37 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\frac {2 \, {\left (8 \, a^{4} - 25 \, a^{2} b^{2} + 15 \, b^{4} - 5 \, {\left (8 \, a^{4} - 3 \, a^{3} b - 22 \, a^{2} b^{2} + 3 \, a b^{3} + 12 \, b^{4}\right )} e^{\left (-2 \, x\right )} + 5 \, {\left (16 \, a^{4} - 15 \, a^{3} b - 32 \, a^{2} b^{2} + 9 \, a b^{3} + 18 \, b^{4}\right )} e^{\left (-4 \, x\right )} + 15 \, {\left (5 \, a^{3} b + 6 \, a^{2} b^{2} - 3 \, a b^{3} - 4 \, b^{4}\right )} e^{\left (-6 \, x\right )} - 15 \, {\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-8 \, x\right )}\right )}}{15 \, {\left (5 \, a^{5} e^{\left (-2 \, x\right )} - 10 \, a^{5} e^{\left (-4 \, x\right )} + 10 \, a^{5} e^{\left (-6 \, x\right )} - 5 \, a^{5} e^{\left (-8 \, x\right )} + a^{5} e^{\left (-10 \, x\right )} - a^{5}\right )}} + \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6}} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{6}} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{6}} \] Input:
integrate(csch(x)^6/(a+b*tanh(x)),x, algorithm="maxima")
Output:
2/15*(8*a^4 - 25*a^2*b^2 + 15*b^4 - 5*(8*a^4 - 3*a^3*b - 22*a^2*b^2 + 3*a* b^3 + 12*b^4)*e^(-2*x) + 5*(16*a^4 - 15*a^3*b - 32*a^2*b^2 + 9*a*b^3 + 18* b^4)*e^(-4*x) + 15*(5*a^3*b + 6*a^2*b^2 - 3*a*b^3 - 4*b^4)*e^(-6*x) - 15*( a^3*b + a^2*b^2 - a*b^3 - b^4)*e^(-8*x))/(5*a^5*e^(-2*x) - 10*a^5*e^(-4*x) + 10*a^5*e^(-6*x) - 5*a^5*e^(-8*x) + a^5*e^(-10*x) - a^5) + (a^4*b - 2*a^ 2*b^3 + b^5)*log(-(a - b)*e^(-2*x) - a - b)/a^6 - (a^4*b - 2*a^2*b^3 + b^5 )*log(e^(-x) + 1)/a^6 - (a^4*b - 2*a^2*b^3 + b^5)*log(e^(-x) - 1)/a^6
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (122) = 244\).
Time = 0.13 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.17 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\frac {{\left (a^{5} b + a^{4} b^{2} - 2 \, a^{3} b^{3} - 2 \, a^{2} b^{4} + a b^{5} + b^{6}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{7} + a^{6} b} - \frac {{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{a^{6}} + \frac {137 \, a^{4} b e^{\left (10 \, x\right )} - 274 \, a^{2} b^{3} e^{\left (10 \, x\right )} + 137 \, b^{5} e^{\left (10 \, x\right )} - 805 \, a^{4} b e^{\left (8 \, x\right )} + 120 \, a^{3} b^{2} e^{\left (8 \, x\right )} + 1490 \, a^{2} b^{3} e^{\left (8 \, x\right )} - 120 \, a b^{4} e^{\left (8 \, x\right )} - 685 \, b^{5} e^{\left (8 \, x\right )} + 1970 \, a^{4} b e^{\left (6 \, x\right )} - 720 \, a^{3} b^{2} e^{\left (6 \, x\right )} - 3100 \, a^{2} b^{3} e^{\left (6 \, x\right )} + 480 \, a b^{4} e^{\left (6 \, x\right )} + 1370 \, b^{5} e^{\left (6 \, x\right )} - 640 \, a^{5} e^{\left (4 \, x\right )} - 1970 \, a^{4} b e^{\left (4 \, x\right )} + 1280 \, a^{3} b^{2} e^{\left (4 \, x\right )} + 3100 \, a^{2} b^{3} e^{\left (4 \, x\right )} - 720 \, a b^{4} e^{\left (4 \, x\right )} - 1370 \, b^{5} e^{\left (4 \, x\right )} + 320 \, a^{5} e^{\left (2 \, x\right )} + 805 \, a^{4} b e^{\left (2 \, x\right )} - 880 \, a^{3} b^{2} e^{\left (2 \, x\right )} - 1490 \, a^{2} b^{3} e^{\left (2 \, x\right )} + 480 \, a b^{4} e^{\left (2 \, x\right )} + 685 \, b^{5} e^{\left (2 \, x\right )} - 64 \, a^{5} - 137 \, a^{4} b + 200 \, a^{3} b^{2} + 274 \, a^{2} b^{3} - 120 \, a b^{4} - 137 \, b^{5}}{60 \, a^{6} {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \] Input:
integrate(csch(x)^6/(a+b*tanh(x)),x, algorithm="giac")
Output:
(a^5*b + a^4*b^2 - 2*a^3*b^3 - 2*a^2*b^4 + a*b^5 + b^6)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^7 + a^6*b) - (a^4*b - 2*a^2*b^3 + b^5)*log(abs(e^ (2*x) - 1))/a^6 + 1/60*(137*a^4*b*e^(10*x) - 274*a^2*b^3*e^(10*x) + 137*b^ 5*e^(10*x) - 805*a^4*b*e^(8*x) + 120*a^3*b^2*e^(8*x) + 1490*a^2*b^3*e^(8*x ) - 120*a*b^4*e^(8*x) - 685*b^5*e^(8*x) + 1970*a^4*b*e^(6*x) - 720*a^3*b^2 *e^(6*x) - 3100*a^2*b^3*e^(6*x) + 480*a*b^4*e^(6*x) + 1370*b^5*e^(6*x) - 6 40*a^5*e^(4*x) - 1970*a^4*b*e^(4*x) + 1280*a^3*b^2*e^(4*x) + 3100*a^2*b^3* e^(4*x) - 720*a*b^4*e^(4*x) - 1370*b^5*e^(4*x) + 320*a^5*e^(2*x) + 805*a^4 *b*e^(2*x) - 880*a^3*b^2*e^(2*x) - 1490*a^2*b^3*e^(2*x) + 480*a*b^4*e^(2*x ) + 685*b^5*e^(2*x) - 64*a^5 - 137*a^4*b + 200*a^3*b^2 + 274*a^2*b^3 - 120 *a*b^4 - 137*b^5)/(a^6*(e^(2*x) - 1)^5)
Time = 2.39 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.82 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx=\frac {2\,\left (a-b\right )\,\left (a\,b-b^2\right )}{a^4\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {8\,\left (4\,a^2-3\,a\,b+b^2\right )}{3\,a^3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {4\,\left (4\,a-b\right )}{a^2\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {32}{5\,a\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {2\,\left (a+b\right )\,\left (a-b\right )\,\left (a\,b-b^2\right )}{a^5\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {b\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{a^6}-\frac {b\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{a^6} \] Input:
int(1/(sinh(x)^6*(a + b*tanh(x))),x)
Output:
(2*(a - b)*(a*b - b^2))/(a^4*(exp(4*x) - 2*exp(2*x) + 1)) - (8*(4*a^2 - 3* a*b + b^2))/(3*a^3*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (4*(4*a - b ))/(a^2*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - 32/(5*a*( 5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*exp(8*x) + exp(10*x) - 1)) - (2 *(a + b)*(a - b)*(a*b - b^2))/(a^5*(exp(2*x) - 1)) + (b*log(a - b + a*exp( 2*x) + b*exp(2*x))*(a + b)^2*(a - b)^2)/a^6 - (b*log(exp(2*x) - 1)*(a + b) ^2*(a - b)^2)/a^6
Time = 0.24 (sec) , antiderivative size = 1407, normalized size of antiderivative = 10.82 \[ \int \frac {\text {csch}^6(x)}{a+b \tanh (x)} \, dx =\text {Too large to display} \] Input:
int(csch(x)^6/(a+b*tanh(x)),x)
Output:
( - 15*e**(10*x)*log(e**x - 1)*a**4*b + 30*e**(10*x)*log(e**x - 1)*a**2*b* *3 - 15*e**(10*x)*log(e**x - 1)*b**5 - 15*e**(10*x)*log(e**x + 1)*a**4*b + 30*e**(10*x)*log(e**x + 1)*a**2*b**3 - 15*e**(10*x)*log(e**x + 1)*b**5 + 15*e**(10*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*a**4*b - 30*e**(10*x)*lo g(e**(2*x)*a + e**(2*x)*b + a - b)*a**2*b**3 + 15*e**(10*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*b**5 - 6*e**(10*x)*a**4*b + 6*e**(10*x)*a**3*b**2 + 6*e**(10*x)*a**2*b**3 - 6*e**(10*x)*a*b**4 + 75*e**(8*x)*log(e**x - 1)*a* *4*b - 150*e**(8*x)*log(e**x - 1)*a**2*b**3 + 75*e**(8*x)*log(e**x - 1)*b* *5 + 75*e**(8*x)*log(e**x + 1)*a**4*b - 150*e**(8*x)*log(e**x + 1)*a**2*b* *3 + 75*e**(8*x)*log(e**x + 1)*b**5 - 75*e**(8*x)*log(e**(2*x)*a + e**(2*x )*b + a - b)*a**4*b + 150*e**(8*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*a* *2*b**3 - 75*e**(8*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*b**5 - 150*e**( 6*x)*log(e**x - 1)*a**4*b + 300*e**(6*x)*log(e**x - 1)*a**2*b**3 - 150*e** (6*x)*log(e**x - 1)*b**5 - 150*e**(6*x)*log(e**x + 1)*a**4*b + 300*e**(6*x )*log(e**x + 1)*a**2*b**3 - 150*e**(6*x)*log(e**x + 1)*b**5 + 150*e**(6*x) *log(e**(2*x)*a + e**(2*x)*b + a - b)*a**4*b - 300*e**(6*x)*log(e**(2*x)*a + e**(2*x)*b + a - b)*a**2*b**3 + 150*e**(6*x)*log(e**(2*x)*a + e**(2*x)* b + a - b)*b**5 + 90*e**(6*x)*a**4*b - 120*e**(6*x)*a**3*b**2 - 30*e**(6*x )*a**2*b**3 + 60*e**(6*x)*a*b**4 + 150*e**(4*x)*log(e**x - 1)*a**4*b - 300 *e**(4*x)*log(e**x - 1)*a**2*b**3 + 150*e**(4*x)*log(e**x - 1)*b**5 + 1...