Integrand size = 13, antiderivative size = 183 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a}-\frac {3 \text {arctanh}(\cosh (x))}{8 b}+\frac {\left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}+\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b^5}-\frac {a \coth (x)}{b^2}+\frac {a \left (a^2+3 b^2\right ) \coth (x)}{b^4}+\frac {a \coth ^3(x)}{3 b^2}+\frac {3 \coth (x) \text {csch}(x)}{8 b}-\frac {\left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}-\frac {\coth (x) \text {csch}^3(x)}{4 b} \] Output:
x/a-3/8*arctanh(cosh(x))/b+1/2*(a^2+3*b^2)*arctanh(cosh(x))/b^3-(a^4+3*a^2 *b^2+3*b^4)*arctanh(cosh(x))/b^5+2*(a^2+b^2)^(5/2)*arctanh((a-b*tanh(1/2*x ))/(a^2+b^2)^(1/2))/a/b^5-a*coth(x)/b^2+a*(a^2+3*b^2)*coth(x)/b^4+1/3*a*co th(x)^3/b^2+3/8*coth(x)*csch(x)/b-1/2*(a^2+3*b^2)*coth(x)*csch(x)/b^3-1/4* coth(x)*csch(x)^3/b
Time = 1.30 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.63 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {\text {csch}(x) (b+a \sinh (x)) \left (192 b^5 x+384 \left (-a^2-b^2\right )^{5/2} \arctan \left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )+32 a^2 b \left (3 a^2+7 b^2\right ) \coth \left (\frac {x}{2}\right )-6 a b^2 \left (4 a^2+9 b^2\right ) \text {csch}^2\left (\frac {x}{2}\right )-3 a b^4 \text {csch}^4\left (\frac {x}{2}\right )-24 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\cosh \left (\frac {x}{2}\right )\right )+24 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\sinh \left (\frac {x}{2}\right )\right )-6 a b^2 \left (4 a^2+9 b^2\right ) \text {sech}^2\left (\frac {x}{2}\right )+3 a b^4 \text {sech}^4\left (\frac {x}{2}\right )-64 a^2 b^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )+4 a^2 b^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+32 a^2 b \left (3 a^2+7 b^2\right ) \tanh \left (\frac {x}{2}\right )\right )}{192 a b^5 (a+b \text {csch}(x))} \] Input:
Integrate[Coth[x]^6/(a + b*Csch[x]),x]
Output:
(Csch[x]*(b + a*Sinh[x])*(192*b^5*x + 384*(-a^2 - b^2)^(5/2)*ArcTan[(a - b *Tanh[x/2])/Sqrt[-a^2 - b^2]] + 32*a^2*b*(3*a^2 + 7*b^2)*Coth[x/2] - 6*a*b ^2*(4*a^2 + 9*b^2)*Csch[x/2]^2 - 3*a*b^4*Csch[x/2]^4 - 24*a*(8*a^4 + 20*a^ 2*b^2 + 15*b^4)*Log[Cosh[x/2]] + 24*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Si nh[x/2]] - 6*a*b^2*(4*a^2 + 9*b^2)*Sech[x/2]^2 + 3*a*b^4*Sech[x/2]^4 - 64* a^2*b^3*Csch[x]^3*Sinh[x/2]^4 + 4*a^2*b^3*Csch[x/2]^4*Sinh[x] + 32*a^2*b*( 3*a^2 + 7*b^2)*Tanh[x/2]))/(192*a*b^5*(a + b*Csch[x]))
Result contains complex when optimal does not.
Time = 0.65 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 25, 4386, 26, 26, 3042, 26, 3376, 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.
| \(\displaystyle \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot (i x)^6}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot (i x)^6}{a+i b \csc (i x)}dx\) |
| \(\Big \downarrow \) 4386 |
| \(\displaystyle -\int -\frac {i \cosh (x) \coth ^5(x)}{i b+i a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int -\frac {i \cosh (x) \coth ^5(x)}{b+a \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\cosh (x) \coth ^5(x)}{a \sinh (x)+b}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \cos (i x)^6}{\sin (i x)^5 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x)^6}{\sin (i x)^5 (b-i a \sin (i x))}dx\) |
| \(\Big \downarrow \) 3376 |
| \(\displaystyle i \int \left (-\frac {i \text {csch}^5(x)}{b}+\frac {i a \text {csch}^4(x)}{b^2}+\frac {i \left (-a^2-3 b^2\right ) \text {csch}^3(x)}{b^3}+\frac {i \left (a^3+3 b^2 a\right ) \text {csch}^2(x)}{b^4}-\frac {i \left (a^4+3 b^2 a^2+3 b^4\right ) \text {csch}(x)}{b^5}-\frac {i}{a}-\frac {\left (a^2+b^2\right )^3}{a b^5 (i b+i a \sinh (x))}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (-\frac {2 i \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {a-b \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a b^5}-\frac {i \left (a^2+3 b^2\right ) \text {arctanh}(\cosh (x))}{2 b^3}-\frac {i a \left (a^2+3 b^2\right ) \coth (x)}{b^4}+\frac {i \left (a^2+3 b^2\right ) \coth (x) \text {csch}(x)}{2 b^3}+\frac {i \left (a^4+3 a^2 b^2+3 b^4\right ) \text {arctanh}(\cosh (x))}{b^5}-\frac {i a \coth ^3(x)}{3 b^2}+\frac {i a \coth (x)}{b^2}-\frac {i x}{a}+\frac {3 i \text {arctanh}(\cosh (x))}{8 b}+\frac {i \coth (x) \text {csch}^3(x)}{4 b}-\frac {3 i \coth (x) \text {csch}(x)}{8 b}\right )\) |
Input:
Int[Coth[x]^6/(a + b*Csch[x]),x]
Output:
I*(((-I)*x)/a + (((3*I)/8)*ArcTanh[Cosh[x]])/b - ((I/2)*(a^2 + 3*b^2)*ArcT anh[Cosh[x]])/b^3 + (I*(a^4 + 3*a^2*b^2 + 3*b^4)*ArcTanh[Cosh[x]])/b^5 - ( (2*I)*(a^2 + b^2)^(5/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b^5 ) + (I*a*Coth[x])/b^2 - (I*a*(a^2 + 3*b^2)*Coth[x])/b^4 - ((I/3)*a*Coth[x] ^3)/b^2 - (((3*I)/8)*Coth[x]*Csch[x])/b + ((I/2)*(a^2 + 3*b^2)*Coth[x]*Csc h[x])/b^3 + ((I/4)*Coth[x]*Csch[x]^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[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(d*sin[ e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x] /; Fr eeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && ( LtQ[m, -1] || (EqQ[m, -1] && GtQ[p, 0]))
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Int[Cos[c + d*x]^m*((b + a*Sin[c + d*x])^n/Sin[c + d*x]^(m + n)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[n] && IntegerQ[m] && (IntegerQ[m/2] || LeQ[m, 1])
Time = 2.58 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.42
| method | result | size |
| default | \(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{4} b^{3}}{4}+\frac {2 a \tanh \left (\frac {x}{2}\right )^{3} b^{2}}{3}+2 a^{2} b \tanh \left (\frac {x}{2}\right )^{2}+4 b^{3} \tanh \left (\frac {x}{2}\right )^{2}+8 \tanh \left (\frac {x}{2}\right ) a^{3}+18 a \,b^{2} \tanh \left (\frac {x}{2}\right )}{16 b^{4}}-\frac {1}{64 b \tanh \left (\frac {x}{2}\right )^{4}}-\frac {4 a^{2}+8 b^{2}}{32 b^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{4}+40 a^{2} b^{2}+30 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 b^{5}}+\frac {a}{24 b^{2} \tanh \left (\frac {x}{2}\right )^{3}}+\frac {a \left (4 a^{2}+9 b^{2}\right )}{8 b^{4} \tanh \left (\frac {x}{2}\right )}+\frac {\left (32 a^{6}+96 a^{4} b^{2}+96 a^{2} b^{4}+32 b^{6}\right ) \operatorname {arctanh}\left (\frac {-2 \tanh \left (\frac {x}{2}\right ) b +2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{16 b^{5} a \sqrt {a^{2}+b^{2}}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\) | \(260\) |
| risch | \(\frac {x}{a}+\frac {-12 a^{2} b \,{\mathrm e}^{7 x}-27 b^{3} {\mathrm e}^{7 x}+24 a^{3} {\mathrm e}^{6 x}+72 a \,b^{2} {\mathrm e}^{6 x}+12 a^{2} b \,{\mathrm e}^{5 x}+3 \,{\mathrm e}^{5 x} b^{3}-72 a^{3} {\mathrm e}^{4 x}-168 a \,b^{2} {\mathrm e}^{4 x}+12 a^{2} b \,{\mathrm e}^{3 x}+3 b^{3} {\mathrm e}^{3 x}+72 a^{3} {\mathrm e}^{2 x}+152 a \,b^{2} {\mathrm e}^{2 x}-12 b \,{\mathrm e}^{x} a^{2}-27 b^{3} {\mathrm e}^{x}-24 a^{3}-56 a \,b^{2}}{12 b^{4} \left ({\mathrm e}^{2 x}-1\right )^{4}}-\frac {\ln \left (1+{\mathrm e}^{x}\right ) a^{4}}{b^{5}}-\frac {5 \ln \left (1+{\mathrm e}^{x}\right ) a^{2}}{2 b^{3}}-\frac {15 \ln \left (1+{\mathrm e}^{x}\right )}{8 b}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}+a^{4} b +2 a^{2} b^{3}+b^{5}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}\right )}{b^{5} a}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}-a^{4} b -2 a^{2} b^{3}-b^{5}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}\right )}{b^{5} a}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a^{4}}{b^{5}}+\frac {5 \ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{2 b^{3}}+\frac {15 \ln \left ({\mathrm e}^{x}-1\right )}{8 b}\) | \(371\) |
Input:
int(coth(x)^6/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
1/16/b^4*(1/4*tanh(1/2*x)^4*b^3+2/3*a*tanh(1/2*x)^3*b^2+2*a^2*b*tanh(1/2*x )^2+4*b^3*tanh(1/2*x)^2+8*tanh(1/2*x)*a^3+18*a*b^2*tanh(1/2*x))-1/64/b/tan h(1/2*x)^4-1/32*(4*a^2+8*b^2)/b^3/tanh(1/2*x)^2+1/16/b^5*(16*a^4+40*a^2*b^ 2+30*b^4)*ln(tanh(1/2*x))+1/24/b^2*a/tanh(1/2*x)^3+1/8*a*(4*a^2+9*b^2)/b^4 /tanh(1/2*x)+1/16*(32*a^6+96*a^4*b^2+96*a^2*b^4+32*b^6)/b^5/a/(a^2+b^2)^(1 /2)*arctanh(1/2*(-2*tanh(1/2*x)*b+2*a)/(a^2+b^2)^(1/2))+1/a*ln(tanh(1/2*x) +1)-1/a*ln(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 3160 vs. \(2 (167) = 334\).
Time = 0.27 (sec) , antiderivative size = 3160, normalized size of antiderivative = 17.27 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \] Input:
integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{6}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(coth(x)**6/(a+b*csch(x)),x)
Output:
Integral(coth(x)**6/(a + b*csch(x)), x)
Time = 0.14 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=-\frac {24 \, a^{3} + 56 \, a b^{2} - 3 \, {\left (4 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \, {\left (9 \, a^{3} + 19 \, a b^{2}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 24 \, {\left (3 \, a^{3} + 7 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (4 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 24 \, {\left (a^{3} + 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} - 3 \, {\left (4 \, a^{2} b + 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{12 \, {\left (4 \, b^{4} e^{\left (-2 \, x\right )} - 6 \, b^{4} e^{\left (-4 \, x\right )} + 4 \, b^{4} e^{\left (-6 \, x\right )} - b^{4} e^{\left (-8 \, x\right )} - b^{4}\right )}} + \frac {x}{a} - \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, b^{5}} + \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, b^{5}} - \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {a e^{\left (-x\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-x\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a b^{5}} \] Input:
integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="maxima")
Output:
-1/12*(24*a^3 + 56*a*b^2 - 3*(4*a^2*b + 9*b^3)*e^(-x) - 8*(9*a^3 + 19*a*b^ 2)*e^(-2*x) + 3*(4*a^2*b + b^3)*e^(-3*x) + 24*(3*a^3 + 7*a*b^2)*e^(-4*x) + 3*(4*a^2*b + b^3)*e^(-5*x) - 24*(a^3 + 3*a*b^2)*e^(-6*x) - 3*(4*a^2*b + 9 *b^3)*e^(-7*x))/(4*b^4*e^(-2*x) - 6*b^4*e^(-4*x) + 4*b^4*e^(-6*x) - b^4*e^ (-8*x) - b^4) + x/a - 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(e^(-x) + 1)/b^ 5 + 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(e^(-x) - 1)/b^5 - (a^6 + 3*a^4*b ^2 + 3*a^2*b^4 + b^6)*log((a*e^(-x) - b - sqrt(a^2 + b^2))/(a*e^(-x) - b + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a*b^5)
Time = 0.13 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.67 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {x}{a} - \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left (e^{x} + 1\right )}{8 \, b^{5}} + \frac {{\left (8 \, a^{4} + 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{8 \, b^{5}} - \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, a e^{x} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{x} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a b^{5}} - \frac {12 \, a^{2} b e^{\left (7 \, x\right )} + 27 \, b^{3} e^{\left (7 \, x\right )} - 24 \, a^{3} e^{\left (6 \, x\right )} - 72 \, a b^{2} e^{\left (6 \, x\right )} - 12 \, a^{2} b e^{\left (5 \, x\right )} - 3 \, b^{3} e^{\left (5 \, x\right )} + 72 \, a^{3} e^{\left (4 \, x\right )} + 168 \, a b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} b e^{\left (3 \, x\right )} - 3 \, b^{3} e^{\left (3 \, x\right )} - 72 \, a^{3} e^{\left (2 \, x\right )} - 152 \, a b^{2} e^{\left (2 \, x\right )} + 12 \, a^{2} b e^{x} + 27 \, b^{3} e^{x} + 24 \, a^{3} + 56 \, a b^{2}}{12 \, b^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} \] Input:
integrate(coth(x)^6/(a+b*csch(x)),x, algorithm="giac")
Output:
x/a - 1/8*(8*a^4 + 20*a^2*b^2 + 15*b^4)*log(e^x + 1)/b^5 + 1/8*(8*a^4 + 20 *a^2*b^2 + 15*b^4)*log(abs(e^x - 1))/b^5 - (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*a*e^x + 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*e^x + 2*b + 2*sqrt (a^2 + b^2)))/(sqrt(a^2 + b^2)*a*b^5) - 1/12*(12*a^2*b*e^(7*x) + 27*b^3*e^ (7*x) - 24*a^3*e^(6*x) - 72*a*b^2*e^(6*x) - 12*a^2*b*e^(5*x) - 3*b^3*e^(5* x) + 72*a^3*e^(4*x) + 168*a*b^2*e^(4*x) - 12*a^2*b*e^(3*x) - 3*b^3*e^(3*x) - 72*a^3*e^(2*x) - 152*a*b^2*e^(2*x) + 12*a^2*b*e^x + 27*b^3*e^x + 24*a^3 + 56*a*b^2)/(b^4*(e^(2*x) - 1)^4)
Time = 3.98 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.97 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {8\,a}{3\,b^2}-\frac {6\,{\mathrm {e}}^x}{b}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {\frac {{\mathrm {e}}^x\,\left (4\,a^2+9\,b^2\right )}{4\,b^3}-\frac {2\,\left (a^4+3\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{2\,x}-1}+\frac {\frac {4\,a}{b^2}-\frac {{\mathrm {e}}^x\,\left (4\,a^2+13\,b^2\right )}{2\,b^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}+\frac {x}{a}+\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (8\,a^4+20\,a^2\,b^2+15\,b^4\right )}{8\,b^5}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (8\,a^4+20\,a^2\,b^2+15\,b^4\right )}{8\,b^5}-\frac {4\,{\mathrm {e}}^x}{b\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {\ln \left (a^3\,\sqrt {{\left (a^2+b^2\right )}^5}-2\,a^7\,b-2\,a\,b^7-6\,a^3\,b^5-6\,a^5\,b^3+a^8\,{\mathrm {e}}^x+4\,b^8\,{\mathrm {e}}^x+2\,a\,b^2\,\sqrt {{\left (a^2+b^2\right )}^5}-4\,b^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}+13\,a^2\,b^6\,{\mathrm {e}}^x+15\,a^4\,b^4\,{\mathrm {e}}^x+7\,a^6\,b^2\,{\mathrm {e}}^x-3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}\right )\,\sqrt {{\left (a^2+b^2\right )}^5}}{a\,b^5}-\frac {\ln \left (a^8\,{\mathrm {e}}^x-2\,a^7\,b-a^3\,\sqrt {{\left (a^2+b^2\right )}^5}-6\,a^3\,b^5-6\,a^5\,b^3-2\,a\,b^7+4\,b^8\,{\mathrm {e}}^x-2\,a\,b^2\,\sqrt {{\left (a^2+b^2\right )}^5}+4\,b^3\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}+13\,a^2\,b^6\,{\mathrm {e}}^x+15\,a^4\,b^4\,{\mathrm {e}}^x+7\,a^6\,b^2\,{\mathrm {e}}^x+3\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^5}\right )\,\sqrt {{\left (a^2+b^2\right )}^5}}{a\,b^5} \] Input:
int(coth(x)^6/(a + b/sinh(x)),x)
Output:
((8*a)/(3*b^2) - (6*exp(x))/b)/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) - ((exp(x)*(4*a^2 + 9*b^2))/(4*b^3) - (2*(a^4 + 3*a^2*b^2))/(a*b^4))/(exp(2* x) - 1) + ((4*a)/b^2 - (exp(x)*(4*a^2 + 13*b^2))/(2*b^3))/(exp(4*x) - 2*ex p(2*x) + 1) + x/a + (log(exp(x) - 1)*(8*a^4 + 15*b^4 + 20*a^2*b^2))/(8*b^5 ) - (log(exp(x) + 1)*(8*a^4 + 15*b^4 + 20*a^2*b^2))/(8*b^5) - (4*exp(x))/( b*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) + (log(a^3*((a^2 + b^2)^5)^(1/2) - 2*a^7*b - 2*a*b^7 - 6*a^3*b^5 - 6*a^5*b^3 + a^8*exp(x) + 4*b^8*exp(x) + 2*a*b^2*((a^2 + b^2)^5)^(1/2) - 4*b^3*exp(x)*((a^2 + b^2)^ 5)^(1/2) + 13*a^2*b^6*exp(x) + 15*a^4*b^4*exp(x) + 7*a^6*b^2*exp(x) - 3*a^ 2*b*exp(x)*((a^2 + b^2)^5)^(1/2))*((a^2 + b^2)^5)^(1/2))/(a*b^5) - (log(a^ 8*exp(x) - 2*a^7*b - a^3*((a^2 + b^2)^5)^(1/2) - 6*a^3*b^5 - 6*a^5*b^3 - 2 *a*b^7 + 4*b^8*exp(x) - 2*a*b^2*((a^2 + b^2)^5)^(1/2) + 4*b^3*exp(x)*((a^2 + b^2)^5)^(1/2) + 13*a^2*b^6*exp(x) + 15*a^4*b^4*exp(x) + 7*a^6*b^2*exp(x ) + 3*a^2*b*exp(x)*((a^2 + b^2)^5)^(1/2))*((a^2 + b^2)^5)^(1/2))/(a*b^5)
Time = 0.24 (sec) , antiderivative size = 1372, normalized size of antiderivative = 7.50 \[ \int \frac {\coth ^6(x)}{a+b \text {csch}(x)} \, dx =\text {Too large to display} \] Input:
int(coth(x)^6/(a+b*csch(x)),x)
Output:
( - 48*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2)) *a**4*i - 96*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b**2*i - 48*e**(8*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/s qrt(a**2 + b**2))*b**4*i + 192*e**(6*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*a**4*i + 384*e**(6*x)*sqrt(a**2 + b**2)*atan((e** x*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b**2*i + 192*e**(6*x)*sqrt(a**2 + b** 2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*b**4*i - 288*e**(4*x)*sqrt(a** 2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*a**4*i - 576*e**(4*x)*s qrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b**2*i - 28 8*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*b**4 *i + 192*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2 ))*a**4*i + 384*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b**2*i + 192*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*a*i + b* i)/sqrt(a**2 + b**2))*b**4*i - 48*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/ sqrt(a**2 + b**2))*a**4*i - 96*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/sqr t(a**2 + b**2))*a**2*b**2*i - 48*sqrt(a**2 + b**2)*atan((e**x*a*i + b*i)/s qrt(a**2 + b**2))*b**4*i + 24*e**(8*x)*log(e**x - 1)*a**5 + 60*e**(8*x)*lo g(e**x - 1)*a**3*b**2 + 45*e**(8*x)*log(e**x - 1)*a*b**4 - 24*e**(8*x)*log (e**x + 1)*a**5 - 60*e**(8*x)*log(e**x + 1)*a**3*b**2 - 45*e**(8*x)*log(e* *x + 1)*a*b**4 + 12*e**(8*x)*a**4*b + 36*e**(8*x)*a**2*b**3 + 24*e**(8*...