\(\int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx\) [124]

Optimal result

Integrand size = 13, antiderivative size = 178 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text {sech}(x))}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {1}{16 (a-b) (1+\text {sech}(x))^2}-\frac {5 a-7 b}{16 (a-b)^2 (1+\text {sech}(x))} \]

[Out]

ln(cosh(x))/a+1/16*(8*a^2+21*a*b+15*b^2)*ln(1-sech(x))/(a+b)^3+1/16*(8*a^2-21*a*b+15*b^2)*ln(1+sech(x))/(a-b)^
3-b^6*ln(a+b*sech(x))/a/(a^2-b^2)^3-1/16/(a+b)/(1-sech(x))^2+1/16*(-5*a-7*b)/(a+b)^2/(1-sech(x))-1/16/(a-b)/(1
+sech(x))^2+1/16*(-5*a+7*b)/(a-b)^2/(1+sech(x))

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3970, 908} \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (\text {sech}(x)+1)}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {5 a-7 b}{16 (a-b)^2 (\text {sech}(x)+1)}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {1}{16 (a-b) (\text {sech}(x)+1)^2}+\frac {\log (\cosh (x))}{a} \]

[In]

Int[Coth[x]^5/(a + b*Sech[x]),x]

[Out]

Log[Cosh[x]]/a + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sech[x]])/(16*(a + b)^3) + ((8*a^2 - 21*a*b + 15*b^2)*Log[
1 + Sech[x]])/(16*(a - b)^3) - (b^6*Log[a + b*Sech[x]])/(a*(a^2 - b^2)^3) - 1/(16*(a + b)*(1 - Sech[x])^2) - (
5*a + 7*b)/(16*(a + b)^2*(1 - Sech[x])) - 1/(16*(a - b)*(1 + Sech[x])^2) - (5*a - 7*b)/(16*(a - b)^2*(1 + Sech
[x]))

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (b^6 \text {Subst}\left (\int \frac {1}{x (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \text {sech}(x)\right )\right ) \\ & = -\left (b^6 \text {Subst}\left (\int \left (\frac {1}{8 b^4 (a+b) (b-x)^3}+\frac {5 a+7 b}{16 b^5 (a+b)^2 (b-x)^2}+\frac {8 a^2+21 a b+15 b^2}{16 b^6 (a+b)^3 (b-x)}+\frac {1}{a b^6 x}+\frac {1}{a (a-b)^3 (a+b)^3 (a+x)}+\frac {1}{8 b^4 (-a+b) (b+x)^3}+\frac {-5 a+7 b}{16 (a-b)^2 b^5 (b+x)^2}+\frac {8 a^2-21 a b+15 b^2}{16 b^6 (-a+b)^3 (b+x)}\right ) \, dx,x,b \text {sech}(x)\right )\right ) \\ & = \frac {\log (\cosh (x))}{a}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text {sech}(x))}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {1}{16 (a-b) (1+\text {sech}(x))^2}-\frac {5 a-7 b}{16 (a-b)^2 (1+\text {sech}(x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.92 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {1}{16} \left (\frac {16 \log (\cosh (x))}{a}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{(a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text {sech}(x))}{(a-b)^3}-\frac {16 b^6 \log (a+b \text {sech}(x))}{a (a-b)^3 (a+b)^3}-\frac {1}{(a+b) (-1+\text {sech}(x))^2}+\frac {5 a+7 b}{(a+b)^2 (-1+\text {sech}(x))}-\frac {1}{(a-b) (1+\text {sech}(x))^2}+\frac {-5 a+7 b}{(a-b)^2 (1+\text {sech}(x))}\right ) \]

[In]

Integrate[Coth[x]^5/(a + b*Sech[x]),x]

[Out]

((16*Log[Cosh[x]])/a + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sech[x]])/(a + b)^3 + ((8*a^2 - 21*a*b + 15*b^2)*Log
[1 + Sech[x]])/(a - b)^3 - (16*b^6*Log[a + b*Sech[x]])/(a*(a - b)^3*(a + b)^3) - 1/((a + b)*(-1 + Sech[x])^2)
+ (5*a + 7*b)/((a + b)^2*(-1 + Sech[x])) - 1/((a - b)*(1 + Sech[x])^2) + (-5*a + 7*b)/((a - b)^2*(1 + Sech[x])
))/16

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {\left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +6 a -8 b \right )^{2}}{64 \left (a -b \right )^{3}}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}-\frac {6 a +8 b}{32 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\left (16 a^{2}+42 a b +30 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{16 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {b^{6} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\)	\(162\)
risch	\(\frac {x}{a}-\frac {x \,a^{2}}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {21 x a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 x \,b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {x \,a^{2}}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}+\frac {21 x a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {15 x \,b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {2 x \,b^{6}}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a}-\frac {\left (-5 a^{2} b \,{\mathrm e}^{6 x}+9 b^{3} {\mathrm e}^{6 x}+16 a^{3} {\mathrm e}^{5 x}-24 a \,b^{2} {\mathrm e}^{5 x}-3 a^{2} b \,{\mathrm e}^{4 x}-b^{3} {\mathrm e}^{4 x}-16 a^{3} {\mathrm e}^{3 x}+32 a \,b^{2} {\mathrm e}^{3 x}-3 a^{2} b \,{\mathrm e}^{2 x}-b^{3} {\mathrm e}^{2 x}+16 a^{3} {\mathrm e}^{x}-24 \,{\mathrm e}^{x} b^{2} a -5 a^{2} b +9 b^{3}\right ) {\mathrm e}^{x}}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 x}-1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{x}-1\right ) a^{2}}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}+\frac {21 \ln \left ({\mathrm e}^{x}-1\right ) a b}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{x}-1\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\ln \left ({\mathrm e}^{x}+1\right ) a^{2}}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {21 \ln \left ({\mathrm e}^{x}+1\right ) a b}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{x}+1\right ) b^{2}}{8 \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) a}\)	\(598\)

[In]

int(coth(x)^5/(a+b*sech(x)),x,method=_RETURNVERBOSE)

[Out]

-1/64*(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b+6*a-8*b)^2/(a-b)^3-1/64/(a+b)/tanh(1/2*x)^4-1/32*(6*a+8*b)/(a+b)^2/tanh
(1/2*x)^2+1/16/(a+b)^3*(16*a^2+42*a*b+30*b^2)*ln(tanh(1/2*x))-1/a*ln(tanh(1/2*x)+1)-1/(a-b)^3*b^6/(a+b)^3/a*ln
(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b+a+b)-1/a*ln(tanh(1/2*x)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5181 vs. \(2 (162) = 324\).

Time = 0.37 (sec) , antiderivative size = 5181, normalized size of antiderivative = 29.11 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^5/(a+b*sech(x)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\coth ^{5}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]

[In]

integrate(coth(x)**5/(a+b*sech(x)),x)

[Out]

Integral(coth(x)**5/(a + b*sech(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (162) = 324\).

Time = 0.22 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.06 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=-\frac {b^{6} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} + {\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} + \frac {x}{a} \]

[In]

integrate(coth(x)^5/(a+b*sech(x)),x, algorithm="maxima")

[Out]

-b^6*log(2*b*e^(-x) + a*e^(-2*x) + a)/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6) + 1/8*(8*a^2 - 21*a*b + 15*b^2)*lo
g(e^(-x) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/8*(8*a^2 + 21*a*b + 15*b^2)*log(e^(-x) - 1)/(a^3 + 3*a^2*b +
3*a*b^2 + b^3) + 1/4*((5*a^2*b - 9*b^3)*e^(-x) - 8*(2*a^3 - 3*a*b^2)*e^(-2*x) + (3*a^2*b + b^3)*e^(-3*x) + 16
*(a^3 - 2*a*b^2)*e^(-4*x) + (3*a^2*b + b^3)*e^(-5*x) - 8*(2*a^3 - 3*a*b^2)*e^(-6*x) + (5*a^2*b - 9*b^3)*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)) + x/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (162) = 324\).

Time = 0.29 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.13 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=-\frac {b^{6} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {3 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 9 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 9 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 5 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 14 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 9 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 8 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 32 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )} - 40 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 28 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )} - 16 \, a^{3} b^{2} + 64 \, a b^{4}}{4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \]

[In]

integrate(coth(x)^5/(a+b*sech(x)),x, algorithm="giac")

[Out]

-b^6*log(abs(a*(e^(-x) + e^x) + 2*b))/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6) + 1/16*(8*a^2 - 21*a*b + 15*b^2)*l
og(e^(-x) + e^x + 2)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/16*(8*a^2 + 21*a*b + 15*b^2)*log(e^(-x) + e^x - 2)/(a
^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(3*a^5*(e^(-x) + e^x)^4 - 9*a^3*b^2*(e^(-x) + e^x)^4 + 9*a*b^4*(e^(-x) + e
^x)^4 - 5*a^4*b*(e^(-x) + e^x)^3 + 14*a^2*b^3*(e^(-x) + e^x)^3 - 9*b^5*(e^(-x) + e^x)^3 - 8*a^5*(e^(-x) + e^x)
^2 + 32*a^3*b^2*(e^(-x) + e^x)^2 - 48*a*b^4*(e^(-x) + e^x)^2 + 12*a^4*b*(e^(-x) + e^x) - 40*a^2*b^3*(e^(-x) +
e^x) + 28*b^5*(e^(-x) + e^x) - 16*a^3*b^2 + 64*a*b^4)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((e^(-x) + e^x)^2 -
4)^2)

Mupad [B] (verification not implemented)

Time = 3.47 (sec) , antiderivative size = 623, normalized size of antiderivative = 3.50 \[ \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx=\frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (8\,a^2+21\,a\,b+15\,b^2\right )}{8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3}-\frac {\frac {2\,\left (4\,a^4-5\,a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (9\,a^2\,b-13\,b^3\right )}{2\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (2\,a^6-5\,a^4\,b^2+3\,a^2\,b^4\right )}{a\,{\left (a^2-b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (5\,a^4\,b-14\,a^2\,b^3+9\,b^5\right )}{4\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {8\,\left (a^4-a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {6\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {x}{a}-\frac {\frac {4\,a}{a^2-b^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^2-b^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (8\,a^2-21\,a\,b+15\,b^2\right )}{8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3}+\frac {b^6\,\ln \left (64\,a^{13}\,{\mathrm {e}}^{2\,x}+64\,a\,b^{12}+64\,a^{13}+159\,a^3\,b^{10}+492\,a^5\,b^8-1214\,a^7\,b^6+1020\,a^9\,b^4-393\,a^{11}\,b^2+128\,b^{13}\,{\mathrm {e}}^x+159\,a^3\,b^{10}\,{\mathrm {e}}^{2\,x}+492\,a^5\,b^8\,{\mathrm {e}}^{2\,x}-1214\,a^7\,b^6\,{\mathrm {e}}^{2\,x}+1020\,a^9\,b^4\,{\mathrm {e}}^{2\,x}-393\,a^{11}\,b^2\,{\mathrm {e}}^{2\,x}+128\,a^{12}\,b\,{\mathrm {e}}^x+64\,a\,b^{12}\,{\mathrm {e}}^{2\,x}+318\,a^2\,b^{11}\,{\mathrm {e}}^x+984\,a^4\,b^9\,{\mathrm {e}}^x-2428\,a^6\,b^7\,{\mathrm {e}}^x+2040\,a^8\,b^5\,{\mathrm {e}}^x-786\,a^{10}\,b^3\,{\mathrm {e}}^x\right )}{-a^7+3\,a^5\,b^2-3\,a^3\,b^4+a\,b^6} \]

[In]

int(coth(x)^5/(a + b/cosh(x)),x)

[Out]

(log(exp(x) - 1)*(21*a*b + 8*a^2 + 15*b^2))/(24*a*b^2 + 24*a^2*b + 8*a^3 + 8*b^3) - ((2*(4*a^4 - 5*a^2*b^2))/(
a*(a^2 - b^2)^2) - (exp(x)*(9*a^2*b - 13*b^3))/(2*(a^2 - b^2)^2))/(exp(4*x) - 2*exp(2*x) + 1) - ((2*(2*a^6 + 3
*a^2*b^4 - 5*a^4*b^2))/(a*(a^2 - b^2)^3) - (exp(x)*(5*a^4*b + 9*b^5 - 14*a^2*b^3))/(4*(a^2 - b^2)^3))/(exp(2*x
) - 1) - ((8*(a^4 - a^2*b^2))/(a*(a^2 - b^2)^2) - (6*exp(x)*(a^2*b - b^3))/(a^2 - b^2)^2)/(3*exp(2*x) - 3*exp(
4*x) + exp(6*x) - 1) - x/a - ((4*a)/(a^2 - b^2) - (4*b*exp(x))/(a^2 - b^2))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6
*x) + exp(8*x) + 1) + (log(exp(x) + 1)*(8*a^2 - 21*a*b + 15*b^2))/(24*a*b^2 - 24*a^2*b + 8*a^3 - 8*b^3) + (b^6
*log(64*a^13*exp(2*x) + 64*a*b^12 + 64*a^13 + 159*a^3*b^10 + 492*a^5*b^8 - 1214*a^7*b^6 + 1020*a^9*b^4 - 393*a
^11*b^2 + 128*b^13*exp(x) + 159*a^3*b^10*exp(2*x) + 492*a^5*b^8*exp(2*x) - 1214*a^7*b^6*exp(2*x) + 1020*a^9*b^
4*exp(2*x) - 393*a^11*b^2*exp(2*x) + 128*a^12*b*exp(x) + 64*a*b^12*exp(2*x) + 318*a^2*b^11*exp(x) + 984*a^4*b^
9*exp(x) - 2428*a^6*b^7*exp(x) + 2040*a^8*b^5*exp(x) - 786*a^10*b^3*exp(x)))/(a*b^6 - a^7 - 3*a^3*b^4 + 3*a^5*
b^2)