\(\int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx\) [124]

Optimal result

Integrand size = 13, antiderivative size = 119 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {csch}(x)}{b^5}+\frac {a \left (a^2+3 b^2\right ) \text {csch}^2(x)}{2 b^4}-\frac {\left (a^2+3 b^2\right ) \text {csch}^3(x)}{3 b^3}+\frac {a \text {csch}^4(x)}{4 b^2}-\frac {\text {csch}^5(x)}{5 b}+\frac {\left (a^2+b^2\right )^3 \log (a+b \text {csch}(x))}{a b^6}+\frac {\log (\sinh (x))}{a} \]

[Out]

-(a^4+3*a^2*b^2+3*b^4)*csch(x)/b^5+1/2*a*(a^2+3*b^2)*csch(x)^2/b^4-1/3*(a^2+3*b^2)*csch(x)^3/b^3+1/4*a*csch(x)
^4/b^2-1/5*csch(x)^5/b+(a^2+b^2)^3*ln(a+b*csch(x))/a/b^6+ln(sinh(x))/a

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 119, 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} \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\frac {\left (a^2+b^2\right )^3 \log (a+b \text {csch}(x))}{a b^6}+\frac {a \left (a^2+3 b^2\right ) \text {csch}^2(x)}{2 b^4}-\frac {\left (a^2+3 b^2\right ) \text {csch}^3(x)}{3 b^3}-\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {csch}(x)}{b^5}+\frac {a \text {csch}^4(x)}{4 b^2}+\frac {\log (\sinh (x))}{a}-\frac {\text {csch}^5(x)}{5 b} \]

[In]

Int[Coth[x]^7/(a + b*Csch[x]),x]

[Out]

-(((a^4 + 3*a^2*b^2 + 3*b^4)*Csch[x])/b^5) + (a*(a^2 + 3*b^2)*Csch[x]^2)/(2*b^4) - ((a^2 + 3*b^2)*Csch[x]^3)/(
3*b^3) + (a*Csch[x]^4)/(4*b^2) - Csch[x]^5/(5*b) + ((a^2 + b^2)^3*Log[a + b*Csch[x]])/(a*b^6) + Log[Sinh[x]]/a

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}& = \frac {\text {Subst}\left (\int \frac {\left (-b^2-x^2\right )^3}{x (a+x)} \, dx,x,b \text {csch}(x)\right )}{b^6} \\ & = \frac {\text {Subst}\left (\int \left (-a^4 \left (1+\frac {3 b^2 \left (a^2+b^2\right )}{a^4}\right )-\frac {b^6}{a x}+a \left (a^2+3 b^2\right ) x-\left (a^2+3 b^2\right ) x^2+a x^3-x^4+\frac {\left (a^2+b^2\right )^3}{a (a+x)}\right ) \, dx,x,b \text {csch}(x)\right )}{b^6} \\ & = -\frac {\left (a^4+3 a^2 b^2+3 b^4\right ) \text {csch}(x)}{b^5}+\frac {a \left (a^2+3 b^2\right ) \text {csch}^2(x)}{2 b^4}-\frac {\left (a^2+3 b^2\right ) \text {csch}^3(x)}{3 b^3}+\frac {a \text {csch}^4(x)}{4 b^2}-\frac {\text {csch}^5(x)}{5 b}+\frac {\left (a^2+b^2\right )^3 \log (a+b \text {csch}(x))}{a b^6}+\frac {\log (\sinh (x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.09 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\frac {-60 b \left (a^4+3 a^2 b^2+3 b^4\right ) \text {csch}(x)+30 a b^2 \left (a^2+3 b^2\right ) \text {csch}^2(x)-20 b^3 \left (a^2+3 b^2\right ) \text {csch}^3(x)+15 a b^4 \text {csch}^4(x)-12 b^5 \text {csch}^5(x)-60 a \left (a^4+3 a^2 b^2+3 b^4\right ) \log (\sinh (x))+\frac {60 \left (a^2+b^2\right )^3 \log (b+a \sinh (x))}{a}}{60 b^6} \]

[In]

Integrate[Coth[x]^7/(a + b*Csch[x]),x]

[Out]

(-60*b*(a^4 + 3*a^2*b^2 + 3*b^4)*Csch[x] + 30*a*b^2*(a^2 + 3*b^2)*Csch[x]^2 - 20*b^3*(a^2 + 3*b^2)*Csch[x]^3 +
15*a*b^4*Csch[x]^4 - 12*b^5*Csch[x]^5 - 60*a*(a^4 + 3*a^2*b^2 + 3*b^4)*Log[Sinh[x]] + (60*(a^2 + b^2)^3*Log[b
+ a*Sinh[x]])/a)/(60*b^6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(111)=222\).

Time = 3.74 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.64

method	result	size
default	\(\frac {\frac {b^{4} \tanh \left (\frac {x}{2}\right )^{5}}{5}+\frac {a \tanh \left (\frac {x}{2}\right )^{4} b^{3}}{2}+\frac {4 a^{2} b^{2} \tanh \left (\frac {x}{2}\right )^{3}}{3}+3 \tanh \left (\frac {x}{2}\right )^{3} b^{4}+4 a^{3} b \tanh \left (\frac {x}{2}\right )^{2}+10 b^{3} \tanh \left (\frac {x}{2}\right )^{2} a +16 a^{4} \tanh \left (\frac {x}{2}\right )+44 a^{2} b^{2} \tanh \left (\frac {x}{2}\right )+38 \tanh \left (\frac {x}{2}\right ) b^{4}}{32 b^{5}}+\frac {\left (32 a^{6}+96 a^{4} b^{2}+96 a^{2} b^{4}+32 b^{6}\right ) \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{32 a \,b^{6}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{160 b \tanh \left (\frac {x}{2}\right )^{5}}-\frac {4 a^{2}+9 b^{2}}{96 b^{3} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {16 a^{4}+44 a^{2} b^{2}+38 b^{4}}{32 b^{5} \tanh \left (\frac {x}{2}\right )}+\frac {a}{64 b^{2} \tanh \left (\frac {x}{2}\right )^{4}}+\frac {a \left (2 a^{2}+5 b^{2}\right )}{16 b^{4} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {a \left (a^{4}+3 a^{2} b^{2}+3 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{b^{6}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}\)	\(314\)
risch	\(-\frac {x}{a}-\frac {2 \,{\mathrm e}^{x} \left (15 a^{4} {\mathrm e}^{8 x}+45 a^{2} b^{2} {\mathrm e}^{8 x}+45 b^{4} {\mathrm e}^{8 x}-15 a^{3} b \,{\mathrm e}^{7 x}-45 a \,b^{3} {\mathrm e}^{7 x}-60 a^{4} {\mathrm e}^{6 x}-160 a^{2} b^{2} {\mathrm e}^{6 x}-120 b^{4} {\mathrm e}^{6 x}+45 a^{3} b \,{\mathrm e}^{5 x}+105 a \,b^{3} {\mathrm e}^{5 x}+90 a^{4} {\mathrm e}^{4 x}+230 a^{2} b^{2} {\mathrm e}^{4 x}+198 b^{4} {\mathrm e}^{4 x}-45 a^{3} b \,{\mathrm e}^{3 x}-105 a \,b^{3} {\mathrm e}^{3 x}-60 a^{4} {\mathrm e}^{2 x}-160 a^{2} b^{2} {\mathrm e}^{2 x}-120 b^{4} {\mathrm e}^{2 x}+15 a^{3} b \,{\mathrm e}^{x}+45 a \,b^{3} {\mathrm e}^{x}+15 a^{4}+45 a^{2} b^{2}+45 b^{4}\right )}{15 b^{5} \left ({\mathrm e}^{2 x}-1\right )^{5}}-\frac {a^{5} \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{6}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{2 x}-1\right )}{b^{2}}+\frac {a^{5} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{6}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{4}}+\frac {3 a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a}\)	\(366\)

[In]

int(coth(x)^7/(a+b*csch(x)),x,method=_RETURNVERBOSE)

[Out]

1/32/b^5*(1/5*b^4*tanh(1/2*x)^5+1/2*a*tanh(1/2*x)^4*b^3+4/3*a^2*b^2*tanh(1/2*x)^3+3*tanh(1/2*x)^3*b^4+4*a^3*b*
tanh(1/2*x)^2+10*b^3*tanh(1/2*x)^2*a+16*a^4*tanh(1/2*x)+44*a^2*b^2*tanh(1/2*x)+38*tanh(1/2*x)*b^4)+1/32/a/b^6*
(32*a^6+96*a^4*b^2+96*a^2*b^4+32*b^6)*ln(-tanh(1/2*x)^2*b+2*a*tanh(1/2*x)+b)-1/a*ln(tanh(1/2*x)+1)-1/160/b/tan
h(1/2*x)^5-1/96/b^3*(4*a^2+9*b^2)/tanh(1/2*x)^3-1/32*(16*a^4+44*a^2*b^2+38*b^4)/b^5/tanh(1/2*x)+1/64*a/b^2/tan
h(1/2*x)^4+1/16*a/b^4*(2*a^2+5*b^2)/tanh(1/2*x)^2-1/b^6*a*(a^4+3*a^2*b^2+3*b^4)*ln(tanh(1/2*x))-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. 4024 vs. \(2 (111) = 222\).

Time = 0.31 (sec) , antiderivative size = 4024, normalized size of antiderivative = 33.82 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^7/(a+b*csch(x)),x, algorithm="fricas")

[Out]

-1/15*(15*b^6*x*cosh(x)^10 + 15*b^6*x*sinh(x)^10 + 30*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x)^9 + 30*(5*b^6*x*co
sh(x) + a^5*b + 3*a^3*b^3 + 3*a*b^5)*sinh(x)^9 - 15*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x)^8 + 15*(45*b^6*x
*cosh(x)^2 - 5*b^6*x - 2*a^4*b^2 - 6*a^2*b^4 + 18*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x))*sinh(x)^8 - 40*(3*a^5
*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^7 + 40*(45*b^6*x*cosh(x)^3 - 3*a^5*b - 8*a^3*b^3 - 6*a*b^5 + 27*(a^5*b + 3*a
^3*b^3 + 3*a*b^5)*cosh(x)^2 - 3*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x))*sinh(x)^7 - 15*b^6*x + 30*(5*b^6*x
+ 3*a^4*b^2 + 7*a^2*b^4)*cosh(x)^6 + 10*(315*b^6*x*cosh(x)^4 + 15*b^6*x + 9*a^4*b^2 + 21*a^2*b^4 + 252*(a^5*b
+ 3*a^3*b^3 + 3*a*b^5)*cosh(x)^3 - 42*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x)^2 - 28*(3*a^5*b + 8*a^3*b^3 +
6*a*b^5)*cosh(x))*sinh(x)^6 + 4*(45*a^5*b + 115*a^3*b^3 + 99*a*b^5)*cosh(x)^5 + 4*(945*b^6*x*cosh(x)^5 + 45*a^
5*b + 115*a^3*b^3 + 99*a*b^5 + 945*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x)^4 - 210*(5*b^6*x + 2*a^4*b^2 + 6*a^2*
b^4)*cosh(x)^3 - 210*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^2 + 45*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x))
*sinh(x)^5 - 30*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x)^4 + 10*(315*b^6*x*cosh(x)^6 - 15*b^6*x - 9*a^4*b^2 -
21*a^2*b^4 + 378*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x)^5 - 105*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x)^4 -
140*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^3 + 45*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x)^2 + 2*(45*a^5*b +
115*a^3*b^3 + 99*a*b^5)*cosh(x))*sinh(x)^4 - 40*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^3 + 40*(45*b^6*x*cosh
(x)^7 + 63*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x)^6 - 3*a^5*b - 8*a^3*b^3 - 6*a*b^5 - 21*(5*b^6*x + 2*a^4*b^2 +
6*a^2*b^4)*cosh(x)^5 - 35*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^4 + 15*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*co
sh(x)^3 + (45*a^5*b + 115*a^3*b^3 + 99*a*b^5)*cosh(x)^2 - 3*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x))*sinh(x)
^3 + 15*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x)^2 + 5*(135*b^6*x*cosh(x)^8 + 216*(a^5*b + 3*a^3*b^3 + 3*a*b^
5)*cosh(x)^7 + 15*b^6*x - 84*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x)^6 + 6*a^4*b^2 + 18*a^2*b^4 - 168*(3*a^5
*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^5 + 90*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x)^4 + 8*(45*a^5*b + 115*a^3*b
^3 + 99*a*b^5)*cosh(x)^3 - 36*(5*b^6*x + 3*a^4*b^2 + 7*a^2*b^4)*cosh(x)^2 - 24*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)
*cosh(x))*sinh(x)^2 + 30*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x) - 15*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(
x)^10 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)*sinh(x)^9 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sinh(x)
^10 - 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^8 - 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 - 9*(a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^3 - (a^6 + 3*a
^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x))*sinh(x)^7 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^6 + 10*(a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6 + 21*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^4 - 14*(a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*cosh(x)^2)*sinh(x)^6 - a^6 - 3*a^4*b^2 - 3*a^2*b^4 - b^6 + 4*(63*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c
osh(x)^5 - 70*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^3 + 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x))*
sinh(x)^5 - 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^4 + 10*(21*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh
(x)^6 - a^6 - 3*a^4*b^2 - 3*a^2*b^4 - b^6 - 35*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^4 + 15*(a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^7 - 7*(a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^5 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^3 - (a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*cosh(x))*sinh(x)^3 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^2 + 5*(9*(a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6)*cosh(x)^8 - 28*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^6 + a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6 + 30*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^4 - 12*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^2)*s
inh(x)^2 + 10*((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^9 - 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^7
+ 6*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^5 - 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x)^3 + (a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*cosh(x))*sinh(x))*log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))) + 15*((a^6 + 3*a^4*b
^2 + 3*a^2*b^4)*cosh(x)^10 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)*sinh(x)^9 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4
)*sinh(x)^10 - 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^8 - 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 - 9*(a^6 + 3*a^4*b^2
+ 3*a^2*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 - (a^6 + 3*a^4*b^2 + 3*a^2*
b^4)*cosh(x))*sinh(x)^7 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 21*(a
^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 - 14*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^6 - a^6 - 3*a^4*b^
2 - 3*a^2*b^4 + 4*(63*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 - 70*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + 1
5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x))*sinh(x)^5 - 10*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 10*(21*(a^6
+ 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 - a^6 - 3*a^4*b^2 - 3*a^2*b^4 - 35*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4
+ 15*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^7 - 7*(a
^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 - (a^6 + 3*a^4*b^2 + 3*a^2*b
^4)*cosh(x))*sinh(x)^3 + 5*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2 + 5*(9*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x
)^8 - 28*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + a^6 + 3*a^4*b^2 + 3*a^2*b^4 + 30*(a^6 + 3*a^4*b^2 + 3*a^2*b
^4)*cosh(x)^4 - 12*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^2 + 10*((a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh
(x)^9 - 4*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^7 + 6*(a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 - 4*(a^6 + 3*a^4
*b^2 + 3*a^2*b^4)*cosh(x)^3 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4)*cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x)
)) + 10*(15*b^6*x*cosh(x)^9 + 27*(a^5*b + 3*a^3*b^3 + 3*a*b^5)*cosh(x)^8 - 12*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4
)*cosh(x)^7 - 28*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^6 + 3*a^5*b + 9*a^3*b^3 + 9*a*b^5 + 18*(5*b^6*x + 3*a
^4*b^2 + 7*a^2*b^4)*cosh(x)^5 + 2*(45*a^5*b + 115*a^3*b^3 + 99*a*b^5)*cosh(x)^4 - 12*(5*b^6*x + 3*a^4*b^2 + 7*
a^2*b^4)*cosh(x)^3 - 12*(3*a^5*b + 8*a^3*b^3 + 6*a*b^5)*cosh(x)^2 + 3*(5*b^6*x + 2*a^4*b^2 + 6*a^2*b^4)*cosh(x
))*sinh(x))/(a*b^6*cosh(x)^10 + 10*a*b^6*cosh(x)*sinh(x)^9 + a*b^6*sinh(x)^10 - 5*a*b^6*cosh(x)^8 + 10*a*b^6*c
osh(x)^6 - 10*a*b^6*cosh(x)^4 + 5*a*b^6*cosh(x)^2 + 5*(9*a*b^6*cosh(x)^2 - a*b^6)*sinh(x)^8 + 40*(3*a*b^6*cosh
(x)^3 - a*b^6*cosh(x))*sinh(x)^7 - a*b^6 + 10*(21*a*b^6*cosh(x)^4 - 14*a*b^6*cosh(x)^2 + a*b^6)*sinh(x)^6 + 4*
(63*a*b^6*cosh(x)^5 - 70*a*b^6*cosh(x)^3 + 15*a*b^6*cosh(x))*sinh(x)^5 + 10*(21*a*b^6*cosh(x)^6 - 35*a*b^6*cos
h(x)^4 + 15*a*b^6*cosh(x)^2 - a*b^6)*sinh(x)^4 + 40*(3*a*b^6*cosh(x)^7 - 7*a*b^6*cosh(x)^5 + 5*a*b^6*cosh(x)^3
- a*b^6*cosh(x))*sinh(x)^3 + 5*(9*a*b^6*cosh(x)^8 - 28*a*b^6*cosh(x)^6 + 30*a*b^6*cosh(x)^4 - 12*a*b^6*cosh(x
)^2 + a*b^6)*sinh(x)^2 + 10*(a*b^6*cosh(x)^9 - 4*a*b^6*cosh(x)^7 + 6*a*b^6*cosh(x)^5 - 4*a*b^6*cosh(x)^3 + a*b
^6*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\coth ^{7}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

integrate(coth(x)**7/(a+b*csch(x)),x)

[Out]

Integral(coth(x)**7/(a + b*csch(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (111) = 222\).

Time = 0.21 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.06 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\frac {2 \, {\left (15 \, {\left (a^{4} + 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-x\right )} - 15 \, {\left (a^{3} b + 3 \, a b^{3}\right )} e^{\left (-2 \, x\right )} - 20 \, {\left (3 \, a^{4} + 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-3 \, x\right )} + 15 \, {\left (3 \, a^{3} b + 7 \, a b^{3}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (45 \, a^{4} + 115 \, a^{2} b^{2} + 99 \, b^{4}\right )} e^{\left (-5 \, x\right )} - 15 \, {\left (3 \, a^{3} b + 7 \, a b^{3}\right )} e^{\left (-6 \, x\right )} - 20 \, {\left (3 \, a^{4} + 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-7 \, x\right )} + 15 \, {\left (a^{3} b + 3 \, a b^{3}\right )} e^{\left (-8 \, x\right )} + 15 \, {\left (a^{4} + 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-9 \, x\right )}\right )}}{15 \, {\left (5 \, b^{5} e^{\left (-2 \, x\right )} - 10 \, b^{5} e^{\left (-4 \, x\right )} + 10 \, b^{5} e^{\left (-6 \, x\right )} - 5 \, b^{5} e^{\left (-8 \, x\right )} + b^{5} e^{\left (-10 \, x\right )} - b^{5}\right )}} + \frac {x}{a} - \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{b^{6}} - \frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a b^{6}} \]

[In]

integrate(coth(x)^7/(a+b*csch(x)),x, algorithm="maxima")

[Out]

2/15*(15*(a^4 + 3*a^2*b^2 + 3*b^4)*e^(-x) - 15*(a^3*b + 3*a*b^3)*e^(-2*x) - 20*(3*a^4 + 8*a^2*b^2 + 6*b^4)*e^(
-3*x) + 15*(3*a^3*b + 7*a*b^3)*e^(-4*x) + 2*(45*a^4 + 115*a^2*b^2 + 99*b^4)*e^(-5*x) - 15*(3*a^3*b + 7*a*b^3)*
e^(-6*x) - 20*(3*a^4 + 8*a^2*b^2 + 6*b^4)*e^(-7*x) + 15*(a^3*b + 3*a*b^3)*e^(-8*x) + 15*(a^4 + 3*a^2*b^2 + 3*b
^4)*e^(-9*x))/(5*b^5*e^(-2*x) - 10*b^5*e^(-4*x) + 10*b^5*e^(-6*x) - 5*b^5*e^(-8*x) + b^5*e^(-10*x) - b^5) + x/
a - (a^5 + 3*a^3*b^2 + 3*a*b^4)*log(e^(-x) + 1)/b^6 - (a^5 + 3*a^3*b^2 + 3*a*b^4)*log(e^(-x) - 1)/b^6 + (a^6 +
3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(-2*b*e^(-x) + a*e^(-2*x) - a)/(a*b^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (111) = 222\).

Time = 0.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.48 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=-\frac {{\left (a^{5} + 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left ({\left | -e^{\left (-x\right )} + e^{x} \right |}\right )}{b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a b^{6}} + \frac {137 \, a^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 411 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 411 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5} + 120 \, a^{4} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 360 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 360 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 120 \, a^{3} b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 360 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 160 \, a^{2} b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 480 \, b^{5} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 240 \, a b^{4} {\left (e^{\left (-x\right )} - e^{x}\right )} + 384 \, b^{5}}{60 \, b^{6} {\left (e^{\left (-x\right )} - e^{x}\right )}^{5}} \]

[In]

integrate(coth(x)^7/(a+b*csch(x)),x, algorithm="giac")

[Out]

-(a^5 + 3*a^3*b^2 + 3*a*b^4)*log(abs(-e^(-x) + e^x))/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(-a*(e^(
-x) - e^x) + 2*b))/(a*b^6) + 1/60*(137*a^5*(e^(-x) - e^x)^5 + 411*a^3*b^2*(e^(-x) - e^x)^5 + 411*a*b^4*(e^(-x)
- e^x)^5 + 120*a^4*b*(e^(-x) - e^x)^4 + 360*a^2*b^3*(e^(-x) - e^x)^4 + 360*b^5*(e^(-x) - e^x)^4 + 120*a^3*b^2
*(e^(-x) - e^x)^3 + 360*a*b^4*(e^(-x) - e^x)^3 + 160*a^2*b^3*(e^(-x) - e^x)^2 + 480*b^5*(e^(-x) - e^x)^2 + 240
*a*b^4*(e^(-x) - e^x) + 384*b^5)/(b^6*(e^(-x) - e^x)^5)

Mupad [B] (verification not implemented)

Time = 2.78 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.66 \[ \int \frac {\coth ^7(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {4\,a}{b^2}-\frac {64\,{\mathrm {e}}^x}{5\,b}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {8\,a}{b^2}-\frac {8\,{\mathrm {e}}^x\,\left (5\,a^2+27\,b^2\right )}{15\,b^3}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {\frac {8\,{\mathrm {e}}^x\,\left (a^2+3\,b^2\right )}{3\,b^3}-\frac {2\,\left (a^4+5\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {x}{a}-\frac {\frac {2\,{\mathrm {e}}^x\,\left (a^4+3\,a^2\,b^2+3\,b^4\right )}{b^5}-\frac {2\,\left (a^4+3\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{2\,x}-1}-\frac {32\,{\mathrm {e}}^x}{5\,b\,\left (5\,{\mathrm {e}}^{2\,x}-10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}-5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}-1\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,x}-1\right )\,\left (a^5+3\,a^3\,b^2+3\,a\,b^4\right )}{b^6}+\frac {\ln \left (2\,b\,{\mathrm {e}}^x-a+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}{a\,b^6} \]

[In]

int(coth(x)^7/(a + b/sinh(x)),x)

[Out]

((4*a)/b^2 - (64*exp(x))/(5*b))/(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1) + ((8*a)/b^2 - (8*exp(x)
*(5*a^2 + 27*b^2))/(15*b^3))/(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1) - ((8*exp(x)*(a^2 + 3*b^2))/(3*b^3) - (2
*(a^4 + 5*a^2*b^2))/(a*b^4))/(exp(4*x) - 2*exp(2*x) + 1) - x/a - ((2*exp(x)*(a^4 + 3*b^4 + 3*a^2*b^2))/b^5 - (
2*(a^4 + 3*a^2*b^2))/(a*b^4))/(exp(2*x) - 1) - (32*exp(x))/(5*b*(5*exp(2*x) - 10*exp(4*x) + 10*exp(6*x) - 5*ex
p(8*x) + exp(10*x) - 1)) - (log(exp(2*x) - 1)*(3*a*b^4 + a^5 + 3*a^3*b^2))/b^6 + (log(2*b*exp(x) - a + a*exp(2
*x))*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))/(a*b^6)