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

Optimal. Leaf size=178 \[ \frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (\text {sech}(x)+1)}{16 (a-b)^3}-\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {5 a-7 b}{16 (a-b)^2 (\text {sech}(x)+1)}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {1}{16 (a-b) (\text {sech}(x)+1)^2}+\frac {\log (\cosh (x))}{a} \]

[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]  time = 0.32, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3885, 894} \[ -\frac {b^6 \log (a+b \text {sech}(x))}{a \left (a^2-b^2\right )^3}+\frac {\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text {sech}(x))}{16 (a+b)^3}+\frac {\left (8 a^2-21 a b+15 b^2\right ) \log (\text {sech}(x)+1)}{16 (a-b)^3}-\frac {5 a+7 b}{16 (a+b)^2 (1-\text {sech}(x))}-\frac {5 a-7 b}{16 (a-b)^2 (\text {sech}(x)+1)}-\frac {1}{16 (a+b) (1-\text {sech}(x))^2}-\frac {1}{16 (a-b) (\text {sech}(x)+1)^2}+\frac {\log (\cosh (x))}{a} \]

Antiderivative was successfully verified.

[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 894

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 3885

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*} \int \frac {\coth ^5(x)}{a+b \text {sech}(x)} \, dx &=-\left (b^6 \operatorname {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 \operatorname {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]  time = 1.03, size = 167, normalized size = 0.94 \[ \frac {1}{64} \left (-\frac {8 \left (a \left (b \left (3 a^4-10 a^2 b^2+15 b^4\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )-8 a \left (a^4-3 a^2 b^2+3 b^4\right ) \log (\sinh (x))\right )+8 b^6 \log (a \cosh (x)+b)\right )}{a (a-b)^3 (a+b)^3}-\frac {\text {csch}^4\left (\frac {x}{2}\right )}{a+b}-\frac {2 (7 a+9 b) \text {csch}^2\left (\frac {x}{2}\right )}{(a+b)^2}-\frac {\text {sech}^4\left (\frac {x}{2}\right )}{a-b}+\frac {2 (7 a-9 b) \text {sech}^2\left (\frac {x}{2}\right )}{(a-b)^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(7*a + 9*b)*Csch[x/2]^2)/(a + b)^2 - Csch[x/2]^4/(a + b) - (8*(8*b^6*Log[b + a*Cosh[x]] + a*(-8*a*(a^4 -
3*a^2*b^2 + 3*b^4)*Log[Sinh[x]] + b*(3*a^4 - 10*a^2*b^2 + 15*b^4)*Log[Tanh[x/2]])))/(a*(a - b)^3*(a + b)^3) +
(2*(7*a - 9*b)*Sech[x/2]^2)/(a - b)^2 - Sech[x/2]^4/(a - b))/64

________________________________________________________________________________________

fricas [B]  time = 0.61, size = 5181, normalized size = 29.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^8 + 8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*sinh(x)^8 -
2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^7 - 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5 - 32*(a^6 - 3*a^4*b^2 + 3*a^2
*b^4 - b^6)*x*cosh(x))*sinh(x)^7 + 16*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x
)*cosh(x)^6 + 2*(16*a^6 - 40*a^4*b^2 + 24*a^2*b^4 + 112*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^2 - 16*(
a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x - 7*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x))*sinh(x)^6 - 2*(3*a^5*b - 2*
a^3*b^3 - a*b^5)*cosh(x)^5 - 2*(3*a^5*b - 2*a^3*b^3 - a*b^5 - 224*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x
)^3 + 21*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^2 - 48*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 +
3*a^2*b^4 - b^6)*x)*cosh(x))*sinh(x)^5 - 16*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 -
b^6)*x)*cosh(x)^4 - 2*(16*a^6 - 48*a^4*b^2 + 32*a^2*b^4 - 280*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^4
+ 35*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^3 - 120*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3
*a^2*b^4 - b^6)*x)*cosh(x)^2 - 24*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x + 5*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh
(x))*sinh(x)^4 - 2*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^3 + 2*(224*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh
(x)^5 - 3*a^5*b + 2*a^3*b^3 + a*b^5 - 35*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^4 + 160*(2*a^6 - 5*a^4*b^2 +
3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^3 - 10*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^2 -
32*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x))*sinh(x)^3 + 16*(2*a^6 -
5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^2 + 2*(112*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*x*cosh(x)^6 + 16*a^6 - 40*a^4*b^2 + 24*a^2*b^4 - 21*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^5 + 12
0*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^4 - 10*(3*a^5*b - 2*a^3*b^
3 - a*b^5)*cosh(x)^3 - 48*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^2
- 16*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x - 3*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x))*sinh(x)^2 + 8*(a^6 - 3*a
^4*b^2 + 3*a^2*b^4 - b^6)*x - 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x) + 8*(b^6*cosh(x)^8 + 8*b^6*cosh(x)*si
nh(x)^7 + b^6*sinh(x)^8 - 4*b^6*cosh(x)^6 + 6*b^6*cosh(x)^4 - 4*b^6*cosh(x)^2 + 4*(7*b^6*cosh(x)^2 - b^6)*sinh
(x)^6 + b^6 + 8*(7*b^6*cosh(x)^3 - 3*b^6*cosh(x))*sinh(x)^5 + 2*(35*b^6*cosh(x)^4 - 30*b^6*cosh(x)^2 + 3*b^6)*
sinh(x)^4 + 8*(7*b^6*cosh(x)^5 - 10*b^6*cosh(x)^3 + 3*b^6*cosh(x))*sinh(x)^3 + 4*(7*b^6*cosh(x)^6 - 15*b^6*cos
h(x)^4 + 9*b^6*cosh(x)^2 - b^6)*sinh(x)^2 + 8*(b^6*cosh(x)^7 - 3*b^6*cosh(x)^5 + 3*b^6*cosh(x)^3 - b^6*cosh(x)
)*sinh(x))*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) - ((8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b
^4 + 15*a*b^5)*cosh(x)^8 + 8*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)*sinh(
x)^7 + (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*sinh(x)^8 - 4*(8*a^6 + 3*a^5*b - 24
*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^6 - 4*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a
^2*b^4 + 15*a*b^5 - 7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^2)*sinh(x)^6
+ 8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5 + 8*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10
*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 - 3*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a
*b^5)*cosh(x))*sinh(x)^5 + 6*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 + 2
*(24*a^6 + 9*a^5*b - 72*a^4*b^2 - 30*a^3*b^3 + 72*a^2*b^4 + 45*a*b^5 + 35*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a
^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 - 30*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*
b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^5
- 10*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 + 3*(8*a^6 + 3*a^5*b - 24*
a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x))*sinh(x)^3 - 4*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^
3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^2 + 4*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5
)*cosh(x)^6 - 8*a^6 - 3*a^5*b + 24*a^4*b^2 + 10*a^3*b^3 - 24*a^2*b^4 - 15*a*b^5 - 15*(8*a^6 + 3*a^5*b - 24*a^4
*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 + 9*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b
^4 + 15*a*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*c
osh(x)^7 - 3*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^5 + 3*(8*a^6 + 3*a^5*
b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 - (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 +
24*a^2*b^4 + 15*a*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - ((8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*
b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^8 + 8*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)
*cosh(x)*sinh(x)^7 + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*sinh(x)^8 - 4*(8*a^6
- 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^6 - 4*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*
a^3*b^3 + 24*a^2*b^4 - 15*a*b^5 - 7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x
)^2)*sinh(x)^6 + 8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5 + 8*(7*(8*a^6 - 3*a^5*b - 2
4*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 - 3*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*
a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x)^5 + 6*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)
*cosh(x)^4 + 2*(24*a^6 - 9*a^5*b - 72*a^4*b^2 + 30*a^3*b^3 + 72*a^2*b^4 - 45*a*b^5 + 35*(8*a^6 - 3*a^5*b - 24*
a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^4 - 30*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a
^2*b^4 - 15*a*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*
b^5)*cosh(x)^5 - 10*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 + 3*(8*a^6 -
3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x)^3 - 4*(8*a^6 - 3*a^5*b - 24*a^4*b
^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^2 + 4*(7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*
b^4 - 15*a*b^5)*cosh(x)^6 - 8*a^6 + 3*a^5*b + 24*a^4*b^2 - 10*a^3*b^3 - 24*a^2*b^4 + 15*a*b^5 - 15*(8*a^6 - 3*
a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^4 + 9*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*
b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4
- 15*a*b^5)*cosh(x)^7 - 3*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^5 + 3*(
8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 - (8*a^6 - 3*a^5*b - 24*a^4*b^2 +
10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(32*(a^6 - 3*a^4*b^2 + 3
*a^2*b^4 - b^6)*x*cosh(x)^7 - 7*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^6 - 5*a^5*b + 14*a^3*b^3 - 9*a*b^5 +
48*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^5 - 5*(3*a^5*b - 2*a^3*b^
3 - a*b^5)*cosh(x)^4 - 32*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^3
- 3*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^2 + 16*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b
^4 - b^6)*x)*cosh(x))*sinh(x))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^8 + 8*(a^7 - 3*a^5*b^2 + 3*a^3*b
^4 - a*b^6)*cosh(x)*sinh(x)^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*sinh(x)^8 + a^7 - 3*a^5*b^2 + 3*a^3*b^4
- a*b^6 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^6 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - 7*(a^7
- 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 -
3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x)^5 + 6*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^
4 + 2*(3*a^7 - 9*a^5*b^2 + 9*a^3*b^4 - 3*a*b^6 + 35*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^4 - 30*(a^7
- 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^5 -
10*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 + 3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x)
^3 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - 7*(a^7 - 3*a
^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^6 + 15*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^4 - 9*(a^7 - 3*a^5*b^
2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^2 + 8*((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^7 - 3*(a^7 - 3*
a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^5 + 3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 - (a^7 - 3*a^5*b^2
+ 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x))

________________________________________________________________________________________

giac [B]  time = 0.14, size = 380, normalized size = 2.13 \[ -\frac {b^{6} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {3 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 9 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 9 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 5 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 14 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 9 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 8 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 32 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )} - 40 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )} + 28 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )} - 16 \, a^{3} b^{2} + 64 \, a b^{4}}{4 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.17, size = 215, normalized size = 1.21 \[ -\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) a}{64 \left (a -b \right )^{2}}+\frac {\left (\tanh ^{4}\left (\frac {x}{2}\right )\right ) b}{64 \left (a -b \right )^{2}}-\frac {3 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) a}{16 \left (a -b \right )^{2}}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b}{4 \left (a -b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {b^{6} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {1}{64 \left (a +b \right ) \tanh \left (\frac {x}{2}\right )^{4}}-\frac {3 a}{16 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b}{4 \left (a +b \right )^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) a^{2}}{\left (a +b \right )^{3}}+\frac {21 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) a b}{8 \left (a +b \right )^{3}}+\frac {15 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) b^{2}}{8 \left (a +b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64/(a-b)^2*tanh(1/2*x)^4*a+1/64/(a-b)^2*tanh(1/2*x)^4*b-3/16/(a-b)^2*tanh(1/2*x)^2*a+1/4/(a-b)^2*tanh(1/2*x
)^2*b-1/a*ln(tanh(1/2*x)-1)-1/(a-b)^3*b^6/(a+b)^3/a*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)-1/a*ln(tanh(1/2*x)
+1)-1/64/(a+b)/tanh(1/2*x)^4-3/16/(a+b)^2/tanh(1/2*x)^2*a-1/4/(a+b)^2/tanh(1/2*x)^2*b+1/(a+b)^3*ln(tanh(1/2*x)
)*a^2+21/8/(a+b)^3*ln(tanh(1/2*x))*a*b+15/8/(a+b)^3*ln(tanh(1/2*x))*b^2

________________________________________________________________________________________

maxima [B]  time = 0.37, size = 366, normalized size = 2.06 \[ -\frac {b^{6} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac {{\left (8 \, a^{2} - 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {{\left (8 \, a^{2} + 21 \, a b + 15 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \, {\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 8 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} + {\left (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-7 \, x\right )}}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} + \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

mupad [B]  time = 2.75, size = 623, normalized size = 3.50 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (8\,a^2+21\,a\,b+15\,b^2\right )}{8\,a^3+24\,a^2\,b+24\,a\,b^2+8\,b^3}-\frac {\frac {2\,\left (4\,a^4-5\,a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (9\,a^2\,b-13\,b^3\right )}{2\,{\left (a^2-b^2\right )}^2}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (2\,a^6-5\,a^4\,b^2+3\,a^2\,b^4\right )}{a\,{\left (a^2-b^2\right )}^3}-\frac {{\mathrm {e}}^x\,\left (5\,a^4\,b-14\,a^2\,b^3+9\,b^5\right )}{4\,{\left (a^2-b^2\right )}^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {8\,\left (a^4-a^2\,b^2\right )}{a\,{\left (a^2-b^2\right )}^2}-\frac {6\,{\mathrm {e}}^x\,\left (a^2\,b-b^3\right )}{{\left (a^2-b^2\right )}^2}}{3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1}-\frac {x}{a}-\frac {\frac {4\,a}{a^2-b^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^2-b^2}}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (8\,a^2-21\,a\,b+15\,b^2\right )}{8\,a^3-24\,a^2\,b+24\,a\,b^2-8\,b^3}+\frac {b^6\,\ln \left (64\,a^{13}\,{\mathrm {e}}^{2\,x}+64\,a\,b^{12}+64\,a^{13}+159\,a^3\,b^{10}+492\,a^5\,b^8-1214\,a^7\,b^6+1020\,a^9\,b^4-393\,a^{11}\,b^2+128\,b^{13}\,{\mathrm {e}}^x+159\,a^3\,b^{10}\,{\mathrm {e}}^{2\,x}+492\,a^5\,b^8\,{\mathrm {e}}^{2\,x}-1214\,a^7\,b^6\,{\mathrm {e}}^{2\,x}+1020\,a^9\,b^4\,{\mathrm {e}}^{2\,x}-393\,a^{11}\,b^2\,{\mathrm {e}}^{2\,x}+128\,a^{12}\,b\,{\mathrm {e}}^x+64\,a\,b^{12}\,{\mathrm {e}}^{2\,x}+318\,a^2\,b^{11}\,{\mathrm {e}}^x+984\,a^4\,b^9\,{\mathrm {e}}^x-2428\,a^6\,b^7\,{\mathrm {e}}^x+2040\,a^8\,b^5\,{\mathrm {e}}^x-786\,a^{10}\,b^3\,{\mathrm {e}}^x\right )}{-a^7+3\,a^5\,b^2-3\,a^3\,b^4+a\,b^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{5}{\relax (x )}}{a + b \operatorname {sech}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________