3.102 \(\int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx\)

Optimal. Leaf size=140 \[ -\frac {\left (a^2-b^2\right )^3 \log (a+b \tanh (x))}{b^7}+\frac {a \left (a^2-3 b^2\right ) \tanh ^3(x)}{3 b^4}-\frac {\left (a^2-3 b^2\right ) \tanh ^4(x)}{4 b^3}+\frac {a \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh (x)}{b^6}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^2(x)}{2 b^5}+\frac {a \tanh ^5(x)}{5 b^2}-\frac {\tanh ^6(x)}{6 b} \]

[Out]

-(a^2-b^2)^3*ln(a+b*tanh(x))/b^7+a*(a^4-3*a^2*b^2+3*b^4)*tanh(x)/b^6-1/2*(a^4-3*a^2*b^2+3*b^4)*tanh(x)^2/b^5+1
/3*a*(a^2-3*b^2)*tanh(x)^3/b^4-1/4*(a^2-3*b^2)*tanh(x)^4/b^3+1/5*a*tanh(x)^5/b^2-1/6*tanh(x)^6/b

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Rubi [A]  time = 0.16, antiderivative size = 140, 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 = {3506, 697} \[ -\frac {\left (a^2-3 b^2\right ) \tanh ^4(x)}{4 b^3}+\frac {a \left (a^2-3 b^2\right ) \tanh ^3(x)}{3 b^4}-\frac {\left (-3 a^2 b^2+a^4+3 b^4\right ) \tanh ^2(x)}{2 b^5}+\frac {a \left (-3 a^2 b^2+a^4+3 b^4\right ) \tanh (x)}{b^6}-\frac {\left (a^2-b^2\right )^3 \log (a+b \tanh (x))}{b^7}+\frac {a \tanh ^5(x)}{5 b^2}-\frac {\tanh ^6(x)}{6 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^8/(a + b*Tanh[x]),x]

[Out]

-(((a^2 - b^2)^3*Log[a + b*Tanh[x]])/b^7) + (a*(a^4 - 3*a^2*b^2 + 3*b^4)*Tanh[x])/b^6 - ((a^4 - 3*a^2*b^2 + 3*
b^4)*Tanh[x]^2)/(2*b^5) + (a*(a^2 - 3*b^2)*Tanh[x]^3)/(3*b^4) - ((a^2 - 3*b^2)*Tanh[x]^4)/(4*b^3) + (a*Tanh[x]
^5)/(5*b^2) - Tanh[x]^6/(6*b)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 3506

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rubi steps

\begin {align*} \int \frac {\text {sech}^8(x)}{a+b \tanh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-\frac {x^2}{b^2}\right )^3}{a+x} \, dx,x,b \tanh (x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^5-3 a^3 b^2+3 a b^4}{b^6}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) x}{b^6}+\frac {a \left (a^2-3 b^2\right ) x^2}{b^6}+\frac {\left (-a^2+3 b^2\right ) x^3}{b^6}+\frac {a x^4}{b^6}-\frac {x^5}{b^6}+\frac {\left (-a^2+b^2\right )^3}{b^6 (a+x)}\right ) \, dx,x,b \tanh (x)\right )}{b}\\ &=-\frac {\left (a^2-b^2\right )^3 \log (a+b \tanh (x))}{b^7}+\frac {a \left (a^4-3 a^2 b^2+3 b^4\right ) \tanh (x)}{b^6}-\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \tanh ^2(x)}{2 b^5}+\frac {a \left (a^2-3 b^2\right ) \tanh ^3(x)}{3 b^4}-\frac {\left (a^2-3 b^2\right ) \tanh ^4(x)}{4 b^3}+\frac {a \tanh ^5(x)}{5 b^2}-\frac {\tanh ^6(x)}{6 b}\\ \end {align*}

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Mathematica [A]  time = 0.64, size = 137, normalized size = 0.98 \[ \frac {2 b^2 \text {sech}^2(x) \left (15 \left (a^2-b^2\right )^2-2 a b \left (5 a^2-9 b^2\right ) \tanh (x)\right )+60 \left (a^2-b^2\right )^3 (\log (\cosh (x))-\log (a \cosh (x)+b \sinh (x)))+3 b^4 \text {sech}^4(x) \left (-5 a^2+4 a b \tanh (x)+5 b^2\right )+4 a b \left (15 a^4-40 a^2 b^2+33 b^4\right ) \tanh (x)+10 b^6 \text {sech}^6(x)}{60 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^8/(a + b*Tanh[x]),x]

[Out]

(60*(a^2 - b^2)^3*(Log[Cosh[x]] - Log[a*Cosh[x] + b*Sinh[x]]) + 10*b^6*Sech[x]^6 + 4*a*b*(15*a^4 - 40*a^2*b^2
+ 33*b^4)*Tanh[x] + 3*b^4*Sech[x]^4*(-5*a^2 + 5*b^2 + 4*a*b*Tanh[x]) + 2*b^2*Sech[x]^2*(15*(a^2 - b^2)^2 - 2*a
*b*(5*a^2 - 9*b^2)*Tanh[x]))/(60*b^7)

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fricas [B]  time = 0.72, size = 5275, normalized size = 37.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8/(a+b*tanh(x)),x, algorithm="fricas")

[Out]

-1/15*(30*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^10 + 300*(a^5*b - a^4*b^2 - 2*a^3*b^
3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)*sinh(x)^9 + 30*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*si
nh(x)^10 + 30*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^8 + 30*(5*a^5*b - 4*a^
4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6 + 45*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)
*cosh(x)^2)*sinh(x)^8 + 240*(15*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^3 + (5*a^5*b -
4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x))*sinh(x)^7 + 20*(15*a^5*b - 9*a^4*b^2 - 40*a^3
*b^3 + 24*a^2*b^4 + 33*a*b^5 - 23*b^6)*cosh(x)^6 + 20*(15*a^5*b - 9*a^4*b^2 - 40*a^3*b^3 + 24*a^2*b^4 + 33*a*b
^5 - 23*b^6 + 315*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^4 + 42*(5*a^5*b - 4*a^4*b^2
- 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^2)*sinh(x)^6 + 30*a^5*b - 80*a^3*b^3 + 66*a*b^5 + 120*(63
*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^5 + 14*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10
*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^3 + (15*a^5*b - 9*a^4*b^2 - 40*a^3*b^3 + 24*a^2*b^4 + 33*a*b^5 - 23*b^6)*c
osh(x))*sinh(x)^5 + 60*(5*a^5*b - 2*a^4*b^2 - 14*a^3*b^3 + 5*a^2*b^4 + 13*a*b^5 - 3*b^6)*cosh(x)^4 + 60*(105*(
a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^6 + 5*a^5*b - 2*a^4*b^2 - 14*a^3*b^3 + 5*a^2*b^
4 + 13*a*b^5 - 3*b^6 + 35*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^4 + 5*(15*
a^5*b - 9*a^4*b^2 - 40*a^3*b^3 + 24*a^2*b^4 + 33*a*b^5 - 23*b^6)*cosh(x)^2)*sinh(x)^4 + 80*(45*(a^5*b - a^4*b^
2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x)^7 + 21*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b
^5 - 6*b^6)*cosh(x)^5 + 5*(15*a^5*b - 9*a^4*b^2 - 40*a^3*b^3 + 24*a^2*b^4 + 33*a*b^5 - 23*b^6)*cosh(x)^3 + 3*(
5*a^5*b - 2*a^4*b^2 - 14*a^3*b^3 + 5*a^2*b^4 + 13*a*b^5 - 3*b^6)*cosh(x))*sinh(x)^3 + 6*(25*a^5*b - 5*a^4*b^2
- 70*a^3*b^3 + 10*a^2*b^4 + 61*a*b^5 - 5*b^6)*cosh(x)^2 + 6*(225*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*
b^5 - b^6)*cosh(x)^8 + 140*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^6 + 25*a^
5*b - 5*a^4*b^2 - 70*a^3*b^3 + 10*a^2*b^4 + 61*a*b^5 - 5*b^6 + 50*(15*a^5*b - 9*a^4*b^2 - 40*a^3*b^3 + 24*a^2*
b^4 + 33*a*b^5 - 23*b^6)*cosh(x)^4 + 60*(5*a^5*b - 2*a^4*b^2 - 14*a^3*b^3 + 5*a^2*b^4 + 13*a*b^5 - 3*b^6)*cosh
(x)^2)*sinh(x)^2 + 15*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^12 + 12*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)
*cosh(x)*sinh(x)^11 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^12 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*c
osh(x)^10 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 11*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^1
0 + 20*(11*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh
(x)^9 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 33*(a^6 - 3
*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 18*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^8 + 24*(33*(
a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 30*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 5*(a^6 - 3*a
^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^7 + 20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 4*(231*(a^6
- 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 5*a^6 - 15*a^4*b^2 + 15*a^2*b^4 - 5*b^6 + 315*(a^6 - 3*a^4*b^2 + 3*
a^2*b^4 - b^6)*cosh(x)^4 + 105*(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 + 24*(33*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 63*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*
cosh(x)^5 + 35*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*
sinh(x)^5 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 15*(33*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh
(x)^8 + 84*(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 + 70*(a^6 - 3*a^4
*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 20*(11*(a^6
- 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^9 + 36*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 42*(a^6 - 3*a^4*
b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a
^2*b^4 - b^6)*cosh(x))*sinh(x)^3 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 6*(11*(a^6 - 3*a^4*b^2 +
3*a^2*b^4 - b^6)*cosh(x)^10 + 45*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 + 70*(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 + 50*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 1
5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^2 + 12*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^1
1 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^9 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 10*(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))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x) - sinh(x))) - 15*((a^6 - 3*a^4*
b^2 + 3*a^2*b^4 - b^6)*cosh(x)^12 + 12*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^11 + (a^6 - 3*a^4*b
^2 + 3*a^2*b^4 - b^6)*sinh(x)^12 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^10 + 6*(a^6 - 3*a^4*b^2 + 3*a
^2*b^4 - b^6 + 11*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^10 + 20*(11*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^9 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*cosh(x)^8 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 33*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^
4 + 18*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^8 + 24*(33*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*c
osh(x)^5 + 30*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*s
inh(x)^7 + 20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 4*(231*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(
x)^6 + 5*a^6 - 15*a^4*b^2 + 15*a^2*b^4 - 5*b^6 + 315*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 105*(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 + 24*(33*(a^6 - 3*a^4*
b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 63*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 35*(a^6 - 3*a^4*b^2 + 3*
a^2*b^4 - b^6)*cosh(x)^3 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^5 + 15*(a^6 - 3*a^4*b^2 + 3*
a^2*b^4 - b^6)*cosh(x)^4 + 15*(33*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 + 84*(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 + 70*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 +
20*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 20*(11*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(
x)^9 + 36*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 42*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 2
0*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^3 + 6
*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 6*(11*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^10 + 45*(a^
6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 + 70*(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 + 50*(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)^2 + 12*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^11 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 -
b^6)*cosh(x)^9 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*co
sh(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))*log(2*cosh(x)/(cosh(x) - sinh(x))) + 12*(25*(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6)*cosh(x
)^9 + 20*(5*a^5*b - 4*a^4*b^2 - 12*a^3*b^3 + 10*a^2*b^4 + 7*a*b^5 - 6*b^6)*cosh(x)^7 + 10*(15*a^5*b - 9*a^4*b^
2 - 40*a^3*b^3 + 24*a^2*b^4 + 33*a*b^5 - 23*b^6)*cosh(x)^5 + 20*(5*a^5*b - 2*a^4*b^2 - 14*a^3*b^3 + 5*a^2*b^4
+ 13*a*b^5 - 3*b^6)*cosh(x)^3 + (25*a^5*b - 5*a^4*b^2 - 70*a^3*b^3 + 10*a^2*b^4 + 61*a*b^5 - 5*b^6)*cosh(x))*s
inh(x))/(b^7*cosh(x)^12 + 12*b^7*cosh(x)*sinh(x)^11 + b^7*sinh(x)^12 + 6*b^7*cosh(x)^10 + 15*b^7*cosh(x)^8 + 2
0*b^7*cosh(x)^6 + 15*b^7*cosh(x)^4 + 6*(11*b^7*cosh(x)^2 + b^7)*sinh(x)^10 + 20*(11*b^7*cosh(x)^3 + 3*b^7*cosh
(x))*sinh(x)^9 + 6*b^7*cosh(x)^2 + 15*(33*b^7*cosh(x)^4 + 18*b^7*cosh(x)^2 + b^7)*sinh(x)^8 + 24*(33*b^7*cosh(
x)^5 + 30*b^7*cosh(x)^3 + 5*b^7*cosh(x))*sinh(x)^7 + b^7 + 4*(231*b^7*cosh(x)^6 + 315*b^7*cosh(x)^4 + 105*b^7*
cosh(x)^2 + 5*b^7)*sinh(x)^6 + 24*(33*b^7*cosh(x)^7 + 63*b^7*cosh(x)^5 + 35*b^7*cosh(x)^3 + 5*b^7*cosh(x))*sin
h(x)^5 + 15*(33*b^7*cosh(x)^8 + 84*b^7*cosh(x)^6 + 70*b^7*cosh(x)^4 + 20*b^7*cosh(x)^2 + b^7)*sinh(x)^4 + 20*(
11*b^7*cosh(x)^9 + 36*b^7*cosh(x)^7 + 42*b^7*cosh(x)^5 + 20*b^7*cosh(x)^3 + 3*b^7*cosh(x))*sinh(x)^3 + 6*(11*b
^7*cosh(x)^10 + 45*b^7*cosh(x)^8 + 70*b^7*cosh(x)^6 + 50*b^7*cosh(x)^4 + 15*b^7*cosh(x)^2 + b^7)*sinh(x)^2 + 1
2*(b^7*cosh(x)^11 + 5*b^7*cosh(x)^9 + 10*b^7*cosh(x)^7 + 10*b^7*cosh(x)^5 + 5*b^7*cosh(x)^3 + b^7*cosh(x))*sin
h(x))

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giac [B]  time = 0.15, size = 593, normalized size = 4.24 \[ -\frac {{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{7} + b^{8}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{7}} - \frac {147 \, a^{6} e^{\left (12 \, x\right )} - 441 \, a^{4} b^{2} e^{\left (12 \, x\right )} + 441 \, a^{2} b^{4} e^{\left (12 \, x\right )} - 147 \, b^{6} e^{\left (12 \, x\right )} + 882 \, a^{6} e^{\left (10 \, x\right )} + 120 \, a^{5} b e^{\left (10 \, x\right )} - 2766 \, a^{4} b^{2} e^{\left (10 \, x\right )} - 240 \, a^{3} b^{3} e^{\left (10 \, x\right )} + 2886 \, a^{2} b^{4} e^{\left (10 \, x\right )} + 120 \, a b^{5} e^{\left (10 \, x\right )} - 1002 \, b^{6} e^{\left (10 \, x\right )} + 2205 \, a^{6} e^{\left (8 \, x\right )} + 600 \, a^{5} b e^{\left (8 \, x\right )} - 7095 \, a^{4} b^{2} e^{\left (8 \, x\right )} - 1440 \, a^{3} b^{3} e^{\left (8 \, x\right )} + 7815 \, a^{2} b^{4} e^{\left (8 \, x\right )} + 840 \, a b^{5} e^{\left (8 \, x\right )} - 2925 \, b^{6} e^{\left (8 \, x\right )} + 2940 \, a^{6} e^{\left (6 \, x\right )} + 1200 \, a^{5} b e^{\left (6 \, x\right )} - 9540 \, a^{4} b^{2} e^{\left (6 \, x\right )} - 3200 \, a^{3} b^{3} e^{\left (6 \, x\right )} + 10740 \, a^{2} b^{4} e^{\left (6 \, x\right )} + 2640 \, a b^{5} e^{\left (6 \, x\right )} - 4780 \, b^{6} e^{\left (6 \, x\right )} + 2205 \, a^{6} e^{\left (4 \, x\right )} + 1200 \, a^{5} b e^{\left (4 \, x\right )} - 7095 \, a^{4} b^{2} e^{\left (4 \, x\right )} - 3360 \, a^{3} b^{3} e^{\left (4 \, x\right )} + 7815 \, a^{2} b^{4} e^{\left (4 \, x\right )} + 3120 \, a b^{5} e^{\left (4 \, x\right )} - 2925 \, b^{6} e^{\left (4 \, x\right )} + 882 \, a^{6} e^{\left (2 \, x\right )} + 600 \, a^{5} b e^{\left (2 \, x\right )} - 2766 \, a^{4} b^{2} e^{\left (2 \, x\right )} - 1680 \, a^{3} b^{3} e^{\left (2 \, x\right )} + 2886 \, a^{2} b^{4} e^{\left (2 \, x\right )} + 1464 \, a b^{5} e^{\left (2 \, x\right )} - 1002 \, b^{6} e^{\left (2 \, x\right )} + 147 \, a^{6} + 120 \, a^{5} b - 441 \, a^{4} b^{2} - 320 \, a^{3} b^{3} + 441 \, a^{2} b^{4} + 264 \, a b^{5} - 147 \, b^{6}}{60 \, b^{7} {\left (e^{\left (2 \, x\right )} + 1\right )}^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8/(a+b*tanh(x)),x, algorithm="giac")

[Out]

-(a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*log(abs(a*e^(2*x) + b*e^(2*x) + a
- b))/(a*b^7 + b^8) + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(e^(2*x) + 1)/b^7 - 1/60*(147*a^6*e^(12*x) - 441
*a^4*b^2*e^(12*x) + 441*a^2*b^4*e^(12*x) - 147*b^6*e^(12*x) + 882*a^6*e^(10*x) + 120*a^5*b*e^(10*x) - 2766*a^4
*b^2*e^(10*x) - 240*a^3*b^3*e^(10*x) + 2886*a^2*b^4*e^(10*x) + 120*a*b^5*e^(10*x) - 1002*b^6*e^(10*x) + 2205*a
^6*e^(8*x) + 600*a^5*b*e^(8*x) - 7095*a^4*b^2*e^(8*x) - 1440*a^3*b^3*e^(8*x) + 7815*a^2*b^4*e^(8*x) + 840*a*b^
5*e^(8*x) - 2925*b^6*e^(8*x) + 2940*a^6*e^(6*x) + 1200*a^5*b*e^(6*x) - 9540*a^4*b^2*e^(6*x) - 3200*a^3*b^3*e^(
6*x) + 10740*a^2*b^4*e^(6*x) + 2640*a*b^5*e^(6*x) - 4780*b^6*e^(6*x) + 2205*a^6*e^(4*x) + 1200*a^5*b*e^(4*x) -
7095*a^4*b^2*e^(4*x) - 3360*a^3*b^3*e^(4*x) + 7815*a^2*b^4*e^(4*x) + 3120*a*b^5*e^(4*x) - 2925*b^6*e^(4*x) +
882*a^6*e^(2*x) + 600*a^5*b*e^(2*x) - 2766*a^4*b^2*e^(2*x) - 1680*a^3*b^3*e^(2*x) + 2886*a^2*b^4*e^(2*x) + 146
4*a*b^5*e^(2*x) - 1002*b^6*e^(2*x) + 147*a^6 + 120*a^5*b - 441*a^4*b^2 - 320*a^3*b^3 + 441*a^2*b^4 + 264*a*b^5
- 147*b^6)/(b^7*(e^(2*x) + 1)^6)

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maple [B]  time = 0.15, size = 925, normalized size = 6.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^8/(a+b*tanh(x)),x)

[Out]

20/b^3/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^4*a^2-2/b^5/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^2*a^4+6/b^3/(tanh(1/2*x)^2+
1)^6*tanh(1/2*x)^2*a^2+2/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)*a^5-6/b^4/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)*a^3+6/b
^2/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)*a+10/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^9*a^5-82/3/b^4/(tanh(1/2*x)^2+1)^6
*tanh(1/2*x)^9*a^3+22/b^2/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^9*a+20/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^7*a^5+6/b
^3/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^10*a^2-8/b^5/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^8*a^4-52/b^4/(tanh(1/2*x)^2+1)
^6*tanh(1/2*x)^7*a^3+212/5/b^2/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^7*a-12/b^5/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^6*a^
4+28/b^3/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^6*a^2+20/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^5*a^5-52/b^4/(tanh(1/2*x
)^2+1)^6*tanh(1/2*x)^5*a^3+212/5/b^2/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^5*a+10/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x
)^3*a^5-82/3/b^4/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^3*a^3+22/b^2/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^3*a+2/b^6/(tanh(
1/2*x)^2+1)^6*tanh(1/2*x)^11*a^5-6/b^4/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^11*a^3+6/b^2/(tanh(1/2*x)^2+1)^6*tanh(1
/2*x)^11*a-2/b^5/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^10*a^4+20/b^3/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^8*a^2-8/b^5/(ta
nh(1/2*x)^2+1)^6*tanh(1/2*x)^4*a^4-1/b^7*ln(a*tanh(1/2*x)^2+2*tanh(1/2*x)*b+a)*a^6+3/b^5*ln(a*tanh(1/2*x)^2+2*
tanh(1/2*x)*b+a)*a^4-3/b^3*ln(a*tanh(1/2*x)^2+2*tanh(1/2*x)*b+a)*a^2-3/b^5*ln(tanh(1/2*x)^2+1)*a^4+3/b^3*ln(ta
nh(1/2*x)^2+1)*a^2-68/3/b/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^6-6/b/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^10-12/b/(tanh(
1/2*x)^2+1)^6*tanh(1/2*x)^8-12/b/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^4-6/b/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^2+1/b^7
*ln(tanh(1/2*x)^2+1)*a^6+1/b*ln(a*tanh(1/2*x)^2+2*tanh(1/2*x)*b+a)-1/b*ln(tanh(1/2*x)^2+1)

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maxima [B]  time = 0.43, size = 386, normalized size = 2.76 \[ \frac {2 \, {\left (15 \, a^{5} - 40 \, a^{3} b^{2} + 33 \, a b^{4} + 3 \, {\left (25 \, a^{5} + 5 \, a^{4} b - 70 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 61 \, a b^{4} + 5 \, b^{5}\right )} e^{\left (-2 \, x\right )} + 30 \, {\left (5 \, a^{5} + 2 \, a^{4} b - 14 \, a^{3} b^{2} - 5 \, a^{2} b^{3} + 13 \, a b^{4} + 3 \, b^{5}\right )} e^{\left (-4 \, x\right )} + 10 \, {\left (15 \, a^{5} + 9 \, a^{4} b - 40 \, a^{3} b^{2} - 24 \, a^{2} b^{3} + 33 \, a b^{4} + 23 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 15 \, {\left (5 \, a^{5} + 4 \, a^{4} b - 12 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 7 \, a b^{4} + 6 \, b^{5}\right )} e^{\left (-8 \, x\right )} + 15 \, {\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} e^{\left (-10 \, x\right )}\right )}}{15 \, {\left (6 \, b^{6} e^{\left (-2 \, x\right )} + 15 \, b^{6} e^{\left (-4 \, x\right )} + 20 \, b^{6} e^{\left (-6 \, x\right )} + 15 \, b^{6} e^{\left (-8 \, x\right )} + 6 \, b^{6} e^{\left (-10 \, x\right )} + b^{6} e^{\left (-12 \, x\right )} + b^{6}\right )}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{7}} + \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^8/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

2/15*(15*a^5 - 40*a^3*b^2 + 33*a*b^4 + 3*(25*a^5 + 5*a^4*b - 70*a^3*b^2 - 10*a^2*b^3 + 61*a*b^4 + 5*b^5)*e^(-2
*x) + 30*(5*a^5 + 2*a^4*b - 14*a^3*b^2 - 5*a^2*b^3 + 13*a*b^4 + 3*b^5)*e^(-4*x) + 10*(15*a^5 + 9*a^4*b - 40*a^
3*b^2 - 24*a^2*b^3 + 33*a*b^4 + 23*b^5)*e^(-6*x) + 15*(5*a^5 + 4*a^4*b - 12*a^3*b^2 - 10*a^2*b^3 + 7*a*b^4 + 6
*b^5)*e^(-8*x) + 15*(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*e^(-10*x))/(6*b^6*e^(-2*x) + 15*b^6*e^
(-4*x) + 20*b^6*e^(-6*x) + 15*b^6*e^(-8*x) + 6*b^6*e^(-10*x) + b^6*e^(-12*x) + b^6) - (a^6 - 3*a^4*b^2 + 3*a^2
*b^4 - b^6)*log(-(a - b)*e^(-2*x) - a - b)/b^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(e^(-2*x) + 1)/b^7

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mupad [B]  time = 1.43, size = 301, normalized size = 2.15 \[ \frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}-\frac {32\,\left (a-5\,b\right )}{5\,b^2\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}-\frac {4\,\left (a^2-4\,a\,b+7\,b^2\right )}{b^3\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^3\,{\left (a-b\right )}^3}{b^7}-\frac {32}{3\,b\,\left (6\,{\mathrm {e}}^{2\,x}+15\,{\mathrm {e}}^{4\,x}+20\,{\mathrm {e}}^{6\,x}+15\,{\mathrm {e}}^{8\,x}+6\,{\mathrm {e}}^{10\,x}+{\mathrm {e}}^{12\,x}+1\right )}-\frac {8\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{3\,b^4\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}-\frac {2\,{\left (a+b\right )}^2\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^6\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {2\,\left (a+b\right )\,\left (a-b\right )\,\left (a^2-2\,a\,b+b^2\right )}{b^5\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^8*(a + b*tanh(x))),x)

[Out]

(log(exp(2*x) + 1)*(a + b)^3*(a - b)^3)/b^7 - (32*(a - 5*b))/(5*b^2*(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) +
5*exp(8*x) + exp(10*x) + 1)) - (4*(a^2 - 4*a*b + 7*b^2))/(b^3*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x)
+ 1)) - (log(a - b + a*exp(2*x) + b*exp(2*x))*(a + b)^3*(a - b)^3)/b^7 - 32/(3*b*(6*exp(2*x) + 15*exp(4*x) +
20*exp(6*x) + 15*exp(8*x) + 6*exp(10*x) + exp(12*x) + 1)) - (8*(a - b)*(a^2 - 2*a*b + b^2))/(3*b^4*(3*exp(2*x)
+ 3*exp(4*x) + exp(6*x) + 1)) - (2*(a + b)^2*(a - b)*(a^2 - 2*a*b + b^2))/(b^6*(exp(2*x) + 1)) - (2*(a + b)*(
a - b)*(a^2 - 2*a*b + b^2))/(b^5*(2*exp(2*x) + exp(4*x) + 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{8}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**8/(a+b*tanh(x)),x)

[Out]

Integral(sech(x)**8/(a + b*tanh(x)), x)

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