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

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

[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)

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Rubi [A]  time = 0.158316, antiderivative size = 140, normalized size of antiderivative = 1., 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 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]

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]

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*}

Mathematica [A]  time = 0.550502, size = 137, normalized size = 0.98 \[ \frac{4 a b \left (-40 a^2 b^2+15 a^4+33 b^4\right ) \tanh (x)+3 b^4 \text{sech}^4(x) \left (-5 a^2+4 a b \tanh (x)+5 b^2\right )+2 b^2 \text{sech}^2(x) \left (15 \left (a^2-b^2\right )^2-2 a b \left (5 a^2-9 b^2\right ) \tanh (x)\right )+60 \left (a^2-b^2\right )^3 (\log (\cosh (x))-\log (a \cosh (x)+b \sinh (x)))+10 b^6 \text{sech}^6(x)}{60 b^7} \]

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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Maple [B]  time = 0.056, size = 925, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/b^6/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^7*a^5-52/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/(tan
h(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/(t
anh(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+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^4-3/b^5*
ln(tanh(1/2*x)^2+1)*a^4+1/b^7*ln(tanh(1/2*x)^2+1)*a^6-1/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^2-68/3/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+3/b^3*ln(tanh(1/2*x)^2+1)*a^2+20/b^3/(tanh(1/2
*x)^2+1)^6*tanh(1/2*x)^8*a^2-8/b^5/(tanh(1/2*x)^2+1)^6*tanh(1/2*x)^4*a^4+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+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 = 1.73043, size = 521, normalized size = 3.72 \begin{align*} \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}} \end{align*}

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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Fricas [B]  time = 3.1661, size = 12186, normalized size = 87.04 \begin{align*} \text{result too large to display} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.23068, size = 801, normalized size = 5.72 \begin{align*} \text{result too large to display} \end{align*}

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)