3.114 \(\int \frac{\tanh ^6(x)}{a+b \text{sech}(x)} \, dx\)

Optimal. Leaf size=187 \[ \frac{a \left (a^2-3 b^2\right ) \tanh (x)}{b^4}-\frac{\left (a^2-3 b^2\right ) \tan ^{-1}(\sinh (x))}{2 b^3}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \tan ^{-1}(\sinh (x))}{b^5}-\frac{\left (a^2-3 b^2\right ) \tanh (x) \text{sech}(x)}{2 b^3}-\frac{a \tanh ^3(x)}{3 b^2}+\frac{a \tanh (x)}{b^2}+\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a b^5}+\frac{x}{a}-\frac{3 \tan ^{-1}(\sinh (x))}{8 b}-\frac{\tanh (x) \text{sech}^3(x)}{4 b}-\frac{3 \tanh (x) \text{sech}(x)}{8 b} \]

[Out]

x/a - (3*ArcTan[Sinh[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTan[Sinh[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTan[
Sinh[x]])/b^5 + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b^5) + (a*Tanh[
x])/b^2 + (a*(a^2 - 3*b^2)*Tanh[x])/b^4 - (3*Sech[x]*Tanh[x])/(8*b) - ((a^2 - 3*b^2)*Sech[x]*Tanh[x])/(2*b^3)
- (Sech[x]^3*Tanh[x])/(4*b) - (a*Tanh[x]^3)/(3*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.293451, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {3898, 2897, 2659, 205, 3770, 3767, 8, 3768} \[ \frac{a \left (a^2-3 b^2\right ) \tanh (x)}{b^4}-\frac{\left (a^2-3 b^2\right ) \tan ^{-1}(\sinh (x))}{2 b^3}-\frac{\left (-3 a^2 b^2+a^4+3 b^4\right ) \tan ^{-1}(\sinh (x))}{b^5}-\frac{\left (a^2-3 b^2\right ) \tanh (x) \text{sech}(x)}{2 b^3}-\frac{a \tanh ^3(x)}{3 b^2}+\frac{a \tanh (x)}{b^2}+\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{a b^5}+\frac{x}{a}-\frac{3 \tan ^{-1}(\sinh (x))}{8 b}-\frac{\tanh (x) \text{sech}^3(x)}{4 b}-\frac{3 \tanh (x) \text{sech}(x)}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^6/(a + b*Sech[x]),x]

[Out]

x/a - (3*ArcTan[Sinh[x]])/(8*b) - ((a^2 - 3*b^2)*ArcTan[Sinh[x]])/(2*b^3) - ((a^4 - 3*a^2*b^2 + 3*b^4)*ArcTan[
Sinh[x]])/b^5 + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(a*b^5) + (a*Tanh[
x])/b^2 + (a*(a^2 - 3*b^2)*Tanh[x])/b^4 - (3*Sech[x]*Tanh[x])/(8*b) - ((a^2 - 3*b^2)*Sech[x]*Tanh[x])/(2*b^3)
- (Sech[x]^3*Tanh[x])/(4*b) - (a*Tanh[x]^3)/(3*b^2)

Rule 3898

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

Rule 2897

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(
m_), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m*(1 - sin[e + f*x]^2)^(p/2), x], x]
/; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[m, 2*n, p/2] && (LtQ[m, -1] || (EqQ[m, -1] && G
tQ[p, 0]))

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rubi steps

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

Mathematica [A]  time = 0.588959, size = 185, normalized size = 0.99 \[ \frac{-12 \left (-20 a^2 b^2+8 a^4+15 b^4\right ) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\frac{48 \left (b^5 x \sqrt{a^2-b^2}-2 \left (a^2-b^2\right )^3 \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2-b^2}}\right )\right )}{a \sqrt{a^2-b^2}}+b \tanh (x) \text{sech}^3(x) \left (4 a \left (9 a^2-17 b^2\right ) \cosh (x)+3 b \left (9 b^2-4 a^2\right ) \cosh (2 x)-12 a^2 b+12 a^3 \cosh (3 x)-28 a b^2 \cosh (3 x)+15 b^3\right )}{48 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^6/(a + b*Sech[x]),x]

[Out]

(-12*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[Tanh[x/2]] + (48*(b^5*Sqrt[a^2 - b^2]*x - 2*(a^2 - b^2)^3*ArcTan[((-
a + b)*Tanh[x/2])/Sqrt[a^2 - b^2]]))/(a*Sqrt[a^2 - b^2]) + b*(-12*a^2*b + 15*b^3 + 4*a*(9*a^2 - 17*b^2)*Cosh[x
] + 3*b*(-4*a^2 + 9*b^2)*Cosh[2*x] + 12*a^3*Cosh[3*x] - 28*a*b^2*Cosh[3*x])*Sech[x]^3*Tanh[x])/(48*b^5)

________________________________________________________________________________________

Maple [B]  time = 0.043, size = 575, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^6/(a+b*sech(x)),x)

[Out]

-15/4/b*arctan(tanh(1/2*x))-2/a*b/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+1/a*ln(tan
h(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-6*a^3/b^3/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+
6*a/b/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-44/3/b^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*
x)^3*a+1/b^3/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a^2-44/3/b^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a-1/b^3/(tanh(1/
2*x)^2+1)^4*tanh(1/2*x)^3*a^2+6/b^4/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*a^3+2/b^4/(tanh(1/2*x)^2+1)^4*tanh(1/2*x
)*a^3-4/b^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*a-1/b^3/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*a^2+2*a^5/b^5/((a+b)*(a-b)
)^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+2/b^4/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7*a^3+1/b^3/(tanh(
1/2*x)^2+1)^4*tanh(1/2*x)^7*a^2-4/b^2/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7*a+6/b^4/(tanh(1/2*x)^2+1)^4*tanh(1/2*x
)^5*a^3-7/4/b/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7-15/4/b/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5+15/4/b/(tanh(1/2*x)^2
+1)^4*tanh(1/2*x)^3+7/4/b/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)-2/b^5*arctan(tanh(1/2*x))*a^4+5/b^3*arctan(tanh(1/2*
x))*a^2

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+b*sech(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 5.43995, size = 12166, normalized size = 65.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+b*sech(x)),x, algorithm="fricas")

[Out]

[1/12*(12*b^5*x*cosh(x)^8 + 12*b^5*x*sinh(x)^8 - 3*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^7 + 3*(32*b^5*x*cosh(x) - 4*a
^3*b^2 + 9*a*b^4)*sinh(x)^7 + 24*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^6 + 3*(112*b^5*x*cosh(x)^2 + 16*b^5*x -
8*a^4*b + 24*a^2*b^3 - 7*(4*a^3*b^2 - 9*a*b^4)*cosh(x))*sinh(x)^6 + 12*b^5*x - 3*(4*a^3*b^2 - a*b^4)*cosh(x)^
5 + 3*(224*b^5*x*cosh(x)^3 - 4*a^3*b^2 + a*b^4 - 21*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^2 + 48*(2*b^5*x - a^4*b + 3*
a^2*b^3)*cosh(x))*sinh(x)^5 - 24*a^4*b + 56*a^2*b^3 + 24*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(x)^4 + 3*(280*b^
5*x*cosh(x)^4 + 24*b^5*x - 24*a^4*b + 56*a^2*b^3 - 35*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^3 + 120*(2*b^5*x - a^4*b +
3*a^2*b^3)*cosh(x)^2 - 5*(4*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^4 + 3*(4*a^3*b^2 - a*b^4)*cosh(x)^3 + 3*(224*b^
5*x*cosh(x)^5 + 4*a^3*b^2 - a*b^4 - 35*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^4 + 160*(2*b^5*x - a^4*b + 3*a^2*b^3)*cos
h(x)^3 - 10*(4*a^3*b^2 - a*b^4)*cosh(x)^2 + 32*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(x))*sinh(x)^3 + 8*(6*b^5*x
- 9*a^4*b + 19*a^2*b^3)*cosh(x)^2 + (336*b^5*x*cosh(x)^6 + 48*b^5*x - 63*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^5 - 72
*a^4*b + 152*a^2*b^3 + 360*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^4 - 30*(4*a^3*b^2 - a*b^4)*cosh(x)^3 + 144*(3
*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(x)^2 + 9*(4*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^2 + 12*((a^4 - 2*a^2*b^2 + b^
4)*cosh(x)^8 + 8*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^7 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^8 + 4*(a^4 - 2*a^
2*b^2 + b^4)*cosh(x)^6 + 4*(a^4 - 2*a^2*b^2 + b^4 + 7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4
- 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^5 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(
x)^4 + 2*(35*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*a^4 - 6*a^2*b^2 + 3*b^4 + 30*(a^4 - 2*a^2*b^2 + b^4)*cosh(x
)^2)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 10*(a^4 - 2*a^2*b^2 + b^4)*c
osh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 4*(7*(a^4 - 2*
a^2*b^2 + b^4)*cosh(x)^6 + 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 9*(a^4 - 2*a^2*b^2 +
b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 3*(a
^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^
2 + a^2*sinh(x)^2 + 2*a*b*cosh(x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) + 2*sqrt(-a^2 + b^2)*(a*cosh(x
) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*cosh(x) + b)*sinh(x) + a)) - 3*((8*a^5 - 2
0*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)*sinh(x)^7 + (8*a^5 - 20*a^3*b^2 +
15*a*b^4)*sinh(x)^8 + 4*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 4*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 7*(8*a^
5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(8*a^5 -
20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^5 + 8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 6*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*
cosh(x)^4 + 2*(24*a^5 - 60*a^3*b^2 + 45*a*b^4 + 35*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 30*(8*a^5 - 20*
a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 10*(8*a^5 - 20*a^3
*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^3 + 4*(8*a^5 - 20*a^3*b^2 + 15
*a*b^4)*cosh(x)^2 + 4*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 15*(8*a^5
- 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 9*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^5 - 20*
a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4
)*cosh(x)^3 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) + 3*(4*a^3*b^2 - 9*a
*b^4)*cosh(x) + (96*b^5*x*cosh(x)^7 - 21*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^6 + 144*(2*b^5*x - a^4*b + 3*a^2*b^3)*c
osh(x)^5 + 12*a^3*b^2 - 27*a*b^4 - 15*(4*a^3*b^2 - a*b^4)*cosh(x)^4 + 96*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(
x)^3 + 9*(4*a^3*b^2 - a*b^4)*cosh(x)^2 + 16*(6*b^5*x - 9*a^4*b + 19*a^2*b^3)*cosh(x))*sinh(x))/(a*b^5*cosh(x)^
8 + 8*a*b^5*cosh(x)*sinh(x)^7 + a*b^5*sinh(x)^8 + 4*a*b^5*cosh(x)^6 + 6*a*b^5*cosh(x)^4 + 4*a*b^5*cosh(x)^2 +
4*(7*a*b^5*cosh(x)^2 + a*b^5)*sinh(x)^6 + a*b^5 + 8*(7*a*b^5*cosh(x)^3 + 3*a*b^5*cosh(x))*sinh(x)^5 + 2*(35*a*
b^5*cosh(x)^4 + 30*a*b^5*cosh(x)^2 + 3*a*b^5)*sinh(x)^4 + 8*(7*a*b^5*cosh(x)^5 + 10*a*b^5*cosh(x)^3 + 3*a*b^5*
cosh(x))*sinh(x)^3 + 4*(7*a*b^5*cosh(x)^6 + 15*a*b^5*cosh(x)^4 + 9*a*b^5*cosh(x)^2 + a*b^5)*sinh(x)^2 + 8*(a*b
^5*cosh(x)^7 + 3*a*b^5*cosh(x)^5 + 3*a*b^5*cosh(x)^3 + a*b^5*cosh(x))*sinh(x)), 1/12*(12*b^5*x*cosh(x)^8 + 12*
b^5*x*sinh(x)^8 - 3*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^7 + 3*(32*b^5*x*cosh(x) - 4*a^3*b^2 + 9*a*b^4)*sinh(x)^7 + 2
4*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^6 + 3*(112*b^5*x*cosh(x)^2 + 16*b^5*x - 8*a^4*b + 24*a^2*b^3 - 7*(4*a^
3*b^2 - 9*a*b^4)*cosh(x))*sinh(x)^6 + 12*b^5*x - 3*(4*a^3*b^2 - a*b^4)*cosh(x)^5 + 3*(224*b^5*x*cosh(x)^3 - 4*
a^3*b^2 + a*b^4 - 21*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^2 + 48*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x))*sinh(x)^5 - 2
4*a^4*b + 56*a^2*b^3 + 24*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(x)^4 + 3*(280*b^5*x*cosh(x)^4 + 24*b^5*x - 24*a
^4*b + 56*a^2*b^3 - 35*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^3 + 120*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^2 - 5*(4*a^
3*b^2 - a*b^4)*cosh(x))*sinh(x)^4 + 3*(4*a^3*b^2 - a*b^4)*cosh(x)^3 + 3*(224*b^5*x*cosh(x)^5 + 4*a^3*b^2 - a*b
^4 - 35*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^4 + 160*(2*b^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^3 - 10*(4*a^3*b^2 - a*b^4)
*cosh(x)^2 + 32*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*cosh(x))*sinh(x)^3 + 8*(6*b^5*x - 9*a^4*b + 19*a^2*b^3)*cosh(x
)^2 + (336*b^5*x*cosh(x)^6 + 48*b^5*x - 63*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^5 - 72*a^4*b + 152*a^2*b^3 + 360*(2*b
^5*x - a^4*b + 3*a^2*b^3)*cosh(x)^4 - 30*(4*a^3*b^2 - a*b^4)*cosh(x)^3 + 144*(3*b^5*x - 3*a^4*b + 7*a^2*b^3)*c
osh(x)^2 + 9*(4*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^2 - 24*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^8 + 8*(a^4 - 2*a^2*b
^2 + b^4)*cosh(x)*sinh(x)^7 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^8 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 4*(a^4
- 2*a^2*b^2 + b^4 + 7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 +
3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^5 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 2*(35*(a^4 - 2*a^2*b^2 +
b^4)*cosh(x)^4 + 3*a^4 - 6*a^2*b^2 + 3*b^4 + 30*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b^
2 + b^4 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 +
b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 4*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 15*(
a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 8*
((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3
+ (a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))*sqrt(a^2 - b^2)*arctan(-(a*cosh(x) + a*sinh(x) + b)/sqrt(a^2 - b^
2)) - 3*((8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)*sinh(x)^7 + (8*
a^5 - 20*a^3*b^2 + 15*a*b^4)*sinh(x)^8 + 4*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 4*(8*a^5 - 20*a^3*b^2 +
15*a*b^4 + 7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh
(x)^3 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^5 + 8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 6*(8*a^5 - 20*a
^3*b^2 + 15*a*b^4)*cosh(x)^4 + 2*(24*a^5 - 60*a^3*b^2 + 45*a*b^4 + 35*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^
4 + 30*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 +
10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^3 + 4*(8*a^
5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2 + 4*(7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 8*a^5 - 20*a^3*b^2 + 1
5*a*b^4 + 15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 9*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^
2 + 8*((8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 3*(8*a^5 - 20
*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) +
3*(4*a^3*b^2 - 9*a*b^4)*cosh(x) + (96*b^5*x*cosh(x)^7 - 21*(4*a^3*b^2 - 9*a*b^4)*cosh(x)^6 + 144*(2*b^5*x - a
^4*b + 3*a^2*b^3)*cosh(x)^5 + 12*a^3*b^2 - 27*a*b^4 - 15*(4*a^3*b^2 - a*b^4)*cosh(x)^4 + 96*(3*b^5*x - 3*a^4*b
+ 7*a^2*b^3)*cosh(x)^3 + 9*(4*a^3*b^2 - a*b^4)*cosh(x)^2 + 16*(6*b^5*x - 9*a^4*b + 19*a^2*b^3)*cosh(x))*sinh(
x))/(a*b^5*cosh(x)^8 + 8*a*b^5*cosh(x)*sinh(x)^7 + a*b^5*sinh(x)^8 + 4*a*b^5*cosh(x)^6 + 6*a*b^5*cosh(x)^4 + 4
*a*b^5*cosh(x)^2 + 4*(7*a*b^5*cosh(x)^2 + a*b^5)*sinh(x)^6 + a*b^5 + 8*(7*a*b^5*cosh(x)^3 + 3*a*b^5*cosh(x))*s
inh(x)^5 + 2*(35*a*b^5*cosh(x)^4 + 30*a*b^5*cosh(x)^2 + 3*a*b^5)*sinh(x)^4 + 8*(7*a*b^5*cosh(x)^5 + 10*a*b^5*c
osh(x)^3 + 3*a*b^5*cosh(x))*sinh(x)^3 + 4*(7*a*b^5*cosh(x)^6 + 15*a*b^5*cosh(x)^4 + 9*a*b^5*cosh(x)^2 + a*b^5)
*sinh(x)^2 + 8*(a*b^5*cosh(x)^7 + 3*a*b^5*cosh(x)^5 + 3*a*b^5*cosh(x)^3 + a*b^5*cosh(x))*sinh(x))]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{6}{\left (x \right )}}{a + b \operatorname{sech}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**6/(a+b*sech(x)),x)

[Out]

Integral(tanh(x)**6/(a + b*sech(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.13604, size = 338, normalized size = 1.81 \begin{align*} \frac{x}{a} - \frac{{\left (8 \, a^{4} - 20 \, a^{2} b^{2} + 15 \, b^{4}\right )} \arctan \left (e^{x}\right )}{4 \, b^{5}} + \frac{2 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \arctan \left (\frac{a e^{x} + b}{\sqrt{a^{2} - b^{2}}}\right )}{\sqrt{a^{2} - b^{2}} a b^{5}} - \frac{12 \, a^{2} b e^{\left (7 \, x\right )} - 27 \, b^{3} e^{\left (7 \, x\right )} + 24 \, a^{3} e^{\left (6 \, x\right )} - 72 \, a b^{2} e^{\left (6 \, x\right )} + 12 \, a^{2} b e^{\left (5 \, x\right )} - 3 \, b^{3} e^{\left (5 \, x\right )} + 72 \, a^{3} e^{\left (4 \, x\right )} - 168 \, a b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} b e^{\left (3 \, x\right )} + 3 \, b^{3} e^{\left (3 \, x\right )} + 72 \, a^{3} e^{\left (2 \, x\right )} - 152 \, a b^{2} e^{\left (2 \, x\right )} - 12 \, a^{2} b e^{x} + 27 \, b^{3} e^{x} + 24 \, a^{3} - 56 \, a b^{2}}{12 \, b^{4}{\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^6/(a+b*sech(x)),x, algorithm="giac")

[Out]

x/a - 1/4*(8*a^4 - 20*a^2*b^2 + 15*b^4)*arctan(e^x)/b^5 + 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*arctan((a*e^x
+ b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a*b^5) - 1/12*(12*a^2*b*e^(7*x) - 27*b^3*e^(7*x) + 24*a^3*e^(6*x) - 72*
a*b^2*e^(6*x) + 12*a^2*b*e^(5*x) - 3*b^3*e^(5*x) + 72*a^3*e^(4*x) - 168*a*b^2*e^(4*x) - 12*a^2*b*e^(3*x) + 3*b
^3*e^(3*x) + 72*a^3*e^(2*x) - 152*a*b^2*e^(2*x) - 12*a^2*b*e^x + 27*b^3*e^x + 24*a^3 - 56*a*b^2)/(b^4*(e^(2*x)
+ 1)^4)