3.109 \(\int \frac{\text{sech}^7(x)}{a+b \tanh (x)} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\left (a^2-b^2\right ) \text{sech}^3(x)}{3 b^3}+\frac{\left (a^2-b^2\right )^2 \text{sech}(x)}{b^5}+\frac{a \left (-20 a^2 b^2+8 a^4+15 b^4\right ) \tan ^{-1}(\sinh (x))}{8 b^6}-\frac{a \left (4 a^2-7 b^2\right ) \tanh (x) \text{sech}(x)}{8 b^4}-\frac{\left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{\cosh (x) (a \tanh (x)+b)}{\sqrt{a^2-b^2}}\right )}{b^6}+\frac{a \tanh (x) \text{sech}^3(x)}{4 b^2}+\frac{\text{sech}^5(x)}{5 b} \]

[Out]

(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(8*b^6) - ((a^2 - b^2)^(5/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x])
)/Sqrt[a^2 - b^2]])/b^6 + ((a^2 - b^2)^2*Sech[x])/b^5 - ((a^2 - b^2)*Sech[x]^3)/(3*b^3) + Sech[x]^5/(5*b) - (a
*(4*a^2 - 7*b^2)*Sech[x]*Tanh[x])/(8*b^4) + (a*Sech[x]^3*Tanh[x])/(4*b^2)

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Rubi [A]  time = 0.272871, antiderivative size = 187, normalized size of antiderivative = 1.19, number of steps used = 14, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {3510, 3486, 3768, 3770, 3509, 206} \[ -\frac{\left (a^2-b^2\right ) \text{sech}^3(x)}{3 b^3}+\frac{\left (a^2-b^2\right )^2 \text{sech}(x)}{b^5}+\frac{a \left (a^2-b^2\right )^2 \tan ^{-1}(\sinh (x))}{b^6}-\frac{a \left (a^2-b^2\right ) \tan ^{-1}(\sinh (x))}{2 b^4}-\frac{a \left (a^2-b^2\right ) \tanh (x) \text{sech}(x)}{2 b^4}-\frac{\left (a^2-b^2\right )^{5/2} \tan ^{-1}\left (\frac{\cosh (x) (a \tanh (x)+b)}{\sqrt{a^2-b^2}}\right )}{b^6}+\frac{3 a \tan ^{-1}(\sinh (x))}{8 b^2}+\frac{a \tanh (x) \text{sech}^3(x)}{4 b^2}+\frac{3 a \tanh (x) \text{sech}(x)}{8 b^2}+\frac{\text{sech}^5(x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^7/(a + b*Tanh[x]),x]

[Out]

(3*a*ArcTan[Sinh[x]])/(8*b^2) - (a*(a^2 - b^2)*ArcTan[Sinh[x]])/(2*b^4) + (a*(a^2 - b^2)^2*ArcTan[Sinh[x]])/b^
6 - ((a^2 - b^2)^(5/2)*ArcTan[(Cosh[x]*(b + a*Tanh[x]))/Sqrt[a^2 - b^2]])/b^6 + ((a^2 - b^2)^2*Sech[x])/b^5 -
((a^2 - b^2)*Sech[x]^3)/(3*b^3) + Sech[x]^5/(5*b) + (3*a*Sech[x]*Tanh[x])/(8*b^2) - (a*(a^2 - b^2)*Sech[x]*Tan
h[x])/(2*b^4) + (a*Sech[x]^3*Tanh[x])/(4*b^2)

Rule 3510

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

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

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]

Rule 3770

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

Rule 3509

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.496309, size = 166, normalized size = 1.06 \[ \frac{30 \left (a \left (-20 a^2 b^2+8 a^4+15 b^4\right ) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-8 \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^2 \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )\right )+10 b^3 \text{sech}^3(x) \left (-4 a^2+3 a b \tanh (x)+4 b^2\right )+15 b \text{sech}(x) \left (\left (7 a b^3-4 a^3 b\right ) \tanh (x)+8 \left (a^2-b^2\right )^2\right )+24 b^5 \text{sech}^5(x)}{120 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^7/(a + b*Tanh[x]),x]

[Out]

(30*(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*ArcTan[Tanh[x/2]] - 8*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^2*ArcTan[(b + a
*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])]) + 24*b^5*Sech[x]^5 + 10*b^3*Sech[x]^3*(-4*a^2 + 4*b^2 + 3*a*b*Tanh[x])
+ 15*b*Sech[x]*(8*(a^2 - b^2)^2 + (-4*a^3*b + 7*a*b^3)*Tanh[x]))/(120*b^6)

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Maple [B]  time = 0.053, size = 715, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^7/(a+b*tanh(x)),x)

[Out]

-14/3/b^3/(tanh(1/2*x)^2+1)^5*a^2-5/b^4*arctan(tanh(1/2*x))*a^3+15/4/b^2*arctan(tanh(1/2*x))*a+2/b^6*arctan(ta
nh(1/2*x))*a^5+6/b/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^8+12/b/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^6+56/3/b/(tanh(1/2*x
)^2+1)^5*tanh(1/2*x)^4+28/3/b/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^2+2/b^5/(tanh(1/2*x)^2+1)^5*a^4+46/15/b/(tanh(1/
2*x)^2+1)^5+2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))+1/b^4/(tanh(1/2*x)^2+1)^5*tanh
(1/2*x)^9*a^3-9/4/b^2/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^9*a+2/b^5/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^8*a^4-6/b^3/(t
anh(1/2*x)^2+1)^5*tanh(1/2*x)^8*a^2+2/b^4/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^7*a^3-5/2/b^2/(tanh(1/2*x)^2+1)^5*ta
nh(1/2*x)^7*a+8/b^5/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^6*a^4-20/b^3/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^6*a^2+12/b^5/
(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^4*a^4-80/3/b^3/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^4*a^2-2/b^4/(tanh(1/2*x)^2+1)^5
*tanh(1/2*x)^3*a^3+5/2/b^2/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^3*a+8/b^5/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^2*a^4-52/
3/b^3/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)^2*a^2-1/b^4/(tanh(1/2*x)^2+1)^5*tanh(1/2*x)*a^3+9/4/b^2/(tanh(1/2*x)^2+1
)^5*tanh(1/2*x)*a-2/b^6/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))*a^6+6/b^4/(a^2-b^2)^
(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))*a^4-6/b^2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+
2*b)/(a^2-b^2)^(1/2))*a^2

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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(sech(x)^7/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.48934, size = 16338, normalized size = 104.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/60*(15*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^9 + 135*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b
^3 + 7*a*b^4 + 8*b^5)*cosh(x)*sinh(x)^8 + 15*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*sinh(x)^9 +
10*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^7 + 10*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b
^3 + 33*a*b^4 + 64*b^5 + 54*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^7 + 70*(18
*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^3 + (48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*
b^4 + 64*b^5)*cosh(x))*sinh(x)^6 + 16*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^5 + 2*(360*a^4*b - 880*a^2*b^3
+ 712*b^5 + 945*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^4 + 105*(48*a^4*b - 12*a^3*b^2 -
112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^2)*sinh(x)^5 + 10*(189*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 +
8*b^5)*cosh(x)^5 + 35*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^3 + 8*(45*a^4*b - 110
*a^2*b^3 + 89*b^5)*cosh(x))*sinh(x)^4 + 10*(48*a^4*b + 12*a^3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)*cosh(x)^3
+ 10*(126*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^6 + 48*a^4*b + 12*a^3*b^2 - 112*a^2*b^
3 - 33*a*b^4 + 64*b^5 + 35*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^4 + 16*(45*a^4*b
- 110*a^2*b^3 + 89*b^5)*cosh(x)^2)*sinh(x)^3 + 10*(54*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cos
h(x)^7 + 21*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^5 + 16*(45*a^4*b - 110*a^2*b^3 +
89*b^5)*cosh(x)^3 + 3*(48*a^4*b + 12*a^3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)*cosh(x))*sinh(x)^2 + 60*((a^4
- 2*a^2*b^2 + b^4)*cosh(x)^10 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^9 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x
)^10 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^8 + 5*(a^4 - 2*a^2*b^2 + b^4 + 9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*s
inh(x)^8 + 40*(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)^7 + 10*(a^4 - 2*
a^2*b^2 + b^4)*cosh(x)^6 + 10*(21*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 14*(a^4 - 2*a^2*
b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 70*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)
^3 + 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^5 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 10*(21*(a^4 - 2*a^
2*b^2 + b^4)*cosh(x)^6 + 35*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 15*(a^4 - 2*a^2*b^2 +
b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 40*(3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 7*(a^4 - 2*a^2*b
^2 + b^4)*cosh(x)^5 + 5*(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)^3 + 5*(a^
4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 5*(9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^8 + 28*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6
+ 30*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 12*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x
)^2 + 10*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^9 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 6*(a^4 - 2*a^2*b^2 + b^4)*
cosh(x)^5 + 4*(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)*l
og(((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 - 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x))
- a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 15*((8*a^5 - 20*a^3*b
^2 + 15*a*b^4)*cosh(x)^10 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)*sinh(x)^9 + (8*a^5 - 20*a^3*b^2 + 15*a*
b^4)*sinh(x)^10 + 5*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 5*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 9*(8*a^5 -
20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(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)^7 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 10*(8*a^5 - 20*a^3*b^2 +
15*a*b^4 + 21*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 14*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(
x)^6 + 4*(63*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 70*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 15*(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 + 10*(8*a^5 - 20*a^3*b^2 + 15*
a*b^4)*cosh(x)^4 + 10*(21*(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 + 35*(8*a^
5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 15*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(8*a^5
- 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 5*(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)^3 + 5*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh
(x)^2 + 5*(9*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 28*(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 + 30*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 12*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*co
sh(x)^2)*sinh(x)^2 + 10*((8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^9 + 4*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)
^7 + 6*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 4*(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)) + 15*(8*a^4*b + 4*a^3*b^2 - 16*a^2*b^3 - 7*a*b^
4 + 8*b^5)*cosh(x) + 5*(27*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^8 + 14*(48*a^4*b - 12*
a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^6 + 24*a^4*b + 12*a^3*b^2 - 48*a^2*b^3 - 21*a*b^4 + 24*b^5
+ 16*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^4 + 6*(48*a^4*b + 12*a^3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)
*cosh(x)^2)*sinh(x))/(b^6*cosh(x)^10 + 10*b^6*cosh(x)*sinh(x)^9 + b^6*sinh(x)^10 + 5*b^6*cosh(x)^8 + 10*b^6*co
sh(x)^6 + 10*b^6*cosh(x)^4 + 5*(9*b^6*cosh(x)^2 + b^6)*sinh(x)^8 + 5*b^6*cosh(x)^2 + 40*(3*b^6*cosh(x)^3 + b^6
*cosh(x))*sinh(x)^7 + 10*(21*b^6*cosh(x)^4 + 14*b^6*cosh(x)^2 + b^6)*sinh(x)^6 + b^6 + 4*(63*b^6*cosh(x)^5 + 7
0*b^6*cosh(x)^3 + 15*b^6*cosh(x))*sinh(x)^5 + 10*(21*b^6*cosh(x)^6 + 35*b^6*cosh(x)^4 + 15*b^6*cosh(x)^2 + b^6
)*sinh(x)^4 + 40*(3*b^6*cosh(x)^7 + 7*b^6*cosh(x)^5 + 5*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x)^3 + 5*(9*b^6*cosh
(x)^8 + 28*b^6*cosh(x)^6 + 30*b^6*cosh(x)^4 + 12*b^6*cosh(x)^2 + b^6)*sinh(x)^2 + 10*(b^6*cosh(x)^9 + 4*b^6*co
sh(x)^7 + 6*b^6*cosh(x)^5 + 4*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x)), 1/60*(15*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^
3 + 7*a*b^4 + 8*b^5)*cosh(x)^9 + 135*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)*sinh(x)^8 +
15*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*sinh(x)^9 + 10*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 +
33*a*b^4 + 64*b^5)*cosh(x)^7 + 10*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5 + 54*(8*a^4*b - 4*a
^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^7 + 70*(18*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b
^4 + 8*b^5)*cosh(x)^3 + (48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x))*sinh(x)^6 + 16*(45*
a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^5 + 2*(360*a^4*b - 880*a^2*b^3 + 712*b^5 + 945*(8*a^4*b - 4*a^3*b^2 - 16
*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^4 + 105*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^
2)*sinh(x)^5 + 10*(189*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^5 + 35*(48*a^4*b - 12*a^3*
b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^3 + 8*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x))*sinh(x)^4 + 10
*(48*a^4*b + 12*a^3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)*cosh(x)^3 + 10*(126*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b
^3 + 7*a*b^4 + 8*b^5)*cosh(x)^6 + 48*a^4*b + 12*a^3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5 + 35*(48*a^4*b - 12*
a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^4 + 16*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^2)*sinh(x)
^3 + 10*(54*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4 + 8*b^5)*cosh(x)^7 + 21*(48*a^4*b - 12*a^3*b^2 - 112*a
^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^5 + 16*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^3 + 3*(48*a^4*b + 12*a^3*
b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)*cosh(x))*sinh(x)^2 + 120*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^10 + 10*(a^4
- 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^9 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^10 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^
8 + 5*(a^4 - 2*a^2*b^2 + b^4 + 9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^8 + 40*(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)^7 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 10*(21*(a^4 -
2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 14*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(
a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 70*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*
sinh(x)^5 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 10*(21*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 35*(a^4 - 2*a^2*
b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b
^2 + b^4 + 40*(3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 5*(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)^3 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 5*(9*(a^
4 - 2*a^2*b^2 + b^4)*cosh(x)^8 + 28*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 30*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 +
a^4 - 2*a^2*b^2 + b^4 + 12*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 10*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)
^9 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 4*(a^4 - 2*a^2*b^2 + b^4)*cos
h(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (
a + b)*sinh(x))) + 15*((8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^10 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)
*sinh(x)^9 + (8*a^5 - 20*a^3*b^2 + 15*a*b^4)*sinh(x)^10 + 5*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 5*(8*a
^5 - 20*a^3*b^2 + 15*a*b^4 + 9*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(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)^7 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b
^4)*cosh(x)^6 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 21*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 14*(8*a^5 -
20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 4*(63*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 70*(8*a^5 - 2
0*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 15*(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 + 10*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 10*(21*(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 + 35*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 15*(8*a^5 - 20*a^3*b^2 + 15*a*b
^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 7*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)
*cosh(x)^5 + 5*(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)^3
+ 5*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2 + 5*(9*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 28*(8*a^5 - 2
0*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 8*a^5 - 20*a^3*b^2 + 15*a*b^4 + 30*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^4
+ 12*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^2 + 10*((8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^9 + 4
*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 6*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 4*(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)) + 15*(8
*a^4*b + 4*a^3*b^2 - 16*a^2*b^3 - 7*a*b^4 + 8*b^5)*cosh(x) + 5*(27*(8*a^4*b - 4*a^3*b^2 - 16*a^2*b^3 + 7*a*b^4
+ 8*b^5)*cosh(x)^8 + 14*(48*a^4*b - 12*a^3*b^2 - 112*a^2*b^3 + 33*a*b^4 + 64*b^5)*cosh(x)^6 + 24*a^4*b + 12*a
^3*b^2 - 48*a^2*b^3 - 21*a*b^4 + 24*b^5 + 16*(45*a^4*b - 110*a^2*b^3 + 89*b^5)*cosh(x)^4 + 6*(48*a^4*b + 12*a^
3*b^2 - 112*a^2*b^3 - 33*a*b^4 + 64*b^5)*cosh(x)^2)*sinh(x))/(b^6*cosh(x)^10 + 10*b^6*cosh(x)*sinh(x)^9 + b^6*
sinh(x)^10 + 5*b^6*cosh(x)^8 + 10*b^6*cosh(x)^6 + 10*b^6*cosh(x)^4 + 5*(9*b^6*cosh(x)^2 + b^6)*sinh(x)^8 + 5*b
^6*cosh(x)^2 + 40*(3*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(x)^7 + 10*(21*b^6*cosh(x)^4 + 14*b^6*cosh(x)^2 + b^6)*s
inh(x)^6 + b^6 + 4*(63*b^6*cosh(x)^5 + 70*b^6*cosh(x)^3 + 15*b^6*cosh(x))*sinh(x)^5 + 10*(21*b^6*cosh(x)^6 + 3
5*b^6*cosh(x)^4 + 15*b^6*cosh(x)^2 + b^6)*sinh(x)^4 + 40*(3*b^6*cosh(x)^7 + 7*b^6*cosh(x)^5 + 5*b^6*cosh(x)^3
+ b^6*cosh(x))*sinh(x)^3 + 5*(9*b^6*cosh(x)^8 + 28*b^6*cosh(x)^6 + 30*b^6*cosh(x)^4 + 12*b^6*cosh(x)^2 + b^6)*
sinh(x)^2 + 10*(b^6*cosh(x)^9 + 4*b^6*cosh(x)^7 + 6*b^6*cosh(x)^5 + 4*b^6*cosh(x)^3 + b^6*cosh(x))*sinh(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)**7/(a+b*tanh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.23608, size = 440, normalized size = 2.8 \begin{align*} \frac{{\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (e^{x}\right )}{4 \, b^{6}} - \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 e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{\sqrt{a^{2} - b^{2}} b^{6}} + \frac{120 \, a^{4} e^{\left (9 \, x\right )} - 60 \, a^{3} b e^{\left (9 \, x\right )} - 240 \, a^{2} b^{2} e^{\left (9 \, x\right )} + 105 \, a b^{3} e^{\left (9 \, x\right )} + 120 \, b^{4} e^{\left (9 \, x\right )} + 480 \, a^{4} e^{\left (7 \, x\right )} - 120 \, a^{3} b e^{\left (7 \, x\right )} - 1120 \, a^{2} b^{2} e^{\left (7 \, x\right )} + 330 \, a b^{3} e^{\left (7 \, x\right )} + 640 \, b^{4} e^{\left (7 \, x\right )} + 720 \, a^{4} e^{\left (5 \, x\right )} - 1760 \, a^{2} b^{2} e^{\left (5 \, x\right )} + 1424 \, b^{4} e^{\left (5 \, x\right )} + 480 \, a^{4} e^{\left (3 \, x\right )} + 120 \, a^{3} b e^{\left (3 \, x\right )} - 1120 \, a^{2} b^{2} e^{\left (3 \, x\right )} - 330 \, a b^{3} e^{\left (3 \, x\right )} + 640 \, b^{4} e^{\left (3 \, x\right )} + 120 \, a^{4} e^{x} + 60 \, a^{3} b e^{x} - 240 \, a^{2} b^{2} e^{x} - 105 \, a b^{3} e^{x} + 120 \, b^{4} e^{x}}{60 \, b^{5}{\left (e^{\left (2 \, x\right )} + 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*arctan(e^x)/b^6 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*arctan((a*e^x + b*
e^x)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*b^6) + 1/60*(120*a^4*e^(9*x) - 60*a^3*b*e^(9*x) - 240*a^2*b^2*e^(9*x) +
105*a*b^3*e^(9*x) + 120*b^4*e^(9*x) + 480*a^4*e^(7*x) - 120*a^3*b*e^(7*x) - 1120*a^2*b^2*e^(7*x) + 330*a*b^3*
e^(7*x) + 640*b^4*e^(7*x) + 720*a^4*e^(5*x) - 1760*a^2*b^2*e^(5*x) + 1424*b^4*e^(5*x) + 480*a^4*e^(3*x) + 120*
a^3*b*e^(3*x) - 1120*a^2*b^2*e^(3*x) - 330*a*b^3*e^(3*x) + 640*b^4*e^(3*x) + 120*a^4*e^x + 60*a^3*b*e^x - 240*
a^2*b^2*e^x - 105*a*b^3*e^x + 120*b^4*e^x)/(b^5*(e^(2*x) + 1)^5)