### 3.623 $$\int \frac{1}{(a \text{sech}(x)+b \tanh (x))^5} \, dx$$

Optimal. Leaf size=95 $-\frac{\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac{4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac{3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac{4 a}{b^5 (a+b \sinh (x))}+\frac{\log (a+b \sinh (x))}{b^5}$

[Out]

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

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Rubi [A]  time = 0.115842, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {4391, 2668, 697} $-\frac{\left (a^2+b^2\right )^2}{4 b^5 (a+b \sinh (x))^4}+\frac{4 a \left (a^2+b^2\right )}{3 b^5 (a+b \sinh (x))^3}-\frac{3 a^2+b^2}{b^5 (a+b \sinh (x))^2}+\frac{4 a}{b^5 (a+b \sinh (x))}+\frac{\log (a+b \sinh (x))}{b^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*Sech[x] + b*Tanh[x])^(-5),x]

[Out]

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

Rule 4391

Int[(u_.)*((b_.)*sec[(c_.) + (d_.)*(x_)]^(n_.) + (a_.)*tan[(c_.) + (d_.)*(x_)]^(n_.))^(p_), x_Symbol] :> Int[A
ctivateTrig[u]*Sec[c + d*x]^(n*p)*(b + a*Sin[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

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

Mathematica [A]  time = 0.229112, size = 83, normalized size = 0.87 $\frac{-\frac{\left (a^2+b^2\right )^2}{4 (a+b \sinh (x))^4}+\frac{4 a \left (a^2+b^2\right )}{3 (a+b \sinh (x))^3}-\frac{3 a^2+b^2}{(a+b \sinh (x))^2}+\frac{4 a}{a+b \sinh (x)}+\log (a+b \sinh (x))}{b^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*Sech[x] + b*Tanh[x])^(-5),x]

[Out]

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

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Maple [B]  time = 0.128, size = 721, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sech(x)+b*tanh(x))^5,x)

[Out]

-1/b^5*ln(tanh(1/2*x)+1)+2/b^4/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^3*tanh(1/2*x)^7-2/(a*tanh(1/2*x)^2-2*ta
nh(1/2*x)*b-a)^4/a*tanh(1/2*x)^7-14/b^3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^2*tanh(1/2*x)^6+6*b/(a*tanh(1/
2*x)^2-2*tanh(1/2*x)*b-a)^4/a^2*tanh(1/2*x)^6-6/b^4/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^3*tanh(1/2*x)^5+10
4/3/b^2/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a*tanh(1/2*x)^5+2/3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4/a*tanh
(1/2*x)^5-8*b^2/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4/a^3*tanh(1/2*x)^5+28/b^3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*
b-a)^4*a^2*tanh(1/2*x)^4-100/3/b/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*tanh(1/2*x)^4-28/3*b/(a*tanh(1/2*x)^2-2
*tanh(1/2*x)*b-a)^4/a^2*tanh(1/2*x)^4+4*b^3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4/a^4*tanh(1/2*x)^4+6/b^4/(a*t
anh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^3*tanh(1/2*x)^3-104/3/b^2/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a*tanh(1/2
*x)^3-2/3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4/a*tanh(1/2*x)^3+8*b^2/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4/a^
3*tanh(1/2*x)^3-14/b^3/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^2*tanh(1/2*x)^2+6*b/(a*tanh(1/2*x)^2-2*tanh(1/2
*x)*b-a)^4/a^2*tanh(1/2*x)^2-2/b^4/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)^4*a^3*tanh(1/2*x)+2/(a*tanh(1/2*x)^2-2*
tanh(1/2*x)*b-a)^4/a*tanh(1/2*x)+1/b^5*ln(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)-1/b^5*ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.2044, size = 401, normalized size = 4.22 \begin{align*} \frac{4 \,{\left (6 \, a b^{3} e^{\left (-x\right )} - 6 \, a b^{3} e^{\left (-7 \, x\right )} + 3 \,{\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, x\right )} + 22 \,{\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-3 \, x\right )} +{\left (25 \, a^{4} - 56 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-4 \, x\right )} - 22 \,{\left (2 \, a^{3} b - a b^{3}\right )} e^{\left (-5 \, x\right )} + 3 \,{\left (9 \, a^{2} b^{2} - b^{4}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \,{\left (8 \, a b^{8} e^{\left (-x\right )} - 8 \, a b^{8} e^{\left (-7 \, x\right )} + b^{9} e^{\left (-8 \, x\right )} + b^{9} + 4 \,{\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-2 \, x\right )} + 8 \,{\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-3 \, x\right )} + 2 \,{\left (8 \, a^{4} b^{5} - 24 \, a^{2} b^{7} + 3 \, b^{9}\right )} e^{\left (-4 \, x\right )} - 8 \,{\left (4 \, a^{3} b^{6} - 3 \, a b^{8}\right )} e^{\left (-5 \, x\right )} + 4 \,{\left (6 \, a^{2} b^{7} - b^{9}\right )} e^{\left (-6 \, x\right )}\right )}} + \frac{x}{b^{5}} + \frac{\log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{b^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="maxima")

[Out]

4/3*(6*a*b^3*e^(-x) - 6*a*b^3*e^(-7*x) + 3*(9*a^2*b^2 - b^4)*e^(-2*x) + 22*(2*a^3*b - a*b^3)*e^(-3*x) + (25*a^
4 - 56*a^2*b^2 + 3*b^4)*e^(-4*x) - 22*(2*a^3*b - a*b^3)*e^(-5*x) + 3*(9*a^2*b^2 - b^4)*e^(-6*x))/(8*a*b^8*e^(-
x) - 8*a*b^8*e^(-7*x) + b^9*e^(-8*x) + b^9 + 4*(6*a^2*b^7 - b^9)*e^(-2*x) + 8*(4*a^3*b^6 - 3*a*b^8)*e^(-3*x) +
2*(8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*e^(-4*x) - 8*(4*a^3*b^6 - 3*a*b^8)*e^(-5*x) + 4*(6*a^2*b^7 - b^9)*e^(-6*x)
) + x/b^5 + log(-2*a*e^(-x) + b*e^(-2*x) - b)/b^5

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Fricas [B]  time = 2.91015, size = 6251, normalized size = 65.8 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="fricas")

[Out]

-1/3*(3*b^4*x*cosh(x)^8 + 3*b^4*x*sinh(x)^8 + 24*(a*b^3*x - a*b^3)*cosh(x)^7 + 24*(b^4*x*cosh(x) + a*b^3*x - a
*b^3)*sinh(x)^7 - 12*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^6 + 12*(7*b^4*x*cosh(x)^2 - 9*a^2*b^2 + b
^4 + (6*a^2*b^2 - b^4)*x + 14*(a*b^3*x - a*b^3)*cosh(x))*sinh(x)^6 - 8*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a
*b^3)*x)*cosh(x)^5 + 8*(21*b^4*x*cosh(x)^3 - 22*a^3*b + 11*a*b^3 + 63*(a*b^3*x - a*b^3)*cosh(x)^2 + 3*(4*a^3*b
- 3*a*b^3)*x - 9*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x))*sinh(x)^5 + 3*b^4*x - 2*(50*a^4 - 112*a^2*b
^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x)*cosh(x)^4 + 2*(105*b^4*x*cosh(x)^4 - 50*a^4 + 112*a^2*b^2 - 6*b
^4 + 420*(a*b^3*x - a*b^3)*cosh(x)^3 - 90*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^2 + 3*(8*a^4 - 24*a^
2*b^2 + 3*b^4)*x - 20*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x))*sinh(x)^4 + 8*(22*a^3*b - 11*a*
b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^3 + 8*(21*b^4*x*cosh(x)^5 + 105*(a*b^3*x - a*b^3)*cosh(x)^4 + 22*a^3*b
- 11*a*b^3 - 30*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^3 - 10*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a
*b^3)*x)*cosh(x)^2 - 3*(4*a^3*b - 3*a*b^3)*x - (50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*
x)*cosh(x))*sinh(x)^3 - 12*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^2 + 4*(21*b^4*x*cosh(x)^6 + 126*(a*
b^3*x - a*b^3)*cosh(x)^5 - 45*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^4 - 27*a^2*b^2 + 3*b^4 - 20*(22*
a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^3 - 3*(50*a^4 - 112*a^2*b^2 + 6*b^4 - 3*(8*a^4 - 24*a^2*b^
2 + 3*b^4)*x)*cosh(x)^2 + 3*(6*a^2*b^2 - b^4)*x + 6*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x))*s
inh(x)^2 - 24*(a*b^3*x - a*b^3)*cosh(x) - 3*(b^4*cosh(x)^8 + b^4*sinh(x)^8 + 8*a*b^3*cosh(x)^7 + 8*(b^4*cosh(x
) + a*b^3)*sinh(x)^7 + 4*(6*a^2*b^2 - b^4)*cosh(x)^6 + 4*(7*b^4*cosh(x)^2 + 14*a*b^3*cosh(x) + 6*a^2*b^2 - b^4
)*sinh(x)^6 + 8*(4*a^3*b - 3*a*b^3)*cosh(x)^5 + 8*(7*b^4*cosh(x)^3 + 21*a*b^3*cosh(x)^2 + 4*a^3*b - 3*a*b^3 +
3*(6*a^2*b^2 - b^4)*cosh(x))*sinh(x)^5 - 8*a*b^3*cosh(x) + 2*(8*a^4 - 24*a^2*b^2 + 3*b^4)*cosh(x)^4 + 2*(35*b^
4*cosh(x)^4 + 140*a*b^3*cosh(x)^3 + 8*a^4 - 24*a^2*b^2 + 3*b^4 + 30*(6*a^2*b^2 - b^4)*cosh(x)^2 + 20*(4*a^3*b
- 3*a*b^3)*cosh(x))*sinh(x)^4 + b^4 - 8*(4*a^3*b - 3*a*b^3)*cosh(x)^3 + 8*(7*b^4*cosh(x)^5 + 35*a*b^3*cosh(x)^
4 - 4*a^3*b + 3*a*b^3 + 10*(6*a^2*b^2 - b^4)*cosh(x)^3 + 10*(4*a^3*b - 3*a*b^3)*cosh(x)^2 + (8*a^4 - 24*a^2*b^
2 + 3*b^4)*cosh(x))*sinh(x)^3 + 4*(6*a^2*b^2 - b^4)*cosh(x)^2 + 4*(7*b^4*cosh(x)^6 + 42*a*b^3*cosh(x)^5 + 15*(
6*a^2*b^2 - b^4)*cosh(x)^4 + 6*a^2*b^2 - b^4 + 20*(4*a^3*b - 3*a*b^3)*cosh(x)^3 + 3*(8*a^4 - 24*a^2*b^2 + 3*b^
4)*cosh(x)^2 - 6*(4*a^3*b - 3*a*b^3)*cosh(x))*sinh(x)^2 + 8*(b^4*cosh(x)^7 + 7*a*b^3*cosh(x)^6 + 3*(6*a^2*b^2
- b^4)*cosh(x)^5 + 5*(4*a^3*b - 3*a*b^3)*cosh(x)^4 - a*b^3 + (8*a^4 - 24*a^2*b^2 + 3*b^4)*cosh(x)^3 - 3*(4*a^3
*b - 3*a*b^3)*cosh(x)^2 + (6*a^2*b^2 - b^4)*cosh(x))*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + 8*(
3*b^4*x*cosh(x)^7 + 21*(a*b^3*x - a*b^3)*cosh(x)^6 - 9*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x)^5 - 3*a
*b^3*x - 5*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^4 + 3*a*b^3 - (50*a^4 - 112*a^2*b^2 + 6*b^4
- 3*(8*a^4 - 24*a^2*b^2 + 3*b^4)*x)*cosh(x)^3 + 3*(22*a^3*b - 11*a*b^3 - 3*(4*a^3*b - 3*a*b^3)*x)*cosh(x)^2 -
3*(9*a^2*b^2 - b^4 - (6*a^2*b^2 - b^4)*x)*cosh(x))*sinh(x))/(b^9*cosh(x)^8 + b^9*sinh(x)^8 + 8*a*b^8*cosh(x)^
7 - 8*a*b^8*cosh(x) + b^9 + 8*(b^9*cosh(x) + a*b^8)*sinh(x)^7 + 4*(6*a^2*b^7 - b^9)*cosh(x)^6 + 4*(7*b^9*cosh(
x)^2 + 14*a*b^8*cosh(x) + 6*a^2*b^7 - b^9)*sinh(x)^6 + 8*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^5 + 8*(7*b^9*cosh(x)^3
+ 21*a*b^8*cosh(x)^2 + 4*a^3*b^6 - 3*a*b^8 + 3*(6*a^2*b^7 - b^9)*cosh(x))*sinh(x)^5 + 2*(8*a^4*b^5 - 24*a^2*b^
7 + 3*b^9)*cosh(x)^4 + 2*(35*b^9*cosh(x)^4 + 140*a*b^8*cosh(x)^3 + 8*a^4*b^5 - 24*a^2*b^7 + 3*b^9 + 30*(6*a^2*
b^7 - b^9)*cosh(x)^2 + 20*(4*a^3*b^6 - 3*a*b^8)*cosh(x))*sinh(x)^4 - 8*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^3 + 8*(7*
b^9*cosh(x)^5 + 35*a*b^8*cosh(x)^4 - 4*a^3*b^6 + 3*a*b^8 + 10*(6*a^2*b^7 - b^9)*cosh(x)^3 + 10*(4*a^3*b^6 - 3*
a*b^8)*cosh(x)^2 + (8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x))*sinh(x)^3 + 4*(6*a^2*b^7 - b^9)*cosh(x)^2 + 4*(7*
b^9*cosh(x)^6 + 42*a*b^8*cosh(x)^5 + 6*a^2*b^7 - b^9 + 15*(6*a^2*b^7 - b^9)*cosh(x)^4 + 20*(4*a^3*b^6 - 3*a*b^
8)*cosh(x)^3 + 3*(8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x)^2 - 6*(4*a^3*b^6 - 3*a*b^8)*cosh(x))*sinh(x)^2 + 8*(
b^9*cosh(x)^7 + 7*a*b^8*cosh(x)^6 - a*b^8 + 3*(6*a^2*b^7 - b^9)*cosh(x)^5 + 5*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^4
+ (8*a^4*b^5 - 24*a^2*b^7 + 3*b^9)*cosh(x)^3 - 3*(4*a^3*b^6 - 3*a*b^8)*cosh(x)^2 + (6*a^2*b^7 - b^9)*cosh(x))*
sinh(x))

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Sympy [A]  time = 28.2562, size = 2166, normalized size = 22.8 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))**5,x)

[Out]

Piecewise((12*a**4*x*sech(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh
(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 12*a**4*log(a*sech(x)/b + tanh(x))*se
ch(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*
a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 12*a**4*log(tanh(x) + 1)*sech(x)**4/(12*a**4*b**5*sech(x)**4
+ 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b*
*9*tanh(x)**4) + 11*a**4*sech(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*
tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 48*a**3*b*x*tanh(x)*sech(x)**3/(1
2*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh
(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 48*a**3*b*log(a*sech(x)/b + tanh(x))*tanh(x)*sech(x)**3/(12*a**4*b**5*s
ech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x
) + 12*b**9*tanh(x)**4) - 48*a**3*b*log(tanh(x) + 1)*tanh(x)*sech(x)**3/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**
6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4)
+ 32*a**3*b*tanh(x)*sech(x)**3/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh
(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 72*a**2*b**2*x*tanh(x)**2*sech(x)**2/
(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*ta
nh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 72*a**2*b**2*log(a*sech(x)/b + tanh(x))*tanh(x)**2*sech(x)**2/(12*a**
4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**
3*sech(x) + 12*b**9*tanh(x)**4) - 72*a**2*b**2*log(tanh(x) + 1)*tanh(x)**2*sech(x)**2/(12*a**4*b**5*sech(x)**4
+ 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b*
*9*tanh(x)**4) + 26*a**2*b**2*tanh(x)**2*sech(x)**2/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3
+ 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 2*a**2*b**2*sech(
x)**2/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b
**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 48*a*b**3*x*tanh(x)**3*sech(x)/(12*a**4*b**5*sech(x)**4 + 48*a*
*3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(
x)**4) + 48*a*b**3*log(a*sech(x)/b + tanh(x))*tanh(x)**3*sech(x)/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(
x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 48*a
*b**3*log(tanh(x) + 1)*tanh(x)**3*sech(x)/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2
*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 8*a*b**3*tanh(x)*sech(x)/(1
2*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh
(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 12*b**4*x*tanh(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*se
ch(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) + 12*b**4*l
og(a*sech(x)/b + tanh(x))*tanh(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7
*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 12*b**4*log(tanh(x) + 1)*tanh(x)
**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**
8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 5*b**4*tanh(x)**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)
*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 6*b**4
*tanh(x)**2/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*sech(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 +
48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4) - 3*b**4/(12*a**4*b**5*sech(x)**4 + 48*a**3*b**6*tanh(x)*se
ch(x)**3 + 72*a**2*b**7*tanh(x)**2*sech(x)**2 + 48*a*b**8*tanh(x)**3*sech(x) + 12*b**9*tanh(x)**4), Ne(b, 0)),
((8*tanh(x)**5/(15*sech(x)**5) - 4*tanh(x)**3/(3*sech(x)**5) + tanh(x)/sech(x)**5)/a**5, True))

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Giac [A]  time = 1.18885, size = 205, normalized size = 2.16 \begin{align*} \frac{\log \left ({\left | -b{\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b^{5}} - \frac{25 \, b^{3}{\left (e^{\left (-x\right )} - e^{x}\right )}^{4} - 104 \, a b^{2}{\left (e^{\left (-x\right )} - e^{x}\right )}^{3} + 168 \, a^{2} b{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 48 \, b^{3}{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 96 \, a^{3}{\left (e^{\left (-x\right )} - e^{x}\right )} - 64 \, a b^{2}{\left (e^{\left (-x\right )} - e^{x}\right )} + 32 \, a^{2} b + 48 \, b^{3}}{12 \,{\left (b{\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a\right )}^{4} b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)+b*tanh(x))^5,x, algorithm="giac")

[Out]

log(abs(-b*(e^(-x) - e^x) + 2*a))/b^5 - 1/12*(25*b^3*(e^(-x) - e^x)^4 - 104*a*b^2*(e^(-x) - e^x)^3 + 168*a^2*b
*(e^(-x) - e^x)^2 + 48*b^3*(e^(-x) - e^x)^2 - 96*a^3*(e^(-x) - e^x) - 64*a*b^2*(e^(-x) - e^x) + 32*a^2*b + 48*
b^3)/((b*(e^(-x) - e^x) - 2*a)^4*b^4)