### 3.123 $$\int \frac{\text{sech}^6(c+d x)}{(a+b \tanh ^2(c+d x))^2} \, dx$$

Optimal. Leaf size=97 $-\frac{(3 a-b) (a+b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac{(a+b)^2 \tanh (c+d x)}{2 a b^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\tanh (c+d x)}{b^2 d}$

[Out]

-((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d) + Tanh[c + d*x]/(b^2*d) + (
(a + b)^2*Tanh[c + d*x])/(2*a*b^2*d*(a + b*Tanh[c + d*x]^2))

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Rubi [A]  time = 0.1295, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.174, Rules used = {3675, 390, 385, 205} $-\frac{(3 a-b) (a+b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{2 a^{3/2} b^{5/2} d}+\frac{(a+b)^2 \tanh (c+d x)}{2 a b^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\tanh (c+d x)}{b^2 d}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

-((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(2*a^(3/2)*b^(5/2)*d) + Tanh[c + d*x]/(b^2*d) + (
(a + b)^2*Tanh[c + d*x])/(2*a*b^2*d*(a + b*Tanh[c + d*x]^2))

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d)*x*(a + b*x^n)^(p +
1))/(a*b*n*(p + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /
; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 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]

Rubi steps

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

Mathematica [A]  time = 0.58541, size = 102, normalized size = 1.05 $\frac{-\frac{(3 a-b) (a+b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} (a+b)^2 \sinh (2 (c+d x))}{a ((a+b) \cosh (2 (c+d x))+a-b)}+2 \sqrt{b} \tanh (c+d x)}{2 b^{5/2} d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^2,x]

[Out]

(-(((3*a - b)*(a + b)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/a^(3/2)) + (Sqrt[b]*(a + b)^2*Sinh[2*(c + d*x)]
)/(a*(a - b + (a + b)*Cosh[2*(c + d*x)])) + 2*Sqrt[b]*Tanh[c + d*x])/(2*b^(5/2)*d)

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Maple [B]  time = 0.092, size = 1283, normalized size = 13.2 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^6/(a+b*tanh(d*x+c)^2)^2,x)

[Out]

1/d/b^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*a*tanh(1/2*d*x+1/2*c)^
3+2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/b*tanh(1/2*d*x+1/2*c)^3+
1/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)^3+1/
d/b^2/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)*a*tanh(1/2*d*x+1/2*c)+2/
d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/b*tanh(1/2*d*x+1/2*c)+1/d/(t
anh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)/a*tanh(1/2*d*x+1/2*c)+3/2/d/b^2*
a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*
b)*a)^(1/2))-3/2/d/b^2*a/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)
-a-2*b)*a)^(1/2))+5/2/d/b*a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/
((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+3/2/d/b^2*a^2/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(
a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+3/2/d/b^2*a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arc
tan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+5/2/d/b*a/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a
+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-1/d/b/((2*(b*(a+b))^(1/2)-a-2
*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+1/2/d/(b*(a+b))^(1/2)/((2*(b*(
a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))+1/d/b/((2*(b*(a
+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+1/2/d/(b*(a+b))^(
1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))+1/2
/d/a/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2))-1/
2/d/(b*(a+b))^(1/2)/a/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arctanh(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)-a-
2*b)*a)^(1/2))*b-1/2/d/a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+
a+2*b)*a)^(1/2))-1/2/d/(b*(a+b))^(1/2)/a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*tanh(1/2*d*x+1/2*c)/((2*
(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))*b+2/d/b^2*tanh(1/2*d*x+1/2*c)/(tanh(1/2*d*x+1/2*c)^2+1)

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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(d*x+c)^6/(a+b*tanh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.59689, size = 6589, normalized size = 67.93 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*tanh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

[-1/4*(4*(3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 16*(3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d
*x + c)^3 + 4*(3*a^3*b + 2*a^2*b^2 - a*b^3)*sinh(d*x + c)^4 + 12*a^3*b + 16*a^2*b^2 + 4*a*b^3 + 8*(3*a^3*b - a
^2*b^2)*cosh(d*x + c)^2 + 8*(3*a^3*b - a^2*b^2 + 3*(3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c
)^2 - ((3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)*sinh(
d*x + c)^5 + (3*a^3 + 5*a^2*b + a*b^2 - b^3)*sinh(d*x + c)^6 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c)
^4 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3 + 15*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
4*(5*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c))*sinh(d
*x + c)^3 + 3*a^3 + 5*a^2*b + a*b^2 - b^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c)^2 + (15*(3*a^3 + 5
*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^4 + 9*a^3 + 3*a^2*b - 5*a*b^2 + b^3 + 6*(9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 + 2*(3*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(9*a^3 + 3*a^2*b -
5*a*b^2 + b^3)*cosh(d*x + c)^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*l
og(((a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*a*b + b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*a*b +
b^2)*sinh(d*x + c)^4 + 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^2 + a^2 - b^2)*
sinh(d*x + c)^2 + a^2 - 6*a*b + b^2 + 4*((a^2 + 2*a*b + b^2)*cosh(d*x + c)^3 + (a^2 - b^2)*cosh(d*x + c))*sinh
(d*x + c) - 4*((a + b)*cosh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a -
b)*sqrt(-a*b))/((a + b)*cosh(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 +
2*(a - b)*cosh(d*x + c)^2 + 2*(3*(a + b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^
3 + (a - b)*cosh(d*x + c))*sinh(d*x + c) + a + b)) + 16*((3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^3 + (3*a^
3*b - a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^6 + 6*(a^3*b^3 + a^2*b^4)*d*
cosh(d*x + c)*sinh(d*x + c)^5 + (a^3*b^3 + a^2*b^4)*d*sinh(d*x + c)^6 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^
4 + (15*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + (3*a^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^4 + (3*a^3*b^3 - a^2*b^
4)*d*cosh(d*x + c)^2 + 4*(5*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^3 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sin
h(d*x + c)^3 + (15*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4 + 6*(3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^2 + (3*a^3*
b^3 - a^2*b^4)*d)*sinh(d*x + c)^2 + (a^3*b^3 + a^2*b^4)*d + 2*(3*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^5 + 2*(3*
a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)), -1/2*(2*(3*a^3*b
+ 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^4 + 8*(3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + 2*(3*a
^3*b + 2*a^2*b^2 - a*b^3)*sinh(d*x + c)^4 + 6*a^3*b + 8*a^2*b^2 + 2*a*b^3 + 4*(3*a^3*b - a^2*b^2)*cosh(d*x + c
)^2 + 4*(3*a^3*b - a^2*b^2 + 3*(3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^3 + 5*a^
2*b + a*b^2 - b^3)*cosh(d*x + c)^6 + 6*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + (3*a^3
+ 5*a^2*b + a*b^2 - b^3)*sinh(d*x + c)^6 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c)^4 + (9*a^3 + 3*a^2*
b - 5*a*b^2 + b^3 + 15*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(3*a^3 + 5*a^2*
b + a*b^2 - b^3)*cosh(d*x + c)^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a^3 +
5*a^2*b + a*b^2 - b^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c)^2 + (15*(3*a^3 + 5*a^2*b + a*b^2 - b^3
)*cosh(d*x + c)^4 + 9*a^3 + 3*a^2*b - 5*a*b^2 + b^3 + 6*(9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c)^2)*sin
h(d*x + c)^2 + 2*(3*(3*a^3 + 5*a^2*b + a*b^2 - b^3)*cosh(d*x + c)^5 + 2*(9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh
(d*x + c)^3 + (9*a^3 + 3*a^2*b - 5*a*b^2 + b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*((a + b)*co
sh(d*x + c)^2 + 2*(a + b)*cosh(d*x + c)*sinh(d*x + c) + (a + b)*sinh(d*x + c)^2 + a - b)*sqrt(a*b)/(a*b)) + 8*
((3*a^3*b + 2*a^2*b^2 - a*b^3)*cosh(d*x + c)^3 + (3*a^3*b - a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^3*b^3 +
a^2*b^4)*d*cosh(d*x + c)^6 + 6*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^5 + (a^3*b^3 + a^2*b^4)*d*si
nh(d*x + c)^6 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^4 + (15*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^2 + (3*a^3*b
^3 - a^2*b^4)*d)*sinh(d*x + c)^4 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^2 + 4*(5*(a^3*b^3 + a^2*b^4)*d*cosh(d
*x + c)^3 + (3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + (15*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^4
+ 6*(3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^2 + (3*a^3*b^3 - a^2*b^4)*d)*sinh(d*x + c)^2 + (a^3*b^3 + a^2*b^4)*
d + 2*(3*(a^3*b^3 + a^2*b^4)*d*cosh(d*x + c)^5 + 2*(3*a^3*b^3 - a^2*b^4)*d*cosh(d*x + c)^3 + (3*a^3*b^3 - a^2*
b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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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(d*x+c)**6/(a+b*tanh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.66475, size = 335, normalized size = 3.45 \begin{align*} -\frac{\frac{{\left (3 \, a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} - b^{2} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac{a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt{a b}}\right ) e^{\left (-2 \, c\right )}}{\sqrt{a b} a b^{2}} + \frac{2 \,{\left (3 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} - b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} + 4 \, a b + b^{2}\right )}}{{\left (a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} a b^{2}}}{2 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*tanh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-1/2*((3*a^2*e^(2*c) + 2*a*b*e^(2*c) - b^2*e^(2*c))*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)
/sqrt(a*b))*e^(-2*c)/(sqrt(a*b)*a*b^2) + 2*(3*a^2*e^(4*d*x + 4*c) + 2*a*b*e^(4*d*x + 4*c) - b^2*e^(4*d*x + 4*c
) + 6*a^2*e^(2*d*x + 2*c) - 2*a*b*e^(2*d*x + 2*c) + 3*a^2 + 4*a*b + b^2)/((a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*
c) + 3*a*e^(4*d*x + 4*c) - b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) - b*e^(2*d*x + 2*c) + a + b)*a*b^2))/d