### 3.128 $$\int \frac{\text{sech}^2(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=96 $\frac{3 \tanh (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}+\frac{\tanh (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}$

[Out]

(3*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*d) + Tanh[c + d*x]/(4*a*d*(a + b*Tanh[c + d*x]^
2)^2) + (3*Tanh[c + d*x])/(8*a^2*d*(a + b*Tanh[c + d*x]^2))

________________________________________________________________________________________

Rubi [A]  time = 0.0754865, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.13, Rules used = {3675, 199, 205} $\frac{3 \tanh (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}+\frac{\tanh (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

(3*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*Sqrt[b]*d) + Tanh[c + d*x]/(4*a*d*(a + b*Tanh[c + d*x]^
2)^2) + (3*Tanh[c + d*x])/(8*a^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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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}^2(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\tanh (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=\frac{\tanh (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{3 \tanh (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} d}+\frac{\tanh (c+d x)}{4 a d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{3 \tanh (c+d x)}{8 a^2 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.748001, size = 77, normalized size = 0.8 $\frac{\frac{\tanh (c+d x) \left (5 a+3 b \tanh ^2(c+d x)\right )}{a^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}}{8 d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

((3*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(a^(5/2)*Sqrt[b]) + (Tanh[c + d*x]*(5*a + 3*b*Tanh[c + d*x]^2))/(
a^2*(a + b*Tanh[c + d*x]^2)^2))/(8*d)

________________________________________________________________________________________

Maple [B]  time = 0.108, size = 764, normalized size = 8. \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)^2/(a+b*tanh(d*x+c)^2)^3,x)

[Out]

5/4/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)^2/a*tanh(1/2*d*x+1/2*c)^
7+15/4/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)^2/a*tanh(1/2*d*x+1/2*
c)^5+3/d/a^2*b/(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)^2*tanh(1/2*d*x+
1/2*c)^5+15/4/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)^2/a*tanh(1/2*d
*x+1/2*c)^3+3/d/a^2*b/(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)^2*tanh(1
/2*d*x+1/2*c)^3+5/4/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)^2/a*tanh
(1/2*d*x+1/2*c)-3/8/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))+3/8/d/a^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))-3/8/d/a^2*b/(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))-3/8/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))-3/8/d/a^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))-3/8/d/a^2*b/(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))

________________________________________________________________________________________

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)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 3.04773, size = 13154, normalized size = 137.02 \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)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 + 24*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*
b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^6 + 20*a^4*b +
52*a^3*b^2 + 44*a^2*b^3 + 12*a*b^4 + 4*(15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 + 4*(15*a^4*
b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4 + 15*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c
)^4 + 16*(5*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*
b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(15*a^4*b + 13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 + 4*(15
*a^4*b + 13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4 + 15*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 6*
(15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 3*((a^4 + 4*a^3*b + 6*a^2*b^2 +
4*a*b^3 + b^4)*cosh(d*x + c)^8 + 8*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^7 +
(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)
^6 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*
x + c)^6 + 8*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b - 2*a*b^3 - b^4
)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*(
a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^4 + 3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4 + 3
0*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4
+ 8*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(
d*x + c)^3 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b
- 2*a*b^3 - b^4)*cosh(d*x + c)^2 + 4*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(a^4
+ 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^4 + a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 3*(3*a^4 + 4*a^3*b + 2*a^2*b^2 +
4*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x +
c)^7 + 3*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^5 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*co
sh(d*x + c)^3 + (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b)*log(((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)) + 8*(3*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 2*(15*a^4*b
- a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^3 + (15*a^4*b + 13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4)*cosh(d*x + c
))*sinh(d*x + c))/((a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^7*b + 4*a^6*
b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^
4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^7
*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5
)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a
^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3
*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*
x + c)^4 + 30*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 +
4*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8
*(7*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^7*b + 2*a^6*b^2 - 2*a^4*b^
4 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*si
nh(d*x + c)^3 + 4*(7*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^7*b + 2*a
^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*
d*cosh(d*x + c)^2 + (a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 6*a^5*
b^3 + 4*a^4*b^4 + a^3*b^5)*d + 8*((a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*
(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3
*a^3*b^5)*d*cosh(d*x + c)^3 + (a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), -1/8*
(2*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 + 12*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*co
sh(d*x + c)*sinh(d*x + c)^5 + 2*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^6 + 10*a^4*b + 26*a^3*
b^2 + 22*a^2*b^3 + 6*a*b^4 + 2*(15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 + 2*(15*a^4*b - a^3*
b^2 + 9*a^2*b^3 + 9*a*b^4 + 15*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*
(5*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh
(d*x + c))*sinh(d*x + c)^3 + 2*(15*a^4*b + 13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 + 2*(15*a^4*b +
13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4 + 15*(5*a^4*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 6*(15*a^4*b
- a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 3*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 +
b^4)*cosh(d*x + c)^8 + 8*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 4
*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*sinh(d*x + c)^8 + 4*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^6 + 4*(a
^4 + 2*a^3*b - 2*a*b^3 - b^4 + 7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6
+ 8*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^3 + 3*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*
x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 + 4*a
^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^4 + 3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4 + 30*(a^4 +
2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 + 8*(7*(
a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^5 + 10*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^
3 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 2*a^3*b - 2*a*b^
3 - b^4)*cosh(d*x + c)^2 + 4*(7*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^6 + 15*(a^4 + 2*a^3*
b - 2*a*b^3 - b^4)*cosh(d*x + c)^4 + a^4 + 2*a^3*b - 2*a*b^3 - b^4 + 3*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3
+ 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cosh(d*x + c)^7 + 3
*(a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c)^5 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x +
c)^3 + (a^4 + 2*a^3*b - 2*a*b^3 - b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b)*arctan(1/2*((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)) + 4*(3*(5*a^4
*b - a^3*b^2 - 9*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 2*(15*a^4*b - a^3*b^2 + 9*a^2*b^3 + 9*a*b^4)*cosh(d*x +
c)^3 + (15*a^4*b + 13*a^3*b^2 - 11*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^7*b + 4*a^6*b^2 + 6*a^
5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cos
h(d*x + c)*sinh(d*x + c)^7 + (a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^7*
b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3
*b^5)*d*cosh(d*x + c)^2 + (a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^7*b + 4*a^6*b^
2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a
^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(
35*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4
- a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 +
4*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b
^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^7*b +
4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7*b + 4*a^6*b^2 + 6*
a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c
)^4 + 3*(3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^7*b + 2*a^6*b^2 - 2*a
^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d + 8*((a^7*b + 4
*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^7*b + 2*a^6*b^2 - 2*a^4*b^4 - a^3*b^5)*d*
cosh(d*x + c)^5 + (3*a^7*b + 4*a^6*b^2 + 2*a^5*b^3 + 4*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^7*b + 2*a^6
*b^2 - 2*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c))]

________________________________________________________________________________________

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)**2/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.91676, size = 432, normalized size = 4.5 \begin{align*} \frac{\frac{3 \, \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 )}{\sqrt{a b} a^{2}} - \frac{2 \,{\left (5 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 9 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 15 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 11 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, a^{3} + 13 \, a^{2} b + 11 \, a b^{2} + 3 \, b^{3}\right )}}{{\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )}{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}^{2}}}{8 \, d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^2/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*(3*arctan(1/2*(a*e^(2*d*x + 2*c) + b*e^(2*d*x + 2*c) + a - b)/sqrt(a*b))/(sqrt(a*b)*a^2) - 2*(5*a^3*e^(6*d
*x + 6*c) - a^2*b*e^(6*d*x + 6*c) - 9*a*b^2*e^(6*d*x + 6*c) - 3*b^3*e^(6*d*x + 6*c) + 15*a^3*e^(4*d*x + 4*c) -
a^2*b*e^(4*d*x + 4*c) + 9*a*b^2*e^(4*d*x + 4*c) + 9*b^3*e^(4*d*x + 4*c) + 15*a^3*e^(2*d*x + 2*c) + 13*a^2*b*e
^(2*d*x + 2*c) - 11*a*b^2*e^(2*d*x + 2*c) - 9*b^3*e^(2*d*x + 2*c) + 5*a^3 + 13*a^2*b + 11*a*b^2 + 3*b^3)/((a^4
+ 2*a^3*b + a^2*b^2)*(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a +
b)^2))/d