### 3.132 $$\int \frac{\text{sech}^6(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=131 $\frac{3 \left (\frac{1}{a^2}-\frac{1}{b^2}\right ) \tanh (c+d x)}{8 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\left (3 a^2-2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2} d}+\frac{(a+b) \tanh (c+d x) \text{sech}^2(c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}$

[Out]

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

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Rubi [A]  time = 0.123816, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.174, Rules used = {3675, 413, 385, 205} $\frac{3 \left (\frac{1}{a^2}-\frac{1}{b^2}\right ) \tanh (c+d x)}{8 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\left (3 a^2-2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{5/2} d}+\frac{(a+b) \tanh (c+d x) \text{sech}^2(c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]])/(8*a^(5/2)*b^(5/2)*d) + ((a + b)*Sech[c + d*
x]^2*Tanh[c + d*x])/(4*a*b*d*(a + b*Tanh[c + d*x]^2)^2) + (3*(a^(-2) - b^(-2))*Tanh[c + d*x])/(8*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 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

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

Mathematica [A]  time = 0.893839, size = 128, normalized size = 0.98 $\frac{\left (3 a^2-2 a b+3 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )-\frac{\sqrt{a} \sqrt{b} (a+b) \sinh (2 (c+d x)) \left (3 \left (a^2-b^2\right ) \cosh (2 (c+d x))+3 a^2-10 a b+3 b^2\right )}{((a+b) \cosh (2 (c+d x))+a-b)^2}}{8 a^{5/2} b^{5/2} d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[c + d*x]^6/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

((3*a^2 - 2*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]] - (Sqrt[a]*Sqrt[b]*(a + b)*(3*a^2 - 10*a*b +
3*b^2 + 3*(a^2 - b^2)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/(a - b + (a + b)*Cosh[2*(c + d*x)])^2)/(8*a^(5/2)*
b^(5/2)*d)

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Maple [B]  time = 0.114, size = 1776, normalized size = 13.6 \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)^3,x)

[Out]

-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))-1/8/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))-1/8/d/(b*(a+b))^(1/2)/a/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arctan(a*ta
nh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))-1/8/d/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))-1/4/d/a/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))-1/8/d/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))+1/4/d/a/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))-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
))-3/8/d*a/b^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))-3/8/d*a/b^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))-9/4/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/b^2*tanh(1/2*d*x+1/2*c)^5-9/4/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2*c)^2*a+4*t
anh(1/2*d*x+1/2*c)^2*b+a)^2*a/b^2*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+3/8/d/a^2/((2*(b*(a+b))^(1/2)-a-2*b)*a)^(1/2)*arct
anh(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)*a
rctan(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^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^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))+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+7/4/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+7/4/d/(tanh(1/2*d*x+1/2*c)^4*a+2*tan
h(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+5/4/d/(tanh(1/2*d*x+1/2*c)^4*a+2*t
anh(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)-7/2/d/(tanh(1/2*d*x+1/2*c)^4*a+2*t
anh(1/2*d*x+1/2*c)^2*a+4*tanh(1/2*d*x+1/2*c)^2*b+a)^2/b*tanh(1/2*d*x+1/2*c)^3+1/2/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)^2/b*tanh(1/2*d*x+1/2*c)+1/2/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)^2/b*tanh(1/2*d*x+1/2*c)^7-7/2/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)^2/b*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)^5-3/4/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/b^2*tanh(1/2*d*x+1/2*c)-3/4/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/b^2*tanh(1/2*d*x+1/2*c)^7

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.86921, size = 11889, normalized size = 90.76 \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)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^6 + 24*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*
cosh(d*x + c)*sinh(d*x + c)^5 + 4*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 + 12*a^4*b + 12*a^3*
b^2 - 12*a^2*b^3 - 12*a*b^4 + 12*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 12*(3*a^4*b - 5
*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4 + 5*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 +
16*(5*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^3 + 3*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*
cosh(d*x + c))*sinh(d*x + c)^3 + 4*(9*a^4*b - 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2 + 4*(9*a^4*b
- 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4 + 15*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 + 18*(3*a^4*b
- 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3
+ 3*b^4)*cosh(d*x + c)^8 + 8*(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)^7 +
(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d
*x + c)^6 + 4*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4 + 7*(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)^6 + 8*(7*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(3*a^4 - 2*
a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)
*cosh(d*x + c)^4 + 2*(35*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 9*a^4 - 12*a^3*b +
22*a^2*b^2 - 12*a*b^3 + 9*b^4 + 30*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^
4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4 + 8*(7*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)
^5 + 10*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^
4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^4 + 4*a^
3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^4 +
3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4 + 3*(9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh
(d*x + c)^2 + 8*((3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(3*a^4 - 2*a^3*b + 2*a*b^
3 - 3*b^4)*cosh(d*x + c)^5 + (9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (3*a^4 - 2*a
^3*b + 2*a*b^3 - 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)*co
sh(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*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^5 + 6*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3
*a*b^4)*cosh(d*x + c)^3 + (9*a^4*b - 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^5*b^
3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7
+ (a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3
+ 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^5*b^3 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^5*b^3 - 2*a^4*b^4
+ 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^5*b^3 - a^3*b^5
)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(a^5*b^3 - a
^3*b^5)*d*cosh(d*x + c)^2 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^5*b^3 - a^3*b^5)*d*c
osh(d*x + c)^2 + 8*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^5*b^3 - a^3*b^5)*d*cosh(d*x +
c)^3 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*
b^5)*d*cosh(d*x + c)^6 + 15*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*co
sh(d*x + c)^2 + (a^5*b^3 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d + 8*((a^5*b^3 + 2*a
^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3
*b^5)*d*cosh(d*x + c)^3 + (a^5*b^3 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(3*a^4*b + a^3*b^2 + a^2
*b^3 + 3*a*b^4)*cosh(d*x + c)^6 + 12*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 2
*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^6 + 6*a^4*b + 6*a^3*b^2 - 6*a^2*b^3 - 6*a*b^4 + 6*(3*a^
4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 6*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4 + 5*(3*a
^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a
*b^4)*cosh(d*x + c)^3 + 3*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(9*a^
4*b - 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2 + 2*(9*a^4*b - 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4 + 15
*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 + 18*(3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(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)^7 + (3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 +
3*b^4)*sinh(d*x + c)^8 + 4*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(3*a^4 - 2*a^3*b + 2*a*b^3
- 3*b^4 + 7*(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)^6 + 8*(7*(3*a^4 +
4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c))*
sinh(d*x + c)^5 + 2*(9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 + 4*a^3*
b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4 + 30*(3*a^
4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^
4 + 8*(7*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b
^4)*cosh(d*x + c)^3 + (9*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3
*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(3*a^4 + 4*a^3*b + 2*a^2*b^2 + 4*a*b^3 + 3*b^4)*cosh(
d*x + c)^6 + 15*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4 + 3*(9
*a^4 - 12*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 + 4*a^3*b + 2*a^
2*b^2 + 4*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (9*a^4 - 12
*a^3*b + 22*a^2*b^2 - 12*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (3*a^4 - 2*a^3*b + 2*a*b^3 - 3*b^4)*cosh(d*x + c))*s
inh(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*(3*a^4*b + a^3*b^2 + a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^5 + 6*(
3*a^4*b - 5*a^3*b^2 + 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (9*a^4*b - 13*a^3*b^2 - 13*a^2*b^3 + 9*a*b^4)*cos
h(d*x + c))*sinh(d*x + c))/((a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 + 2*a^4*b^4 + a^3*b
^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^5*b^3 - a^3*b^5
)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^5*b^3 - a^3*b^5)*d)*sinh(d*x
+ c)^6 + 2*(3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cos
h(d*x + c)^3 + 3*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*
d*cosh(d*x + c)^4 + 30*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x
+ c)^4 + 4*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 1
0*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3
+ 4*(7*(a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^
5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^5*b^3 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^5*b^3 + 2*a^4*b
^4 + a^3*b^5)*d + 8*((a^5*b^3 + 2*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^5*b^3 - a^3*b^5)*d*cosh(d*x + c)
^5 + (3*a^5*b^3 - 2*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^5*b^3 - a^3*b^5)*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)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.93837, size = 456, normalized size = 3.48 \begin{align*} \frac{\frac{{\left (3 \, a^{2} e^{\left (2 \, c\right )} - 2 \, a b e^{\left (2 \, c\right )} + 3 \, 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^{2} b^{2}} + \frac{2 \,{\left (3 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 15 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 13 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 13 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - 3 \, b^{3}\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} a^{2} b^{2}}}{8 \, 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)^3,x, algorithm="giac")

[Out]

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