3.130 $$\int \frac{\text{sech}^4(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx$$

Optimal. Leaf size=115 $-\frac{(a-3 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} d}-\frac{(a-3 b) \tanh (c+d x)}{8 a^2 b d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(a+b) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}$

[Out]

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

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Rubi [A]  time = 0.0968225, antiderivative size = 115, 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, 385, 199, 205} $-\frac{(a-3 b) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2} d}-\frac{(a-3 b) \tanh (c+d x)}{8 a^2 b d \left (a+b \tanh ^2(c+d x)\right )}+\frac{(a+b) \tanh (c+d x)}{4 a b d \left (a+b \tanh ^2(c+d x)\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

Mathematica [A]  time = 0.962973, size = 115, normalized size = 1. $\frac{\frac{\sqrt{a} \sinh (2 (c+d x)) \left (\left (a^2+4 a b+3 b^2\right ) \cosh (2 (c+d x))+a^2+6 a b-3 b^2\right )}{b ((a+b) \cosh (2 (c+d x))+a-b)^2}+\frac{(3 b-a) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{b^{3/2}}}{8 a^{5/2} d}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[c + d*x]^4/(a + b*Tanh[c + d*x]^2)^3,x]

[Out]

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

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Maple [B]  time = 0.111, size = 1270, normalized size = 11. \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)^4/(a+b*tanh(d*x+c)^2)^3,x)

[Out]

1/4/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/b*tanh(1/2*d*x+1/2*c)^
7+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+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/b*tanh(1/2*d*x+1/2
*c)^5+11/4/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+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/b*tanh(1/
2*d*x+1/2*c)^3+11/4/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+1/4/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/
b*tanh(1/2*d*x+1/2*c)+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)+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))-1/8/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))-1/4/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/(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))+1/8/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))-1/4/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)*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/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))

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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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.97443, size = 12604, normalized size = 109.6 \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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(4*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 + 24*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)
*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^6 + 4*a^4*b + 20*a^3*
b^2 + 28*a^2*b^3 + 12*a*b^4 + 4*(3*a^4*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 + 4*(3*a^4*b + 7*a
^3*b^2 - 3*a^2*b^3 + 9*a*b^4 + 15*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 1
6*(5*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (3*a^4*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4)*cos
h(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4*b + 13*a^3*b^2 + a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 + 4*(3*a^4*b + 13*a
^3*b^2 + a^2*b^3 - 9*a*b^4 + 15*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 6*(3*a^4*b + 7*a^3*b
^2 - 3*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c
)^8 + 8*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4
)*sinh(d*x + c)^8 + 4*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 4*(a^4 - 2*a^3*b - 4*a^2
*b^2 + 2*a*b^3 + 3*b^4 + 7*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 -
6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + 3*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c))*
sinh(d*x + c)^5 + 2*(3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 6*a^2*b^2 - 8*a*b^3 -
3*b^4)*cosh(d*x + c)^4 + 3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4 + 30*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^
4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4 + 8*(7*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*
b^4)*cosh(d*x + c)^5 + 10*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (3*a^4 - 8*a^3*b - 2
*a^2*b^2 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x +
c)^2 + 4*(7*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 15*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*
b^4)*cosh(d*x + c)^4 + a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4 + 3*(3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4)*c
osh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^7 + 3*(a^4 - 2*a^3*b -
4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + (3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4)*cosh(d*x + c)^3 + (a^4 -
2*a^3*b - 4*a^2*b^2 + 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)*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))*si
nh(d*x + c) + a + b)) + 8*(3*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 2*(3*a^4*b + 7*a^3*b^2
- 3*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^3 + (3*a^4*b + 13*a^3*b^2 + a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x +
c))/((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*
b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^6*
b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh
(d*x + c)^2 + (a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^6*b^2 + a^5*b^3 + a^4*b^4 +
3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^6*b^2
+ a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^
3*b^5)*d*cosh(d*x + c)^4 + 30*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + a^5*b^3
+ a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(
7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d
*cosh(d*x + c)^3 + (3*a^6*b^2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6*b^
2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x
+ c)^4 + 3*(3*a^6*b^2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3
*b^5)*d)*sinh(d*x + c)^2 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d + 8*((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4
+ a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^6*b^2 + a^5
*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(
d*x + c)), -1/8*(2*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^6 + 12*(a^4*b - a^3*b^2 - 5*a^2*b^3 -
3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*sinh(d*x + c)^6 + 2*a^4*b
+ 10*a^3*b^2 + 14*a^2*b^3 + 6*a*b^4 + 2*(3*a^4*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^4 + 2*(3*a^4
*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4 + 15*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x +
c)^4 + 8*(5*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^3 + (3*a^4*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b
^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(3*a^4*b + 13*a^3*b^2 + a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2 + 2*(3*a^4*b
+ 13*a^3*b^2 + a^2*b^3 - 9*a*b^4 + 15*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^4 + 6*(3*a^4*b +
7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(
d*x + c)^8 + 8*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 - 6*a^2*b^2 - 8*a*b^3
- 3*b^4)*sinh(d*x + c)^8 + 4*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 4*(a^4 - 2*a^3*b
- 4*a^2*b^2 + 2*a*b^3 + 3*b^4 + 7*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*
(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + 3*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x
+ c))*sinh(d*x + c)^5 + 2*(3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4)*cosh(d*x + c)^4 + 2*(35*(a^4 - 6*a^2*b^2 - 8*
a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4 + 30*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3
+ 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4 + 8*(7*(a^4 - 6*a^2*b^2 - 8*a*b
^3 - 3*b^4)*cosh(d*x + c)^5 + 10*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + (3*a^4 - 8*a^
3*b - 2*a^2*b^2 - 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh
(d*x + c)^2 + 4*(7*(a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 15*(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b
^3 + 3*b^4)*cosh(d*x + c)^4 + a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4 + 3*(3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9
*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 - 6*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^7 + 3*(a^4 - 2*a
^3*b - 4*a^2*b^2 + 2*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + (3*a^4 - 8*a^3*b - 2*a^2*b^2 - 9*b^4)*cosh(d*x + c)^3 +
(a^4 - 2*a^3*b - 4*a^2*b^2 + 2*a*b^3 + 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
*(a^4*b - a^3*b^2 - 5*a^2*b^3 - 3*a*b^4)*cosh(d*x + c)^5 + 2*(3*a^4*b + 7*a^3*b^2 - 3*a^2*b^3 + 9*a*b^4)*cosh(
d*x + c)^3 + (3*a^4*b + 13*a^3*b^2 + a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^6*b^2 + 3*a^5*b^3 +
3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*
x + c)^7 + (a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^
3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^6*b^2 + a^5
*b^3 - a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^6*b^2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4
+ 8*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b
^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 3
0*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*d)*s
inh(d*x + c)^4 + 4*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a
^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^6*b^
2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 +
a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^6*b^2 + a^5
*b^3 + a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (
a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d + 8*((a^6*b^2 + 3*a^5*b^3 + 3*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^
7 + 3*(a^6*b^2 + a^5*b^3 - a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^6*b^2 + a^5*b^3 + a^4*b^4 + 3*a^3*b^5)*
d*cosh(d*x + c)^3 + (a^6*b^2 + a^5*b^3 - a^4*b^4 - 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)**4/(a+b*tanh(d*x+c)**2)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.9162, size = 448, normalized size = 3.9 \begin{align*} -\frac{\frac{{\left (a e^{\left (2 \, c\right )} - 3 \, b 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} + \frac{2 \,{\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} - a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 5 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 7 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 13 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 5 \, a^{2} b + 7 \, a b^{2} + 3 \, b^{3}\right )}}{{\left (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)^4/(a+b*tanh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

-1/8*((a*e^(2*c) - 3*b*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*(a^3*e^(6*d*x + 6*c) - a^2*b*e^(6*d*x + 6*c) - 5*a*b^2*e^(6*d*x + 6*c) - 3*b^3*e^(6*d*x
+ 6*c) + 3*a^3*e^(4*d*x + 4*c) + 7*a^2*b*e^(4*d*x + 4*c) - 3*a*b^2*e^(4*d*x + 4*c) + 9*b^3*e^(4*d*x + 4*c) +
3*a^3*e^(2*d*x + 2*c) + 13*a^2*b*e^(2*d*x + 2*c) + a*b^2*e^(2*d*x + 2*c) - 9*b^3*e^(2*d*x + 2*c) + a^3 + 5*a^2
*b + 7*a*b^2 + 3*b^3)/((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