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

Optimal. Leaf size=144 \[ \frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{5/2}}+\frac {b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \]

[Out]

1/8*(8*a^2+8*a*b+3*b^2)*arctan(sinh(d*x+c)*(a+b)^(1/2)/a^(1/2))/a^(5/2)/(a+b)^(5/2)/d+1/4*b*cosh(d*x+c)^2*sinh
(d*x+c)/a/(a+b)/d/(a+(a+b)*sinh(d*x+c)^2)^2+3/8*b*(2*a+b)*sinh(d*x+c)/a^2/(a+b)^2/d/(a+(a+b)*sinh(d*x+c)^2)

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Rubi [A]  time = 0.14, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3676, 413, 385, 205} \[ \frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \sinh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d (a+b)^{5/2}}+\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left ((a+b) \sinh ^2(c+d x)+a\right )}+\frac {b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left ((a+b) \sinh ^2(c+d x)+a\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 3676

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> With[{ff = F
reeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b*(ff*x)^n + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 -
ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
IntegerQ[n/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]  time = 1.18, size = 134, normalized size = 0.93 \[ \frac {\frac {2 \sqrt {a} b \sinh (c+d x) \left (\left (8 a^2+11 a b+3 b^2\right ) \cosh (2 (c+d x))+8 a^2-a b-3 b^2\right )}{(a+b)^2 ((a+b) \cosh (2 (c+d x))+a-b)^2}-\frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \text {csch}(c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}}{8 a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Csch[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2)) + (2*Sqrt[a]*b*(8*a^2
- a*b - 3*b^2 + (8*a^2 + 11*a*b + 3*b^2)*Cosh[2*(c + d*x)])*Sinh[c + d*x])/((a + b)^2*(a - b + (a + b)*Cosh[2*
(c + d*x)])^2))/(8*a^(5/2)*d)

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fricas [B]  time = 0.56, size = 7909, normalized size = 54.92 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^7 + 28*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3
+ 3*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^6 + 4*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^7 +
4*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^5 + 4*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4
+ 21*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(8*a^4*b + 19*a^3
*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^3 + (8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sin
h(d*x + c)^4 - 4*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^3 - 4*(8*a^4*b - 5*a^3*b^2 - 22*a^
2*b^3 - 9*a*b^4 - 35*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 - 10*(8*a^4*b - 5*a^3*b^2 -
22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 4*(21*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*
cosh(d*x + c)^5 + 10*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^3 - 3*(8*a^4*b - 5*a^3*b^2 - 2
2*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 - ((8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh
(d*x + c)^8 + 8*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^4 + 24
*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*co
sh(d*x + c)^6 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4 + 7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3
+ 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x
+ c)^3 + 3*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 8*a^3*
b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*
cosh(d*x + c)^4 + 24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4 + 30*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3
- 3*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4 + 8*(7*(8*a^4 +
24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)
*cosh(d*x + c)^3 + (24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^
4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 +
3*b^4)*cosh(d*x + c)^6 + 15*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 8*a^4 + 8*a^3*b
- 5*a^2*b^2 - 8*a*b^3 - 3*b^4 + 3*(24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(8*a^4 + 8*a^3*b - 5*a^2*b
^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (
8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 - a*b)*log(((a + b)*cos
h(d*x + c)^4 + 4*(a + b)*cosh(d*x + c)*sinh(d*x + c)^3 + (a + b)*sinh(d*x + c)^4 - 2*(3*a + b)*cosh(d*x + c)^2
+ 2*(3*(a + b)*cosh(d*x + c)^2 - 3*a - b)*sinh(d*x + c)^2 + 4*((a + b)*cosh(d*x + c)^3 - (3*a + b)*cosh(d*x +
c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)
^2 - 1)*sinh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + 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)) - 4*(8*a^4
*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c) + 4*(7*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cos
h(d*x + c)^6 - 8*a^4*b - 19*a^3*b^2 - 14*a^2*b^3 - 3*a*b^4 + 5*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*co
sh(d*x + c)^4 - 3*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/((a^8 + 5*a^7*b
+ 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^8 + 8*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b
^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b
^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x +
c)^6 + 4*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 3*a^7*
b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^6 + 2*(3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b
^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^
3*b^5)*d*cosh(d*x + c)^3 + 3*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*si
nh(d*x + c)^5 + 2*(35*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(
a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + (3*a^8 + 7*a^7*b + 6*a^6*b^2
+ 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4
- a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*
x + c)^5 + 10*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^8 + 7*a^7
*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^8 + 5*a^7*b + 1
0*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^6 + 15*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 -
3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d
*cosh(d*x + c)^2 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^8 + 5
*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d + 8*((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^
4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*
x + c)^5 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^8 + 3*a^7*
b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(8*a^4*b + 19*a^3*b^2
+ 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^7 + 14*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)*sinh
(d*x + c)^6 + 2*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*sinh(d*x + c)^7 + 2*(8*a^4*b - 5*a^3*b^2 - 22*a^
2*b^3 - 9*a*b^4)*cosh(d*x + c)^5 + 2*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4 + 21*(8*a^4*b + 19*a^3*b^2 +
14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*c
osh(d*x + c)^3 + (8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(8*a^4*b - 5*
a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^3 - 2*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4 - 35*(8*a^4*b
+ 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^4 - 10*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*
x + c)^2)*sinh(d*x + c)^3 + 2*(21*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^5 + 10*(8*a^4*b
- 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^3 - 3*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x
+ c))*sinh(d*x + c)^2 + ((8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^8 + 8*(8*a^4 + 24*a^
3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3
+ 3*b^4)*sinh(d*x + c)^8 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^6 + 4*(8*a^4 + 8*a
^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4 + 7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^2)*sin
h(d*x + c)^6 + 8*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3 + 3*(8*a^4 + 8*a^3*b -
5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*
b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^4 + 24*a^4 + 8*a
^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4 + 30*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^2)*sin
h(d*x + c)^4 + 8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4 + 8*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b
^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^3 + (24*a^4 + 8
*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*
b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^6 + 15*(8
*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4
+ 3*(24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8*a^4 + 24*a^3*b
+ 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x
+ c)^5 + (24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (8*a^4 + 8*a^3*b - 5*a^2*b^2 -
8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*((a + b)*cosh(d*x + c)^3 + 3*(a + b)
*cosh(d*x + c)*sinh(d*x + c)^2 + (a + b)*sinh(d*x + c)^3 + (3*a - b)*cosh(d*x + c) + (3*(a + b)*cosh(d*x + c)^
2 + 3*a - b)*sinh(d*x + c))/sqrt(a^2 + a*b)) + ((8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x +
c)^8 + 8*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^4 + 24*a^3*b
+ 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*sinh(d*x + c)^8 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x
+ c)^6 + 4*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4 + 7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^
4)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^3
+ 3*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 8*a^3*b + 17*
a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*
x + c)^4 + 24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4 + 30*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^
4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4 + 8*(7*(8*a^4 + 24*a^3*
b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^5 + 10*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d
*x + c)^3 + (24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4 + 8*a
^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*
cosh(d*x + c)^6 + 15*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c)^4 + 8*a^4 + 8*a^3*b - 5*a^2
*b^2 - 8*a*b^3 - 3*b^4 + 3*(24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 8*((8*a^4 + 24*a^3*b + 27*a^2*b^2 + 14*a*b^3 + 3*b^4)*cosh(d*x + c)^7 + 3*(8*a^4 + 8*a^3*b - 5*a^2*b^2 - 8*
a*b^3 - 3*b^4)*cosh(d*x + c)^5 + (24*a^4 + 8*a^3*b + 17*a^2*b^2 + 18*a*b^3 + 9*b^4)*cosh(d*x + c)^3 + (8*a^4 +
8*a^3*b - 5*a^2*b^2 - 8*a*b^3 - 3*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*arctan(1/2*sqrt(a^2 + a*
b)*(cosh(d*x + c) + sinh(d*x + c))/a) - 2*(8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c) + 2*(7*(
8*a^4*b + 19*a^3*b^2 + 14*a^2*b^3 + 3*a*b^4)*cosh(d*x + c)^6 - 8*a^4*b - 19*a^3*b^2 - 14*a^2*b^3 - 3*a*b^4 + 5
*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)*cosh(d*x + c)^4 - 3*(8*a^4*b - 5*a^3*b^2 - 22*a^2*b^3 - 9*a*b^4)
*cosh(d*x + c)^2)*sinh(d*x + c))/((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x +
c)^8 + 8*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a
^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*sinh(d*x + c)^8 + 4*(a^8 + 3*a^7*b + 2*a^6*b^2
- 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^6 + 4*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*
b^4 + a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d)*sinh(d*x +
c)^6 + 2*(3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^8 + 5*
a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^3 + 3*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^
5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3
+ 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^4 + 30*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d
*cosh(d*x + c)^2 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a
^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^2 + 8*(7*(a^8 + 5*a^7*b + 10*a^6*b
^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^5 + 10*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b
^4 - a^3*b^5)*d*cosh(d*x + c)^3 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x
+ c))*sinh(d*x + c)^3 + 4*(7*(a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^
6 + 15*(a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^4 + 3*(3*a^8 + 7*a^7*b +
6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^4 + 3*a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*
a^4*b^4 - a^3*b^5)*d)*sinh(d*x + c)^2 + (a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d + 8*
((a^8 + 5*a^7*b + 10*a^6*b^2 + 10*a^5*b^3 + 5*a^4*b^4 + a^3*b^5)*d*cosh(d*x + c)^7 + 3*(a^8 + 3*a^7*b + 2*a^6*
b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x + c)^5 + (3*a^8 + 7*a^7*b + 6*a^6*b^2 + 6*a^5*b^3 + 7*a^4*b^
4 + 3*a^3*b^5)*d*cosh(d*x + c)^3 + (a^8 + 3*a^7*b + 2*a^6*b^2 - 2*a^5*b^3 - 3*a^4*b^4 - a^3*b^5)*d*cosh(d*x +
c))*sinh(d*x + c))]

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giac [B]  time = 0.45, size = 1137, normalized size = 7.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((8*a^5 - 72*a^4*b - 37*a^3*b^2 + 10*a^2*b^3 + 15*a*b^4 + (40*a^4 - 40*a^3*b - 57*a^2*b^2 - 22*a*b^3 + 3*b
^4)*sqrt(-a*b))*sqrt(a^2 - b^2 + 2*sqrt(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a^5*e^(
2*c) + a^4*b*e^(2*c) - a^3*b^2*e^(2*c) - a^2*b^3*e^(2*c) + sqrt((a^5*e^(2*c) + a^4*b*e^(2*c) - a^3*b^2*e^(2*c)
- a^2*b^3*e^(2*c))^2 - (a^5*e^(4*c) + 3*a^4*b*e^(4*c) + 3*a^3*b^2*e^(4*c) + a^2*b^3*e^(4*c))*(a^5 + 3*a^4*b +
3*a^3*b^2 + a^2*b^3)))/(a^5*e^(4*c) + 3*a^4*b*e^(4*c) + 3*a^3*b^2*e^(4*c) + a^2*b^3*e^(4*c))))*e^(-2*c)/(a^10
- 11*a^9*b - 39*a^8*b^2 - 27*a^7*b^3 + 27*a^6*b^4 + 39*a^5*b^5 + 11*a^4*b^6 - a^3*b^7 + 2*(3*a^9 + 2*a^8*b -
19*a^7*b^2 - 36*a^6*b^3 - 19*a^5*b^4 + 2*a^4*b^5 + 3*a^3*b^6)*sqrt(-a*b)) + (8*a^5 - 72*a^4*b - 37*a^3*b^2 + 1
0*a^2*b^3 + 15*a*b^4 - (40*a^4 - 40*a^3*b - 57*a^2*b^2 - 22*a*b^3 + 3*b^4)*sqrt(-a*b))*sqrt(a^2 - b^2 - 2*sqrt
(-a*b)*(a + b))*abs(a*e^(2*c) + b*e^(2*c))*arctan(e^(d*x)/sqrt((a^5*e^(2*c) + a^4*b*e^(2*c) - a^3*b^2*e^(2*c)
- a^2*b^3*e^(2*c) - sqrt((a^5*e^(2*c) + a^4*b*e^(2*c) - a^3*b^2*e^(2*c) - a^2*b^3*e^(2*c))^2 - (a^5*e^(4*c) +
3*a^4*b*e^(4*c) + 3*a^3*b^2*e^(4*c) + a^2*b^3*e^(4*c))*(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2*b^3)))/(a^5*e^(4*c) +
3*a^4*b*e^(4*c) + 3*a^3*b^2*e^(4*c) + a^2*b^3*e^(4*c))))*e^(-2*c)/(a^10 - 11*a^9*b - 39*a^8*b^2 - 27*a^7*b^3 +
27*a^6*b^4 + 39*a^5*b^5 + 11*a^4*b^6 - a^3*b^7 - 2*(3*a^9 + 2*a^8*b - 19*a^7*b^2 - 36*a^6*b^3 - 19*a^5*b^4 +
2*a^4*b^5 + 3*a^3*b^6)*sqrt(-a*b)) + 2*(8*a^2*b*e^(7*d*x + 7*c) + 11*a*b^2*e^(7*d*x + 7*c) + 3*b^3*e^(7*d*x +
7*c) + 8*a^2*b*e^(5*d*x + 5*c) - 13*a*b^2*e^(5*d*x + 5*c) - 9*b^3*e^(5*d*x + 5*c) - 8*a^2*b*e^(3*d*x + 3*c) +
13*a*b^2*e^(3*d*x + 3*c) + 9*b^3*e^(3*d*x + 3*c) - 8*a^2*b*e^(d*x + c) - 11*a*b^2*e^(d*x + c) - 3*b^3*e^(d*x +
c))/((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

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maple [B]  time = 0.45, size = 1676, normalized size = 11.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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/(a^2+2*a*b+b^2)*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*b^2/a
/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^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/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-29/4/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/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5-3/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^2*b^3/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^5+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/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2
*c)^3+29/4/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/(a^2+2*a*
b+b^2)*tanh(1/2*d*x+1/2*c)^3+3/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^2*b^3/(a^2+2*a*b+b^2)*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*ta
nh(1/2*d*x+1/2*c)^2*b+a)^2*b/(a^2+2*a*b+b^2)*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*b^2/a/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)-1/d/(a^2+2*a*b+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))-1/d/a/(a^2+2
*a*b+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
))*b-3/8/d/a^2/(a^2+2*a*b+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))*b^2+1/d/(a^2+2*a*b+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))*b+1/d/a/(a^2+2*a*b+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))*b^2+3/8/d/a^2/(a^2+2
*a*b+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))*b^3+1/d/(a^2+2*a*b+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))+1/d/a/(a^2+2*a*b+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))*b+3/8/d/a^2/(a^2+2*a*b+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))*b^2+1/d/(a^2+2*a*b+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))*b+1/d/a/(
a^2+2*a*b+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))*b^2+3/8/d/a^2/(a^2+2*a*b+b^2)/(b*(a+b))^(1/2)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2)*arcta
n(a*tanh(1/2*d*x+1/2*c)/((2*(b*(a+b))^(1/2)+a+2*b)*a)^(1/2))*b^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (8 \, a^{2} b e^{\left (7 \, c\right )} + 11 \, a b^{2} e^{\left (7 \, c\right )} + 3 \, b^{3} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (8 \, a^{2} b e^{\left (5 \, c\right )} - 13 \, a b^{2} e^{\left (5 \, c\right )} - 9 \, b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} - {\left (8 \, a^{2} b e^{\left (3 \, c\right )} - 13 \, a b^{2} e^{\left (3 \, c\right )} - 9 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (8 \, a^{2} b e^{c} + 11 \, a b^{2} e^{c} + 3 \, b^{3} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{6} d + 4 \, a^{5} b d + 6 \, a^{4} b^{2} d + 4 \, a^{3} b^{3} d + a^{2} b^{4} d + {\left (a^{6} d e^{\left (8 \, c\right )} + 4 \, a^{5} b d e^{\left (8 \, c\right )} + 6 \, a^{4} b^{2} d e^{\left (8 \, c\right )} + 4 \, a^{3} b^{3} d e^{\left (8 \, c\right )} + a^{2} b^{4} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{6} d e^{\left (6 \, c\right )} + 2 \, a^{5} b d e^{\left (6 \, c\right )} - 2 \, a^{3} b^{3} d e^{\left (6 \, c\right )} - a^{2} b^{4} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (3 \, a^{6} d e^{\left (4 \, c\right )} + 4 \, a^{5} b d e^{\left (4 \, c\right )} + 2 \, a^{4} b^{2} d e^{\left (4 \, c\right )} + 4 \, a^{3} b^{3} d e^{\left (4 \, c\right )} + 3 \, a^{2} b^{4} d e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 4 \, {\left (a^{6} d e^{\left (2 \, c\right )} + 2 \, a^{5} b d e^{\left (2 \, c\right )} - 2 \, a^{3} b^{3} d e^{\left (2 \, c\right )} - a^{2} b^{4} d e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}} + 2 \, \int \frac {{\left (8 \, a^{2} e^{\left (3 \, c\right )} + 8 \, a b e^{\left (3 \, c\right )} + 3 \, b^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (8 \, a^{2} e^{c} + 8 \, a b e^{c} + 3 \, b^{2} e^{c}\right )} e^{\left (d x\right )}}{8 \, {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} + {\left (a^{5} e^{\left (4 \, c\right )} + 3 \, a^{4} b e^{\left (4 \, c\right )} + 3 \, a^{3} b^{2} e^{\left (4 \, c\right )} + a^{2} b^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{5} e^{\left (2 \, c\right )} + a^{4} b e^{\left (2 \, c\right )} - a^{3} b^{2} e^{\left (2 \, c\right )} - a^{2} b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((8*a^2*b*e^(7*c) + 11*a*b^2*e^(7*c) + 3*b^3*e^(7*c))*e^(7*d*x) + (8*a^2*b*e^(5*c) - 13*a*b^2*e^(5*c) - 9*
b^3*e^(5*c))*e^(5*d*x) - (8*a^2*b*e^(3*c) - 13*a*b^2*e^(3*c) - 9*b^3*e^(3*c))*e^(3*d*x) - (8*a^2*b*e^c + 11*a*
b^2*e^c + 3*b^3*e^c)*e^(d*x))/(a^6*d + 4*a^5*b*d + 6*a^4*b^2*d + 4*a^3*b^3*d + a^2*b^4*d + (a^6*d*e^(8*c) + 4*
a^5*b*d*e^(8*c) + 6*a^4*b^2*d*e^(8*c) + 4*a^3*b^3*d*e^(8*c) + a^2*b^4*d*e^(8*c))*e^(8*d*x) + 4*(a^6*d*e^(6*c)
+ 2*a^5*b*d*e^(6*c) - 2*a^3*b^3*d*e^(6*c) - a^2*b^4*d*e^(6*c))*e^(6*d*x) + 2*(3*a^6*d*e^(4*c) + 4*a^5*b*d*e^(4
*c) + 2*a^4*b^2*d*e^(4*c) + 4*a^3*b^3*d*e^(4*c) + 3*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^6*d*e^(2*c) + 2*a^5*b*
d*e^(2*c) - 2*a^3*b^3*d*e^(2*c) - a^2*b^4*d*e^(2*c))*e^(2*d*x)) + 2*integrate(1/8*((8*a^2*e^(3*c) + 8*a*b*e^(3
*c) + 3*b^2*e^(3*c))*e^(3*d*x) + (8*a^2*e^c + 8*a*b*e^c + 3*b^2*e^c)*e^(d*x))/(a^5 + 3*a^4*b + 3*a^3*b^2 + a^2
*b^3 + (a^5*e^(4*c) + 3*a^4*b*e^(4*c) + 3*a^3*b^2*e^(4*c) + a^2*b^3*e^(4*c))*e^(4*d*x) + 2*(a^5*e^(2*c) + a^4*
b*e^(2*c) - a^3*b^2*e^(2*c) - a^2*b^3*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)*(a + b*tanh(c + d*x)^2)^3),x)

[Out]

int(1/(cosh(c + d*x)*(a + b*tanh(c + d*x)^2)^3), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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