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

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

[Out]

1/8*(3*a^2+8*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/b^(5/2)/(a+b)^(5/2)/d-1/4*a*sech(d*x+c)^2*tan
h(d*x+c)/b/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2-3/8*a*(a+2*b)*tanh(d*x+c)/b^2/(a+b)^2/d/(a+b-b*tanh(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 = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {4146, 413, 385, 208} \[ \frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} d (a+b)^{5/2}}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a \tanh (c+d x) \text {sech}^2(c+d x)}{4 b d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[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 4146

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

Rubi steps

\begin {align*} \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{\left (a+b-b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-a-4 b+(3 a+4 b) x^2}{\left (a+b-b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 b (a+b) d}\\ &=-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}+\frac {\left (3 a^2+8 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{8 b^2 (a+b)^2 d}\\ &=\frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} (a+b)^{5/2} d}-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 1.01, size = 125, normalized size = 0.87 \[ \frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {a \sqrt {b} \sinh (2 (c+d x)) \left (3 a^2+3 a (a+2 b) \cosh (2 (c+d x))+16 a b+16 b^2\right )}{(a+b)^2 (a \cosh (2 (c+d x))+a+2 b)^2}}{8 b^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16
*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*(c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])
^2))/(8*b^(5/2)*d)

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fricas [B]  time = 0.53, size = 5887, normalized size = 40.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^6 + 24*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3
+ 8*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^5 + 4*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*sinh(d*x + c)^6 +
12*a^4*b + 36*a^3*b^2 + 24*a^2*b^3 + 12*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)*cosh(d*x + c)^
4 + 12*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5 + 5*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^3 + 3*(
3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(9*a^4*b + 49*a^3*b^
2 + 80*a^2*b^3 + 40*a*b^4)*cosh(d*x + c)^2 + 4*(9*a^4*b + 49*a^3*b^2 + 80*a^2*b^3 + 40*a*b^4 + 15*(3*a^4*b + 1
1*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^4 + 18*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(3*a^4 + 8*a^3*b + 8*a^
2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^4 + 8*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(3*a^4 + 14*a^3*b + 2
4*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^6 + 4*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3 + 7*(3*a^4 + 8*a^3*b + 8*a
^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 14*
a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3
+ 64*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 9*a^4 + 48*a^3*b + 112*a^2*b
^2 + 128*a*b^3 + 64*b^4 + 30*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a
^4 + 8*a^3*b + 8*a^2*b^2 + 8*(7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(3*a^4 + 14*a^3*b + 24*a^2*
b^2 + 16*a*b^3)*cosh(d*x + c)^3 + (9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*
x + c)^3 + 4*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*c
osh(d*x + c)^6 + 15*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^4 + 3*a^4 + 14*a^3*b + 24*a^2*b^2
+ 16*a*b^3 + 3*(9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3
*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7 + 3*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^5 + (
9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c)^3 + (3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b
^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a*b + b^2)*log((a^2*cosh(d*x + c)^4 + 4*a^2*cosh(d*x + c)*sinh(d*x + c)
^3 + a^2*sinh(d*x + c)^4 + 2*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*a^2*cosh(d*x + c)^2 + a^2 + 2*a*b)*sinh(d*x
+ c)^2 + a^2 + 8*a*b + 8*b^2 + 4*(a^2*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c) - 4*(a*cosh
(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(a*b + b^2))/(a*cosh(d*x + c)
^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh(d*x + c
)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a)) + 8*(3*(3
*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^5 + 6*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4
+ 16*b^5)*cosh(d*x + c)^3 + (9*a^4*b + 49*a^3*b^2 + 80*a^2*b^3 + 40*a*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^5
*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*co
sh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*sinh(d*x + c)^8 + 4*(a^5*b^3 + 5*a
^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)
*d*cosh(d*x + c)^2 + (a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d)*sinh(d*x + c)^6 + 2*(3*a^5*b^3
+ 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c)^4 + 8*(7*(a^5*b^3 + 3*a^4*b^4 + 3*
a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^3 + 3*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x +
c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^4 + 30*(a^5*b^3 + 5*a^
4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^2 + (3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6
+ 32*a*b^7 + 8*b^8)*d)*sinh(d*x + c)^4 + 4*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x
+ c)^2 + 8*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^5 + 10*(a^5*b^3 + 5*a^4*b^4 + 9*a^3
*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^3 + (3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 +
8*b^8)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^6
+ 15*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^4 + 3*(3*a^5*b^3 + 17*a^4*b^4 + 4
1*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c)^2 + (a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 +
2*a*b^7)*d)*sinh(d*x + c)^2 + (a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d + 8*((a^5*b^3 + 3*a^4*b^4 + 3*a^3*
b^5 + a^2*b^6)*d*cosh(d*x + c)^7 + 3*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^5
+ (3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c)^3 + (a^5*b^3 + 5*a^4*
b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(3*a^4*b + 11*a^3*b^2 + 16*a^2*
b^3 + 8*a*b^4)*cosh(d*x + c)^6 + 12*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)*sinh(d*x + c)^
5 + 2*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*sinh(d*x + c)^6 + 6*a^4*b + 18*a^3*b^2 + 12*a^2*b^3 + 6*(3
*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)*cosh(d*x + c)^4 + 6*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 +
40*a*b^4 + 16*b^5 + 5*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(
3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^3 + 3*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4
+ 16*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(9*a^4*b + 49*a^3*b^2 + 80*a^2*b^3 + 40*a*b^4)*cosh(d*x + c)^2 +
2*(9*a^4*b + 49*a^3*b^2 + 80*a^2*b^3 + 40*a*b^4 + 15*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x +
c)^4 + 18*(3*a^4*b + 17*a^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((3*a^4
+ 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^8 + 8*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^
4 + 8*a^3*b + 8*a^2*b^2)*sinh(d*x + c)^8 + 4*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^6 + 4*(3
*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3 + 7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8
*(7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + 3*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)
)*sinh(d*x + c)^5 + 2*(9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c)^4 + 2*(35*(3*a^4 + 8
*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + 9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4 + 30*(3*a^4 + 14*a^3
*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 3*a^4 + 8*a^3*b + 8*a^2*b^2 + 8*(7*(3*a^4 + 8*a
^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + 10*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^3 + (9*a^4 + 4
8*a^3*b + 112*a^2*b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(3*a^4 + 14*a^3*b + 24*a^2*b^2
+ 16*a*b^3)*cosh(d*x + c)^2 + 4*(7*(3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 15*(3*a^4 + 14*a^3*b + 24*a
^2*b^2 + 16*a*b^3)*cosh(d*x + c)^4 + 3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3 + 3*(9*a^4 + 48*a^3*b + 112*a^2*
b^2 + 128*a*b^3 + 64*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^4 + 8*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^7
+ 3*(3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c)^5 + (9*a^4 + 48*a^3*b + 112*a^2*b^2 + 128*a*b^3 +
64*b^4)*cosh(d*x + c)^3 + (3*a^4 + 14*a^3*b + 24*a^2*b^2 + 16*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a*b
- b^2)*arctan(1/2*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a + 2*b)*sqrt(-a*
b - b^2)/(a*b + b^2)) + 4*(3*(3*a^4*b + 11*a^3*b^2 + 16*a^2*b^3 + 8*a*b^4)*cosh(d*x + c)^5 + 6*(3*a^4*b + 17*a
^3*b^2 + 38*a^2*b^3 + 40*a*b^4 + 16*b^5)*cosh(d*x + c)^3 + (9*a^4*b + 49*a^3*b^2 + 80*a^2*b^3 + 40*a*b^4)*cosh
(d*x + c))*sinh(d*x + c))/((a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^8 + 8*(a^5*b^3 + 3*a^4*
b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*sin
h(d*x + c)^8 + 4*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^6 + 4*(7*(a^5*b^3 + 3
*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^2 + (a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d)
*sinh(d*x + c)^6 + 2*(3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c)^4 +
8*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^3 + 3*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a
^2*b^6 + 2*a*b^7)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(
d*x + c)^4 + 30*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^2 + (3*a^5*b^3 + 17*a^
4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d)*sinh(d*x + c)^4 + 4*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 +
7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^2 + 8*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^5 +
10*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^3 + (3*a^5*b^3 + 17*a^4*b^4 + 41*a^
3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^
5 + a^2*b^6)*d*cosh(d*x + c)^6 + 15*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c)^4
+ 3*(3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh(d*x + c)^2 + (a^5*b^3 + 5*a^4
*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d)*sinh(d*x + c)^2 + (a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d + 8
*((a^5*b^3 + 3*a^4*b^4 + 3*a^3*b^5 + a^2*b^6)*d*cosh(d*x + c)^7 + 3*(a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b
^6 + 2*a*b^7)*d*cosh(d*x + c)^5 + (3*a^5*b^3 + 17*a^4*b^4 + 41*a^3*b^5 + 51*a^2*b^6 + 32*a*b^7 + 8*b^8)*d*cosh
(d*x + c)^3 + (a^5*b^3 + 5*a^4*b^4 + 9*a^3*b^5 + 7*a^2*b^6 + 2*a*b^7)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [B]  time = 1.75, size = 302, normalized size = 2.10 \[ \frac {\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (3 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 8 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 9 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 42 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 72 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 40 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{3} + 6 \, a^{2} b\right )}}{{\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*((3*a^2 + 8*a*b + 8*b^2)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^2*b^2 + 2*a*b^3 +
b^4)*sqrt(-a*b - b^2)) + 2*(3*a^3*e^(6*d*x + 6*c) + 8*a^2*b*e^(6*d*x + 6*c) + 8*a*b^2*e^(6*d*x + 6*c) + 9*a^3*
e^(4*d*x + 4*c) + 42*a^2*b*e^(4*d*x + 4*c) + 72*a*b^2*e^(4*d*x + 4*c) + 48*b^3*e^(4*d*x + 4*c) + 9*a^3*e^(2*d*
x + 2*c) + 40*a^2*b*e^(2*d*x + 2*c) + 40*a*b^2*e^(2*d*x + 2*c) + 3*a^3 + 6*a^2*b)/((a^2*b^2 + 2*a*b^3 + b^4)*(
a*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) + 4*b*e^(2*d*x + 2*c) + a)^2))/d

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maple [B]  time = 0.34, size = 1245, normalized size = 8.65 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x)

[Out]

-3/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+
b)^2*a^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+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1
/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/b/(a+b)*tanh(1/2*d*x+1/2*c)^7*a-9/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*ta
nh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^3/(a+b)^2/b^2*tanh(1/2*d*x+1/
2*c)^5-13/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)
^2*b+a+b)^2/(a+b)^2/b*tanh(1/2*d*x+1/2*c)^5*a^2+2/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/
2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^5*a-9/4/d/(tanh(1/2*d*x+1/2*c)^4
*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^3/(a+b)^2/b^2*tanh(1/2
*d*x+1/2*c)^3-13/4/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)^2*b+a+b)^2/(a+b)^2/b*tanh(1/2*d*x+1/2*c)^3*a^2+2/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*
tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)^2*tanh(1/2*d*x+1/2*c)^3*a-3/4/d/(tanh(1/2*d*x+1
/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*a^2/(a+b)/b^2*tan
h(1/2*d*x+1/2*c)-2/d/(tanh(1/2*d*x+1/2*c)^4*a+b*tanh(1/2*d*x+1/2*c)^4+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x
+1/2*c)^2*b+a+b)^2/b/(a+b)*tanh(1/2*d*x+1/2*c)*a-3/16/d/b^(5/2)/(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*ta
nh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)-(a+b)^(1/2))*a^2-1/2/d/b^(3/2)/(a^2+2*a*b+b^2)*a/(a+b)^(1/2)
*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)-(a+b)^(1/2))-1/2/d/(a^2+2*a*b+b^2)/b^(1/2
)/(a+b)^(1/2)*ln(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)-(a+b)^(1/2))+3/16/d/b^(5/2)/
(a^2+2*a*b+b^2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*c)+(a+b)^(1/2))*a^
2+1/2/d/b^(3/2)/(a^2+2*a*b+b^2)*a/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(1/2*d*x+1/2*
c)+(a+b)^(1/2))+1/2/d/(a^2+2*a*b+b^2)/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*b^(1/2)*tanh(
1/2*d*x+1/2*c)+(a+b)^(1/2))

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maxima [B]  time = 1.37, size = 395, normalized size = 2.74 \[ -\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, a^{3} + 6 \, a^{2} b + {\left (9 \, a^{3} + 40 \, a^{2} b + 40 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{3} + 14 \, a^{2} b + 24 \, a b^{2} + 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 27 \, a^{2} b^{4} + 24 \, a b^{5} + 8 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^6/(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/16*(3*a^2 + 8*a*b + 8*b^2)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a +
2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) - 1/4*(3*a^3 + 6*a^2*b + (9*a^3 + 40*
a^2*b + 40*a*b^2)*e^(-2*d*x - 2*c) + 3*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3)*e^(-4*d*x - 4*c) + (3*a^3 + 8*a^
2*b + 8*a*b^2)*e^(-6*d*x - 6*c))/((a^4*b^2 + 2*a^3*b^3 + a^2*b^4 + 4*(a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^
5)*e^(-2*d*x - 2*c) + 2*(3*a^4*b^2 + 14*a^3*b^3 + 27*a^2*b^4 + 24*a*b^5 + 8*b^6)*e^(-4*d*x - 4*c) + 4*(a^4*b^2
+ 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^5)*e^(-6*d*x - 6*c) + (a^4*b^2 + 2*a^3*b^3 + a^2*b^4)*e^(-8*d*x - 8*c))*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^6*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

int(1/(cosh(c + d*x)^6*(a + b/cosh(c + d*x)^2)^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{6}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**6/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**6/(a + b*sech(c + d*x)**2)**3, x)

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