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

Optimal. Leaf size=125 \[ \frac {(a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{3/2} d (a+b)^{5/2}}+\frac {(a+4 b) \tanh (c+d x)}{8 b d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a \tanh (c+d x)}{4 b d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 125, 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, 385, 199, 208} \[ \frac {(a+4 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{3/2} d (a+b)^{5/2}}+\frac {(a+4 b) \tanh (c+d x)}{8 b d (a+b)^2 \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {a \tanh (c+d x)}{4 b d (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 3.39, size = 250, normalized size = 2.00 \[ \frac {\text {sech}^6(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (-\frac {4 (a+b) \text {sech}(2 c) ((a+2 b) \sinh (2 c)-a \sinh (2 d x))}{a}+\frac {\text {sech}(2 c) ((a+4 b) \sinh (2 c)-(a-2 b) \sinh (2 d x)) (a \cosh (2 (c+d x))+a+2 b)}{b}+\frac {(a+4 b) (\cosh (2 c)-\sinh (2 c)) (a \cosh (2 (c+d x))+a+2 b)^2 \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )}{b \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )}{64 d (a+b)^2 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^6*(((a + 4*b)*ArcTanh[(Sech[d*x]*(Cosh[2*c] - Sinh[2*c])*((a +
2*b)*Sinh[d*x] - a*Sinh[2*c + d*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4])]*(a + 2*b + a*Cosh[2*(c + d
*x)])^2*(Cosh[2*c] - Sinh[2*c]))/(b*Sqrt[a + b]*Sqrt[b*(Cosh[c] - Sinh[c])^4]) - (4*(a + b)*Sech[2*c]*((a + 2*
b)*Sinh[2*c] - a*Sinh[2*d*x]))/a + ((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[2*c]*((a + 4*b)*Sinh[2*c] - (a - 2*b)
*Sinh[2*d*x]))/b))/(64*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3)

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fricas [B]  time = 0.52, size = 5447, normalized size = 43.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

[1/16*(4*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^6 + 24*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)*si
nh(d*x + c)^5 + 4*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*sinh(d*x + c)^6 + 4*a^4*b - 4*a^3*b^2 - 8*a^2*b^3 + 4*(3*a^4
*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c)^4 + 4*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^
4 - 16*b^5 + 15*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(5*(a^4*b + 5*a^3*b^2 +
4*a^2*b^3)*cosh(d*x + c)^3 + (3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c))*sinh(d*x + c
)^3 + 4*(3*a^4*b - a^3*b^2 - 20*a^2*b^3 - 16*a*b^4)*cosh(d*x + c)^2 + 4*(3*a^4*b - a^3*b^2 - 20*a^2*b^3 - 16*a
*b^4 + 15*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^4 + 6*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16
*b^5)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((a^4 + 4*a^3*b)*cosh(d*x + c)^8 + 8*(a^4 + 4*a^3*b)*cosh(d*x + c)*si
nh(d*x + c)^7 + (a^4 + 4*a^3*b)*sinh(d*x + c)^8 + 4*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 4*(a^4 + 6*a
^3*b + 8*a^2*b^2 + 7*(a^4 + 4*a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^3 +
3*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*c
osh(d*x + c)^4 + 2*(35*(a^4 + 4*a^3*b)*cosh(d*x + c)^4 + 3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3 + 30*(a^4 +
6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + a^4 + 4*a^3*b + 8*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^5 +
10*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^3 + (3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c))*si
nh(d*x + c)^3 + 4*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^2 + 4*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^6 + 15*(a^4
+ 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^4 + a^4 + 6*a^3*b + 8*a^2*b^2 + 3*(3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((a^4 + 4*a^3*b)*cosh(d*x + c)^7 + 3*(a^4 + 6*a^3*b + 8*a^2*b^2)*cos
h(d*x + c)^5 + (3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c)^3 + (a^4 + 6*a^3*b + 8*a^2*b^2)*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*s
inh(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*c
osh(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*(a^4*b + 5*a
^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^5 + 2*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c)^3
+ (3*a^4*b - a^3*b^2 - 20*a^2*b^3 - 16*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^
5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d)*sinh(d*x + c)^6 + 2*(3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^
4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 +
7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8
*a*b^7)*d)*sinh(d*x + c)^4 + 4*(a^6*b^2 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^
3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^3 + (3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^
7)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^4 + 3*(3*a^6*b^2 + 17*a^5*b^3 + 41*a
^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*d*cosh(d*x + c)^2 + (a^6*b^2 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 +
2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x +
c)^5 + (3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*d*cosh(d*x + c)^3 + (a^6*b^2
+ 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c))*sinh(d*x + c)), 1/8*(2*(a^4*b + 5*a^3*b^2 +
4*a^2*b^3)*cosh(d*x + c)^6 + 12*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(a^4*b + 5*
a^3*b^2 + 4*a^2*b^3)*sinh(d*x + c)^6 + 2*a^4*b - 2*a^3*b^2 - 4*a^2*b^3 + 2*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 -
24*a*b^4 - 16*b^5)*cosh(d*x + c)^4 + 2*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5 + 15*(a^4*b + 5*a^
3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(5*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^3 + (
3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c))*sinh(d*x + c)^3 + 2*(3*a^4*b - a^3*b^2 - 2
0*a^2*b^3 - 16*a*b^4)*cosh(d*x + c)^2 + 2*(3*a^4*b - a^3*b^2 - 20*a^2*b^3 - 16*a*b^4 + 15*(a^4*b + 5*a^3*b^2 +
4*a^2*b^3)*cosh(d*x + c)^4 + 6*(3*a^4*b + 5*a^3*b^2 - 6*a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + ((a^4 + 4*a^3*b)*cosh(d*x + c)^8 + 8*(a^4 + 4*a^3*b)*cosh(d*x + c)*sinh(d*x + c)^7 + (a^4 + 4*a^3*b
)*sinh(d*x + c)^8 + 4*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^6 + 4*(a^4 + 6*a^3*b + 8*a^2*b^2 + 7*(a^4 + 4*
a^3*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^3 + 3*(a^4 + 6*a^3*b + 8*a^2*b^2)
*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c)^4 + 2*(35*(a^4 +
4*a^3*b)*cosh(d*x + c)^4 + 3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3 + 30*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x
+ c)^2)*sinh(d*x + c)^4 + a^4 + 4*a^3*b + 8*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^5 + 10*(a^4 + 6*a^3*b + 8*a^2*b^2
)*cosh(d*x + c)^3 + (3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a^4 + 6*a^3
*b + 8*a^2*b^2)*cosh(d*x + c)^2 + 4*(7*(a^4 + 4*a^3*b)*cosh(d*x + c)^6 + 15*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d
*x + c)^4 + a^4 + 6*a^3*b + 8*a^2*b^2 + 3*(3*a^4 + 20*a^3*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^2 + 8*((a^4 + 4*a^3*b)*cosh(d*x + c)^7 + 3*(a^4 + 6*a^3*b + 8*a^2*b^2)*cosh(d*x + c)^5 + (3*a^4 + 20*a^3
*b + 40*a^2*b^2 + 32*a*b^3)*cosh(d*x + c)^3 + (a^4 + 6*a^3*b + 8*a^2*b^2)*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*(a^4*b + 5*a^3*b^2 + 4*a^2*b^3)*cosh(d*x + c)^5 + 2*(3*a^4*b + 5*a^3*b^2 - 6*
a^2*b^3 - 24*a*b^4 - 16*b^5)*cosh(d*x + c)^3 + (3*a^4*b - a^3*b^2 - 20*a^2*b^3 - 16*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d)*sinh(d*x +
c)^6 + 2*(3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5
+ 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^2 + (3*a^6*b^2 + 17*a^5*b
^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*d)*sinh(d*x + c)^4 + 4*(a^6*b^2 + 5*a^5*b^3 + 9*a^4*b^4 +
7*a^3*b^5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^3 + (3*a^6*b^2 + 17*a^5*b^3 +
41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*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 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(
d*x + c)^4 + 3*(3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^2*b^6 + 8*a*b^7)*d*cosh(d*x + c)^2 + (
a^6*b^2 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*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 + 5*a^5*b^3 + 9*
a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)^5 + (3*a^6*b^2 + 17*a^5*b^3 + 41*a^4*b^4 + 51*a^3*b^5 + 32*a^
2*b^6 + 8*a*b^7)*d*cosh(d*x + c)^3 + (a^6*b^2 + 5*a^5*b^3 + 9*a^4*b^4 + 7*a^3*b^5 + 2*a^2*b^6)*d*cosh(d*x + c)
)*sinh(d*x + c))]

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giac [B]  time = 1.73, size = 274, normalized size = 2.19 \[ \frac {\frac {{\left (a + 4 \, b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sqrt {-a b - b^{2}}} + \frac {2 \, {\left (a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 4 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 16 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 16 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} - 2 \, a^{2} b\right )}}{{\left (a^{3} b + 2 \, a^{2} b^{2} + a b^{3}\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}^{2}}}{8 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/8*((a + 4*b)*arctan(1/2*(a*e^(2*d*x + 2*c) + a + 2*b)/sqrt(-a*b - b^2))/((a^2*b + 2*a*b^2 + b^3)*sqrt(-a*b -
b^2)) + 2*(a^3*e^(6*d*x + 6*c) + 4*a^2*b*e^(6*d*x + 6*c) + 3*a^3*e^(4*d*x + 4*c) + 2*a^2*b*e^(4*d*x + 4*c) -
8*a*b^2*e^(4*d*x + 4*c) - 16*b^3*e^(4*d*x + 4*c) + 3*a^3*e^(2*d*x + 2*c) - 4*a^2*b*e^(2*d*x + 2*c) - 16*a*b^2*
e^(2*d*x + 2*c) + a^3 - 2*a^2*b)/((a^3*b + 2*a^2*b^2 + a*b^3)*(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 = 0.28, size = 1084, normalized size = 8.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^4/(a+b*sech(d*x+c)^2)^3,x)

[Out]

-1/4/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/b/(a+b)*tanh(1/2*d*x+1/2*c)^7*a+1/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)*tanh(1/2*d*x+1/2*c)^7-3/4/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+b)^2/b*tanh(1/2*d*x+1/2*c)^5*a^2+5/4
/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*tanh(1/2*d*x+1/2*c)^5*a-1/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)^2*tanh(1/2*d*x+1/2*c)^5-3/4/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+b)^2/b*tanh(1/2*d*x+1/2*c)^3*a^2+5/4
/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*tanh(1/2*d*x+1/2*c)^3*a-1/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)^2*tanh(1/2*d*x+1/2*c)^3-1/4/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/b/(a+b)*tanh(1/2*d*x+1/2*c)*a+1/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)*t
anh(1/2*d*x+1/2*c)-1/16/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))+1/16/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))-1/4/d/(a^2+2*a*b+b^2)/b^(1/2)/(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))+1/4/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 = 0.53, size = 369, normalized size = 2.95 \[ -\frac {{\left (a + 4 \, b\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {a^{3} - 2 \, a^{2} b + {\left (3 \, a^{3} - 4 \, a^{2} b - 16 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (3 \, a^{3} + 2 \, a^{2} b - 8 \, a b^{2} - 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} + 4 \, a^{2} b\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3} + 4 \, {\left (a^{5} b + 4 \, a^{4} b^{2} + 5 \, a^{3} b^{3} + 2 \, a^{2} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{5} b + 14 \, a^{4} b^{2} + 27 \, a^{3} b^{3} + 24 \, a^{2} b^{4} + 8 \, a b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{5} b + 4 \, a^{4} b^{2} + 5 \, a^{3} b^{3} + 2 \, a^{2} b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} b + 2 \, a^{4} b^{2} + a^{3} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^4\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3),x)

[Out]

int(1/(cosh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(d*x+c)**4/(a+b*sech(d*x+c)**2)**3,x)

[Out]

Integral(sech(c + d*x)**4/(a + b*sech(c + d*x)**2)**3, x)

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