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

Optimal. Leaf size=142 \[ -\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{5/2} d (a+b)^{5/2}}-\frac {b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.14, antiderivative size = 142, 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 = {4147, 413, 385, 205} \[ \frac {\left (8 a^2+8 a b+3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{5/2} d (a+b)^{5/2}}-\frac {3 b (2 a+b) \sinh (c+d x)}{8 a^2 d (a+b)^2 \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {b \sinh (c+d x) \cosh ^2(c+d x)}{4 a d (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(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 + b + a*Sinh[c + d*x]^2)^2) - (3*b*(2*a + b)*Sinh[c + d*x])/(8*a^2*(
a + b)^2*d*(a + b + a*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 4147

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = Fr
eeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[ExpandToSum[b + 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 \text {sech}^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\left (a+b+a 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+b+a \sinh ^2(c+d x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {4 a+b+(4 a+3 b) x^2}{\left (a+b+a 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+b+a \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+b+a \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+b+a 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} \sinh (c+d x)}{\sqrt {a+b}}\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+b+a \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+b+a \sinh ^2(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.94, size = 214, normalized size = 1.51 \[ \frac {\text {sech}^5(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac {\left (8 a^2+8 a b+3 b^2\right ) (\sinh (c)-\cosh (c)) \text {sech}(c+d x) (a \cosh (2 (c+d x))+a+2 b)^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} (\sinh (c)+\cosh (c)) \text {csch}(c+d x)}{\sqrt {a}}\right )}{\sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}}+8 \sqrt {a} b^2 (a+b) \tanh (c+d x)-2 \sqrt {a} b (8 a+5 b) \tanh (c+d x) (a \cosh (2 (c+d x))+a+2 b)\right )}{64 a^{5/2} d (a+b)^2 \left (a+b \text {sech}^2(c+d x)\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^5*(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*S
qrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(a + 2*b + a*Cosh[2*(c + d*x)])^2*Sech[c + d*x]*(-Cos
h[c] + Sinh[c]))/(Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]) + 8*Sqrt[a]*b^2*(a + b)*Tanh[c + d*x] - 2*Sqrt[a]*b
*(8*a + 5*b)*(a + 2*b + a*Cosh[2*(c + d*x)])*Tanh[c + d*x]))/(64*a^(5/2)*(a + b)^2*d*(a + b*Sech[c + d*x]^2)^3
)

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fricas [B]  time = 0.54, size = 6806, normalized size = 47.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(4*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^7 + 28*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^6 + 4*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*sinh(d*x + c)^7 + 4*(8*a^4*b + 37*a^3*b^2 + 41*a^2
*b^3 + 12*a*b^4)*cosh(d*x + c)^5 + 4*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4 + 21*(8*a^4*b + 13*a^3*b^2
+ 5*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 20*(7*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^3 + (8*
a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 4*(8*a^4*b + 37*a^3*b^2 + 41*a^2*
b^3 + 12*a*b^4)*cosh(d*x + c)^3 - 4*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4 - 35*(8*a^4*b + 13*a^3*b^2 +
5*a^2*b^3)*cosh(d*x + c)^4 - 10*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)
^3 + 4*(21*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^5 + 10*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b
^4)*cosh(d*x + c)^3 - 3*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 + ((8*a^
4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*
a^4 + 8*a^3*b + 3*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(
8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3 + 7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8
*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^5 + 2*(24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 8*
a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4 + 30*(8*a^4 + 24*a^3*
b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a^4 + 8*a^3*b + 3*a^2*b^2 + 8*(7*(8*a^4 + 8*a^3
*b + 3*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 88*
a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6
*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 24*a^3*b + 19*a^2*b
^2 + 6*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3 + 3*(24*a^4 + 88*a^3*b + 137*a^2*b^2 +
88*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(8
*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^5 + (24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4
)*cosh(d*x + c)^3 + (8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a^2 - a*b)*l
og((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2
+ 2*(3*a*cosh(d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2*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)*s
inh(d*x + c) - cosh(d*x + c))*sqrt(-a^2 - a*b) + a)/(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)) - 4*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c
) + 4*(7*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^6 - 8*a^4*b - 13*a^3*b^2 - 5*a^2*b^3 + 5*(8*a^4*b +
37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^4 - 3*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*
x + c)^2)*sinh(d*x + c))/((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^8 + 3*a^7*b + 3*a^6*b
^2 + a^5*b^3)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a
^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^
3)*d*cosh(d*x + c)^2 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^8 + 17*
a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^8 + 3*a^7*b + 3*a^6*b^2
+ a^5*b^3)*d*cosh(d*x + c)^3 + 3*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x
+ c)^5 + 2*(35*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^
5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d
)*sinh(d*x + c)^4 + 4*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^8 + 3*a^
7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^5 + 10*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d
*x + c)^3 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x +
c)^3 + 4*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^
3 + 2*a^4*b^4)*d*cosh(d*x + c)^4 + 3*(3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*c
osh(d*x + c)^2 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^2 + (a^8 + 3*a^7*b + 3*a
^6*b^2 + a^5*b^3)*d + 8*((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 3*(a^8 + 5*a^7*b + 9*a^6*b^
2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^5 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^
3*b^5)*d*cosh(d*x + c)^3 + (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c))
, -1/8*(2*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^7 + 14*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x
+ c)*sinh(d*x + c)^6 + 2*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*sinh(d*x + c)^7 + 2*(8*a^4*b + 37*a^3*b^2 + 41*a^2
*b^3 + 12*a*b^4)*cosh(d*x + c)^5 + 2*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4 + 21*(8*a^4*b + 13*a^3*b^2
+ 5*a^2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^3 + (8*
a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^4 - 2*(8*a^4*b + 37*a^3*b^2 + 41*a^2*
b^3 + 12*a*b^4)*cosh(d*x + c)^3 - 2*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4 - 35*(8*a^4*b + 13*a^3*b^2 +
5*a^2*b^3)*cosh(d*x + c)^4 - 10*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)
^3 + 2*(21*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^5 + 10*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b
^4)*cosh(d*x + c)^3 - 3*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c))*sinh(d*x + c)^2 - ((8*a^
4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^8 + 8*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*
a^4 + 8*a^3*b + 3*a^2*b^2)*sinh(d*x + c)^8 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(
8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3 + 7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8
*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^3 + 3*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))
*sinh(d*x + c)^5 + 2*(24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 8*
a^3*b + 3*a^2*b^2)*cosh(d*x + c)^4 + 24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4 + 30*(8*a^4 + 24*a^3*
b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*a^4 + 8*a^3*b + 3*a^2*b^2 + 8*(7*(8*a^4 + 8*a^3
*b + 3*a^2*b^2)*cosh(d*x + c)^5 + 10*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 88*
a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6
*a*b^3)*cosh(d*x + c)^2 + 4*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 24*a^3*b + 19*a^2*b
^2 + 6*a*b^3)*cosh(d*x + c)^4 + 8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3 + 3*(24*a^4 + 88*a^3*b + 137*a^2*b^2 +
88*a*b^3 + 24*b^4)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(8
*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^5 + (24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4
)*cosh(d*x + c)^3 + (8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*sqrt(a^2 + a*b)*ar
ctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c
) + (3*a*cosh(d*x + c)^2 + 3*a + 4*b)*sinh(d*x + c))/sqrt(a^2 + a*b)) - ((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*
x + c)^8 + 8*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (8*a^4 + 8*a^3*b + 3*a^2*b^2)*sinh(
d*x + c)^8 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^6 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 +
6*a*b^3 + 7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 8*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2
)*cosh(d*x + c)^3 + 3*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(24*a^4 + 8
8*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c)^4 + 2*(35*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)
^4 + 24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4 + 30*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^4 + 8*a^4 + 8*a^3*b + 3*a^2*b^2 + 8*(7*(8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^5
+ 10*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^3 + (24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 +
24*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^2 + 4*(7*(
8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^6 + 15*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3)*cosh(d*x + c)^4 +
8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6*a*b^3 + 3*(24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c
)^2)*sinh(d*x + c)^2 + 8*((8*a^4 + 8*a^3*b + 3*a^2*b^2)*cosh(d*x + c)^7 + 3*(8*a^4 + 24*a^3*b + 19*a^2*b^2 + 6
*a*b^3)*cosh(d*x + c)^5 + (24*a^4 + 88*a^3*b + 137*a^2*b^2 + 88*a*b^3 + 24*b^4)*cosh(d*x + c)^3 + (8*a^4 + 24*
a^3*b + 19*a^2*b^2 + 6*a*b^3)*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 + b)) - 2*(8*a^4*b + 13*a^3*b^2 + 5*a^2*b^3)*cosh(d*x + c) + 2*(7*(8*a^4*b + 13*a^
3*b^2 + 5*a^2*b^3)*cosh(d*x + c)^6 - 8*a^4*b - 13*a^3*b^2 - 5*a^2*b^3 + 5*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 +
12*a*b^4)*cosh(d*x + c)^4 - 3*(8*a^4*b + 37*a^3*b^2 + 41*a^2*b^3 + 12*a*b^4)*cosh(d*x + c)^2)*sinh(d*x + c))/
((a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^8 + 8*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x
+ c)*sinh(d*x + c)^7 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*sinh(d*x + c)^8 + 4*(a^8 + 5*a^7*b + 9*a^6*b^2
+ 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^6 + 4*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^2 + (a
^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^6 + 2*(3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a
^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^4 + 8*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c
)^3 + 3*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*(a^8 + 3*
a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^4 + 30*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh
(d*x + c)^2 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d)*sinh(d*x + c)^4 + 4*(a^
8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^2 + 8*(7*(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3
)*d*cosh(d*x + c)^5 + 10*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c)^3 + (3*a^8 + 17*a
^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^8 + 3*a^7*
b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^6 + 15*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x
+ c)^4 + 3*(3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^2 + (a^8 + 5
*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d)*sinh(d*x + c)^2 + (a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d + 8*(
(a^8 + 3*a^7*b + 3*a^6*b^2 + a^5*b^3)*d*cosh(d*x + c)^7 + 3*(a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4
)*d*cosh(d*x + c)^5 + (3*a^8 + 17*a^7*b + 41*a^6*b^2 + 51*a^5*b^3 + 32*a^4*b^4 + 8*a^3*b^5)*d*cosh(d*x + c)^3
+ (a^8 + 5*a^7*b + 9*a^6*b^2 + 7*a^5*b^3 + 2*a^4*b^4)*d*cosh(d*x + c))*sinh(d*x + c))]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was
done assuming [a,b]=[84,-86]Warning, need to choose a branch for the root of a polynomial with parameters. Thi
s might be wrong.The choice was done assuming [a,b]=[-42,-12]Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [a,b]=[-43,-99]Warning, need to
choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assumin
g [a,b]=[-28,94]Warning, need to choose a branch for the root of a polynomial with parameters. This might be w
rong.The choice was done assuming [a,b]=[-7,46]Warning, need to choose a branch for the root of a polynomial w
ith parameters. This might be wrong.The choice was done assuming [a,b]=[-35,-99]Undef/Unsigned Inf encountered
in limitEvaluation time: 1.18Limit: Max order reached or unable to make series expansion Error: Bad Argument
Value

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maple [B]  time = 0.40, size = 1172, normalized size = 8.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)/a*tanh(1/2*d*x+1/2*c)^7+3/4/d/a^2*b^2/(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+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)^2*tanh(1/2*d*x+1/2*c)^5-13
/4/d/a*b^2/(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)^5-9/4/d/a^2*b^3/(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)^5-2/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)^2*tanh(1/2*d*x+1/2*
c)^3+13/4/d/a*b^2/(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+9/4/d/a^2*b^3/(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)^3-2/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)/a*tanh(1/2*d
*x+1/2*c)-3/4/d/a^2*b^2/(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)+1/d/(a^2+2*a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1
/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^(1/2))+1/d/a^(3/2)*b/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2
)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^(1/2))+3/8/d/a^(5/2)*b^2/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1
/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^(1/2))+1/d/(a^2+2*a*b+b^2)/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*
tanh(1/2*d*x+1/2*c)+2*b^(1/2))/a^(1/2))+1/d/a^(3/2)*b/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*ta
nh(1/2*d*x+1/2*c)+2*b^(1/2))/a^(1/2))+3/8/d/a^(5/2)*b^2/(a^2+2*a*b+b^2)/(a+b)^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*
tanh(1/2*d*x+1/2*c)+2*b^(1/2))/a^(1/2))

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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 )} + 5 \, a b^{2} e^{\left (7 \, c\right )}\right )} e^{\left (7 \, d x\right )} + {\left (8 \, a^{2} b e^{\left (5 \, c\right )} + 29 \, a b^{2} e^{\left (5 \, c\right )} + 12 \, b^{3} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} - {\left (8 \, a^{2} b e^{\left (3 \, c\right )} + 29 \, a b^{2} e^{\left (3 \, c\right )} + 12 \, b^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (8 \, a^{2} b e^{c} + 5 \, a b^{2} e^{c}\right )} e^{\left (d x\right )}}{4 \, {\left (a^{6} d + 2 \, a^{5} b d + a^{4} b^{2} d + {\left (a^{6} d e^{\left (8 \, c\right )} + 2 \, a^{5} b d e^{\left (8 \, c\right )} + a^{4} b^{2} d e^{\left (8 \, c\right )}\right )} e^{\left (8 \, d x\right )} + 4 \, {\left (a^{6} d e^{\left (6 \, c\right )} + 4 \, a^{5} b d e^{\left (6 \, c\right )} + 5 \, a^{4} b^{2} d e^{\left (6 \, c\right )} + 2 \, a^{3} b^{3} d e^{\left (6 \, c\right )}\right )} e^{\left (6 \, d x\right )} + 2 \, {\left (3 \, a^{6} d e^{\left (4 \, c\right )} + 14 \, a^{5} b d e^{\left (4 \, c\right )} + 27 \, a^{4} b^{2} d e^{\left (4 \, c\right )} + 24 \, a^{3} b^{3} d e^{\left (4 \, c\right )} + 8 \, 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 )} + 4 \, a^{5} b d e^{\left (2 \, c\right )} + 5 \, a^{4} b^{2} d e^{\left (2 \, c\right )} + 2 \, a^{3} b^{3} 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} + 2 \, a^{4} b + a^{3} b^{2} + {\left (a^{5} e^{\left (4 \, c\right )} + 2 \, a^{4} b e^{\left (4 \, c\right )} + a^{3} b^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + 2 \, {\left (a^{5} e^{\left (2 \, c\right )} + 4 \, a^{4} b e^{\left (2 \, c\right )} + 5 \, a^{3} b^{2} e^{\left (2 \, c\right )} + 2 \, 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*sech(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/4*((8*a^2*b*e^(7*c) + 5*a*b^2*e^(7*c))*e^(7*d*x) + (8*a^2*b*e^(5*c) + 29*a*b^2*e^(5*c) + 12*b^3*e^(5*c))*e^
(5*d*x) - (8*a^2*b*e^(3*c) + 29*a*b^2*e^(3*c) + 12*b^3*e^(3*c))*e^(3*d*x) - (8*a^2*b*e^c + 5*a*b^2*e^c)*e^(d*x
))/(a^6*d + 2*a^5*b*d + a^4*b^2*d + (a^6*d*e^(8*c) + 2*a^5*b*d*e^(8*c) + a^4*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^6
*d*e^(6*c) + 4*a^5*b*d*e^(6*c) + 5*a^4*b^2*d*e^(6*c) + 2*a^3*b^3*d*e^(6*c))*e^(6*d*x) + 2*(3*a^6*d*e^(4*c) + 1
4*a^5*b*d*e^(4*c) + 27*a^4*b^2*d*e^(4*c) + 24*a^3*b^3*d*e^(4*c) + 8*a^2*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^6*d*e^
(2*c) + 4*a^5*b*d*e^(2*c) + 5*a^4*b^2*d*e^(2*c) + 2*a^3*b^3*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 + 2*a^4*b
+ a^3*b^2 + (a^5*e^(4*c) + 2*a^4*b*e^(4*c) + a^3*b^2*e^(4*c))*e^(4*d*x) + 2*(a^5*e^(2*c) + 4*a^4*b*e^(2*c) +
5*a^3*b^2*e^(2*c) + 2*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 \[ \int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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