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

Optimal. Leaf size=128 \[ \frac {\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{16 d}+\frac {b (8 a+5 b) \tanh (c+d x) \text {sech}^3(c+d x)}{24 d}+\frac {b \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 d} \]

[Out]

1/16*(8*a^2+12*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/16*(8*a^2+12*a*b+5*b^2)*sech(d*x+c)*tanh(d*x+c)/d+1/24*b*(8*
a+5*b)*sech(d*x+c)^3*tanh(d*x+c)/d+1/6*b*sech(d*x+c)^5*(a+b+a*sinh(d*x+c)^2)*tanh(d*x+c)/d

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Rubi [A]  time = 0.15, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4147, 413, 385, 199, 203} \[ \frac {\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{16 d}+\frac {b (8 a+5 b) \tanh (c+d x) \text {sech}^3(c+d x)}{24 d}+\frac {b \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + ((8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x
])/(16*d) + (b*(8*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x])/(24*d) + (b*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]
^2)*Tanh[c + d*x])/(6*d)

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 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 \text {sech}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+a x^2\right )^2}{\left (1+x^2\right )^4} \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+b) (6 a+5 b)+3 a (2 a+b) x^2}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac {b (8 a+5 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{8 d}\\ &=\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {b (8 a+5 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac {\left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {\left (8 a^2+12 a b+5 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {b (8 a+5 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 104, normalized size = 0.81 \[ \frac {3 \left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))+3 \left (8 a^2+12 a b+5 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)+2 b (12 a+5 b) \tanh (c+d x) \text {sech}^3(c+d x)+8 b^2 \tanh (c+d x) \text {sech}^5(c+d x)}{48 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(8*a^2 + 12*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]] + 3*(8*a^2 + 12*a*b + 5*b^2)*Sech[c + d*x]*Tanh[c + d*x] + 2
*b*(12*a + 5*b)*Sech[c + d*x]^3*Tanh[c + d*x] + 8*b^2*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d)

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fricas [B]  time = 0.43, size = 2946, normalized size = 23.02 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^11 + 33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^10
+ 3*(8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^11 + (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^9 + (165*(8*a^2 + 1
2*a*b + 5*b^2)*cosh(d*x + c)^2 + 72*a^2 + 204*a*b + 85*b^2)*sinh(d*x + c)^9 + 9*(55*(8*a^2 + 12*a*b + 5*b^2)*c
osh(d*x + c)^3 + (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c))*sinh(d*x + c)^8 + 6*(8*a^2 + 28*a*b + 33*b^2)*cosh
(d*x + c)^7 + 6*(165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 6*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^2
+ 8*a^2 + 28*a*b + 33*b^2)*sinh(d*x + c)^7 + 42*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 2*(72*a^2 + 204
*a*b + 85*b^2)*cosh(d*x + c)^3 + (8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 6*(8*a^2 + 28*a*b
+ 33*b^2)*cosh(d*x + c)^5 + 6*(231*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 21*(72*a^2 + 204*a*b + 85*b^2)*c
osh(d*x + c)^4 + 21*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2 - 8*a^2 - 28*a*b - 33*b^2)*sinh(d*x + c)^5 + 6*(
165*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 21*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^5 + 35*(8*a^2 + 28
*a*b + 33*b^2)*cosh(d*x + c)^3 - 5*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (72*a^2 + 204*a*
b + 85*b^2)*cosh(d*x + c)^3 + (495*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 84*(72*a^2 + 204*a*b + 85*b^2)*c
osh(d*x + c)^6 + 210*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^4 - 60*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^2
- 72*a^2 - 204*a*b - 85*b^2)*sinh(d*x + c)^3 + 3*(55*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^9 + 12*(72*a^2 + 2
04*a*b + 85*b^2)*cosh(d*x + c)^7 + 42*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^5 - 20*(8*a^2 + 28*a*b + 33*b^2)
*cosh(d*x + c)^3 - (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 3*((8*a^2 + 12*a*b + 5*b^2)*co
sh(d*x + c)^12 + 12*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)*sinh(d*x + c)^11 + (8*a^2 + 12*a*b + 5*b^2)*sinh(d*
x + c)^12 + 6*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^10 + 6*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a
^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^10 + 20*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 3*(8*a^2 + 12*a*b +
5*b^2)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 15*(33*(8*a^2 + 12*a*b +
5*b^2)*cosh(d*x + c)^4 + 18*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^
8 + 24*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 30*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(8*a^2 +
12*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 4*(231*(8*a^2
+ 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 315*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 105*(8*a^2 + 12*a*b + 5*b^2
)*cosh(d*x + c)^2 + 40*a^2 + 60*a*b + 25*b^2)*sinh(d*x + c)^6 + 24*(33*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^
7 + 63*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 + 35*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 5*(8*a^2 + 12*
a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^4 + 15*(33*(8*a^2 + 12
*a*b + 5*b^2)*cosh(d*x + c)^8 + 84*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 70*(8*a^2 + 12*a*b + 5*b^2)*cosh
(d*x + c)^4 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*x + c)^4 + 20*(11*(
8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^9 + 36*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 42*(8*a^2 + 12*a*b + 5
*b^2)*cosh(d*x + c)^5 + 20*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + 3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)
)*sinh(d*x + c)^3 + 6*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 6*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^
10 + 45*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^8 + 70*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^6 + 50*(8*a^2 + 1
2*a*b + 5*b^2)*cosh(d*x + c)^4 + 15*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^2 + 8*a^2 + 12*a*b + 5*b^2)*sinh(d*
x + c)^2 + 8*a^2 + 12*a*b + 5*b^2 + 12*((8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^11 + 5*(8*a^2 + 12*a*b + 5*b^2)
*cosh(d*x + c)^9 + 10*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^7 + 10*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^5 +
5*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c)^3 + (8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c))*sinh(d*x + c))*arctan(co
sh(d*x + c) + sinh(d*x + c)) - 3*(8*a^2 + 12*a*b + 5*b^2)*cosh(d*x + c) + 3*(11*(8*a^2 + 12*a*b + 5*b^2)*cosh(
d*x + c)^10 + 3*(72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^8 + 14*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^6 - 1
0*(8*a^2 + 28*a*b + 33*b^2)*cosh(d*x + c)^4 - (72*a^2 + 204*a*b + 85*b^2)*cosh(d*x + c)^2 - 8*a^2 - 12*a*b - 5
*b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^12 + 12*d*cosh(d*x + c)*sinh(d*x + c)^11 + d*sinh(d*x + c)^12 + 6*d*cosh
(d*x + c)^10 + 6*(11*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^10 + 20*(11*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*s
inh(d*x + c)^9 + 15*d*cosh(d*x + c)^8 + 15*(33*d*cosh(d*x + c)^4 + 18*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^8 +
24*(33*d*cosh(d*x + c)^5 + 30*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^7 + 20*d*cosh(d*x + c)^6 +
4*(231*d*cosh(d*x + c)^6 + 315*d*cosh(d*x + c)^4 + 105*d*cosh(d*x + c)^2 + 5*d)*sinh(d*x + c)^6 + 24*(33*d*co
sh(d*x + c)^7 + 63*d*cosh(d*x + c)^5 + 35*d*cosh(d*x + c)^3 + 5*d*cosh(d*x + c))*sinh(d*x + c)^5 + 15*d*cosh(d
*x + c)^4 + 15*(33*d*cosh(d*x + c)^8 + 84*d*cosh(d*x + c)^6 + 70*d*cosh(d*x + c)^4 + 20*d*cosh(d*x + c)^2 + d)
*sinh(d*x + c)^4 + 20*(11*d*cosh(d*x + c)^9 + 36*d*cosh(d*x + c)^7 + 42*d*cosh(d*x + c)^5 + 20*d*cosh(d*x + c)
^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^3 + 6*d*cosh(d*x + c)^2 + 6*(11*d*cosh(d*x + c)^10 + 45*d*cosh(d*x + c)^
8 + 70*d*cosh(d*x + c)^6 + 50*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^2 + 12*(d*cosh(d*x +
c)^11 + 5*d*cosh(d*x + c)^9 + 10*d*cosh(d*x + c)^7 + 10*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 + d*cosh(d*x
+ c))*sinh(d*x + c) + d)

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giac [B]  time = 0.16, size = 293, normalized size = 2.29 \[ \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (8 \, a^{2} + 12 \, a b + 5 \, b^{2}\right )} + \frac {4 \, {\left (24 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 36 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 15 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 192 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 160 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 384 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 960 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 528 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(8*a^2 + 12*a*b + 5*b^2) + 4*(24*a^2*(e^(d*x +
c) - e^(-d*x - c))^5 + 36*a*b*(e^(d*x + c) - e^(-d*x - c))^5 + 15*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 192*a^
2*(e^(d*x + c) - e^(-d*x - c))^3 + 384*a*b*(e^(d*x + c) - e^(-d*x - c))^3 + 160*b^2*(e^(d*x + c) - e^(-d*x - c
))^3 + 384*a^2*(e^(d*x + c) - e^(-d*x - c)) + 960*a*b*(e^(d*x + c) - e^(-d*x - c)) + 528*b^2*(e^(d*x + c) - e^
(-d*x - c)))/((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3)/d

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maple [A]  time = 0.46, size = 169, normalized size = 1.32 \[ \frac {a^{2} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {a^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {a b \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{2 d}+\frac {3 a b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{4 d}+\frac {3 a b \arctan \left ({\mathrm e}^{d x +c}\right )}{2 d}+\frac {b^{2} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{5}}{6 d}+\frac {5 b^{2} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{24 d}+\frac {5 b^{2} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{16 d}+\frac {5 b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*a^2*sech(d*x+c)*tanh(d*x+c)+1/d*a^2*arctan(exp(d*x+c))+1/2/d*a*b*tanh(d*x+c)*sech(d*x+c)^3+3/4/d*a*b*sec
h(d*x+c)*tanh(d*x+c)+3/2/d*a*b*arctan(exp(d*x+c))+1/6/d*b^2*tanh(d*x+c)*sech(d*x+c)^5+5/24/d*b^2*tanh(d*x+c)*s
ech(d*x+c)^3+5/16/d*b^2*sech(d*x+c)*tanh(d*x+c)+5/8/d*b^2*arctan(exp(d*x+c))

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maxima [B]  time = 0.42, size = 348, normalized size = 2.72 \[ -\frac {1}{24} \, b^{2} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{2} \, a b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - a^{2} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*b^2*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) + 85*e^(-3*d*x - 3*c) + 198*e^(-5*d*x - 5*c) - 198*e^(
-7*d*x - 7*c) - 85*e^(-9*d*x - 9*c) - 15*e^(-11*d*x - 11*c))/(d*(6*e^(-2*d*x - 2*c) + 15*e^(-4*d*x - 4*c) + 20
*e^(-6*d*x - 6*c) + 15*e^(-8*d*x - 8*c) + 6*e^(-10*d*x - 10*c) + e^(-12*d*x - 12*c) + 1))) - 1/2*a*b*(3*arctan
(e^(-d*x - c))/d - (3*e^(-d*x - c) + 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) - 3*e^(-7*d*x - 7*c))/(d*(4*e^(
-2*d*x - 2*c) + 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) + 1))) - a^2*(arctan(e^(-d*x - c))/
d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1)))

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mupad [B]  time = 1.57, size = 569, normalized size = 4.45 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (8\,a^2\,\sqrt {d^2}+5\,b^2\,\sqrt {d^2}+12\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4+192\,a^3\,b+224\,a^2\,b^2+120\,a\,b^3+25\,b^4}}\right )\,\sqrt {64\,a^4+192\,a^3\,b+224\,a^2\,b^2+120\,a\,b^3+25\,b^4}}{8\,\sqrt {d^2}}-\frac {\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {2\,a^2\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}+\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (3\,a^2+8\,a\,b+8\,b^2\right )}{3\,d}+\frac {8\,a\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (a+2\,b\right )}{3\,d}+\frac {8\,a\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (a+2\,b\right )}{3\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a\,b-11\,b^2\right )}{3\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {16\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-16\,a^2+12\,a\,b+5\,b^2\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (20\,a\,b-b^2\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((exp(d*x)*exp(c)*(8*a^2*(d^2)^(1/2) + 5*b^2*(d^2)^(1/2) + 12*a*b*(d^2)^(1/2)))/(d*(120*a*b^3 + 192*a^3*b
+ 64*a^4 + 25*b^4 + 224*a^2*b^2)^(1/2)))*(120*a*b^3 + 192*a^3*b + 64*a^4 + 25*b^4 + 224*a^2*b^2)^(1/2))/(8*(d
^2)^(1/2)) - ((2*a^2*exp(c + d*x))/(3*d) + (2*a^2*exp(9*c + 9*d*x))/(3*d) + (4*exp(5*c + 5*d*x)*(8*a*b + 3*a^2
+ 8*b^2))/(3*d) + (8*a*exp(3*c + 3*d*x)*(a + 2*b))/(3*d) + (8*a*exp(7*c + 7*d*x)*(a + 2*b))/(3*d))/(6*exp(2*c
+ 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c
+ 12*d*x) + 1) + (2*exp(c + d*x)*(4*a*b - 11*b^2))/(3*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c +
6*d*x) + exp(8*c + 8*d*x) + 1)) + (16*b^2*exp(c + d*x))/(3*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*e
xp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) + (exp(c + d*x)*(12*a*b + 8*a^2 + 5*b^2))/(8*d
*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(12*a*b - 16*a^2 + 5*b^2))/(12*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*
x) + 1)) - (exp(c + d*x)*(20*a*b - b^2))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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