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

Optimal. Leaf size=147 \[ \frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{48 d}+\frac {b \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{6 d}+\frac {5 b (2 a+b) \tanh (c+d x) \text {sech}^3(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{24 d} \]

[Out]

1/16*(2*a+b)*(8*a^2+8*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/48*b*(44*a^2+44*a*b+15*b^2)*sech(d*x+c)*tanh(d*x+c)/d
+5/24*b*(2*a+b)*sech(d*x+c)^3*(a+b+a*sinh(d*x+c)^2)*tanh(d*x+c)/d+1/6*b*sech(d*x+c)^5*(a+b+a*sinh(d*x+c)^2)^2*
tanh(d*x+c)/d

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Rubi [A]  time = 0.14, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {4147, 413, 526, 385, 203} \[ \frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \tanh (c+d x) \text {sech}(c+d x)}{48 d}+\frac {b \tanh (c+d x) \text {sech}^5(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )^2}{6 d}+\frac {5 b (2 a+b) \tanh (c+d x) \text {sech}^3(c+d x) \left (a \sinh ^2(c+d x)+a+b\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a + b)*(8*a^2 + 8*a*b + 5*b^2)*ArcTan[Sinh[c + d*x]])/(16*d) + (b*(44*a^2 + 44*a*b + 15*b^2)*Sech[c + d*x]
*Tanh[c + d*x])/(48*d) + (5*b*(2*a + b)*Sech[c + d*x]^3*(a + b + a*Sinh[c + d*x]^2)*Tanh[c + d*x])/(24*d) + (b
*Sech[c + d*x]^5*(a + b + a*Sinh[c + d*x]^2)^2*Tanh[c + d*x])/(6*d)

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 526

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(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 - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*(p + 1) + (b*e - a*f)*(n*q + 1))*x
^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && GtQ[q, 0]

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}(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+a x^2\right )^3}{\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 )^2 \tanh (c+d x)}{6 d}+\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+a x^2\right ) \left ((a+b) (6 a+5 b)+a (6 a+b) x^2\right )}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{6 d}\\ &=\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}-\frac {\operatorname {Subst}\left (\int \frac {-(a+b) \left (24 a^2+34 a b+15 b^2\right )-a \left (24 a^2+14 a b+5 b^2\right ) x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{24 d}\\ &=\frac {b \left (44 a^2+44 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}+\frac {\left ((2 a+b) \left (8 a^2+8 a b+5 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{16 d}\\ &=\frac {(2 a+b) \left (8 a^2+8 a b+5 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{16 d}+\frac {b \left (44 a^2+44 a b+15 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{48 d}+\frac {5 b (2 a+b) \text {sech}^3(c+d x) \left (a+b+a \sinh ^2(c+d x)\right ) \tanh (c+d x)}{24 d}+\frac {b \text {sech}^5(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^2 \tanh (c+d x)}{6 d}\\ \end {align*}

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Mathematica [C]  time = 9.84, size = 1430, normalized size = 9.73 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Coth[c + d*x]^6*Csch[c + d*x]*(a + b*Sech[c + d*x]^2)^3*(117228825*(a + b)^3*Sinh[c + d*x]^2 + 274542345*a*(a
+ b)^2*Sinh[c + d*x]^4 + 70189350*(a + b)^3*Sinh[c + d*x]^4 + 215549775*a^2*(a + b)*Sinh[c + d*x]^6 + 1689515
10*a*(a + b)^2*Sinh[c + d*x]^6 + 4093425*(a + b)^3*Sinh[c + d*x]^6 + 58009455*a^3*Sinh[c + d*x]^8 + 135323370*
a^2*(a + b)*Sinh[c + d*x]^8 + 9514449*a*(a + b)^2*Sinh[c + d*x]^8 + 36772890*a^3*Sinh[c + d*x]^10 + 7808535*a^
2*(a + b)*Sinh[c + d*x]^10 - 75520*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d
*x]^2]*Sinh[c + d*x]^10 - 13824*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c
+ d*x]^2]*Sinh[c + d*x]^10 - 1024*(a + b)^3*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2},
-Sinh[c + d*x]^2]*Sinh[c + d*x]^10 + 2160711*a^3*Sinh[c + d*x]^12 - 189696*a*(a + b)^2*HypergeometricPFQ[{3/2,
2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 38400*a*(a + b)^2*HypergeometricPFQ[{3/2,
2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 3072*a*(a + b)^2*HypergeometricPFQ[{3
/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^12 - 158976*a^2*(a + b)*Hypergeo
metricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^14 - 35328*a^2*(a + b)*Hypergeom
etricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^14 - 3072*a^2*(a + b)*Hyper
geometricPFQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^14 - 44800*a^3*Hy
pergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 - 10752*a^3*Hypergeomet
ricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 - 1024*a^3*HypergeometricP
FQ[{3/2, 2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 1, 11/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^16 + 117228825*(a + b)^3*Ar
cTanh[Sqrt[-Sinh[c + d*x]^2]]*Sqrt[-Sinh[c + d*x]^2] + 215549775*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*S
inh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 260465625*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*
Sqrt[-Sinh[c + d*x]^2] + 17069535*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x
]^2] + 58009455*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 207173295*a^2*(a
+ b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 41427855*a*(a + b)^2*ArcTanh[Sqr
t[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 56109375*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sin
h[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 33756345*a^2*(a + b)*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqr
t[-Sinh[c + d*x]^2] + 210735*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2
] + 9261945*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] + 174825*a^2*(a + b)*A
rcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10*Sqrt[-Sinh[c + d*x]^2] + 48825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]
^2]]*Sinh[c + d*x]^12*Sqrt[-Sinh[c + d*x]^2] - 274542345*a*(a + b)^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c
+ d*x]^2)^(3/2) - 109265625*(a + b)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(-Sinh[c + d*x]^2)^(3/2) + 142065*(a + b
)^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4*(-Sinh[c + d*x]^2)^(3/2)))/(90720*d*(a + 2*b + a*Cosh[2*c
+ 2*d*x])^3)

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fricas [B]  time = 0.47, size = 3465, normalized size = 23.57 \[ \text {result too large to display} \]

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]

1/24*(3*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^11 + 33*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d
*x + c)^10 + 3*(24*a^2*b + 18*a*b^2 + 5*b^3)*sinh(d*x + c)^11 + (216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)
^9 + (216*a^2*b + 306*a*b^2 + 85*b^3 + 165*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*
(55*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + (216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c))*sinh(d*x +
c)^8 + 18*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^7 + 18*(55*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^
4 + 8*a^2*b + 14*a*b^2 + 11*b^3 + 2*(216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 42*(33
*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 2*(216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^3 + 3*(8*a^2
*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 18*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^5 + 18
*(77*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 7*(216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^4 - 8*a^
2*b - 14*a*b^2 - 11*b^3 + 21*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 18*(55*(24*a^2*b
+ 18*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 7*(216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^5 + 35*(8*a^2*b + 14*a
*b^2 + 11*b^3)*cosh(d*x + c)^3 - 5*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - (216*a^2*b +
306*a*b^2 + 85*b^3)*cosh(d*x + c)^3 + (495*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 84*(216*a^2*b + 30
6*a*b^2 + 85*b^3)*cosh(d*x + c)^6 + 630*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^4 - 216*a^2*b - 306*a*b^2
- 85*b^3 - 180*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 3*(55*(24*a^2*b + 18*a*b^2 + 5
*b^3)*cosh(d*x + c)^9 + 12*(216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^7 + 126*(8*a^2*b + 14*a*b^2 + 11*b^3
)*cosh(d*x + c)^5 - 60*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^3 - (216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d
*x + c))*sinh(d*x + c)^2 + 3*((16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^12 + 12*(16*a^3 + 24*a^2*b
+ 18*a*b^2 + 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^11 + (16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*sinh(d*x + c)^12 +
6*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 6*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3 + 11*(16*
a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 + 20*(11*(16*a^3 + 24*a^2*b + 18*a*b^2 +
5*b^3)*cosh(d*x + c)^3 + 3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 15*(16*a^3
+ 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^8 + 15*(33*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4
+ 16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3 + 18*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^8 + 24*(33*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 30*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*
b^3)*cosh(d*x + c)^3 + 5*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 20*(16*a^3 +
24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 4*(231*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 +
315*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 80*a^3 + 120*a^2*b + 90*a*b^2 + 25*b^3 + 105*(16
*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 24*(33*(16*a^3 + 24*a^2*b + 18*a*b^2 +
5*b^3)*cosh(d*x + c)^7 + 63*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 35*(16*a^3 + 24*a^2*b + 1
8*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 5*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + 1
5*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^4 + 15*(33*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(
d*x + c)^8 + 84*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 70*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*
b^3)*cosh(d*x + c)^4 + 16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3 + 20*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^4 + 20*(11*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 36*(16*a^3 + 24*a
^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 42*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 20*(16*
a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^3 + 3*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c))*s
inh(d*x + c)^3 + 16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3 + 6*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)
^2 + 6*(11*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 45*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)
*cosh(d*x + c)^8 + 70*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^6 + 50*(16*a^3 + 24*a^2*b + 18*a*b^
2 + 5*b^3)*cosh(d*x + c)^4 + 16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3 + 15*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^2 + 12*((16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^11 + 5*(16*a^3 + 2
4*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^9 + 10*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 10*(
16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^5 + 5*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)
^3 + (16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)
) - 3*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c) + 3*(11*(24*a^2*b + 18*a*b^2 + 5*b^3)*cosh(d*x + c)^10 + 3*(
216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(d*x + c)^8 + 42*(8*a^2*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^6 - 30*(8*a^2
*b + 14*a*b^2 + 11*b^3)*cosh(d*x + c)^4 - 24*a^2*b - 18*a*b^2 - 5*b^3 - (216*a^2*b + 306*a*b^2 + 85*b^3)*cosh(
d*x + c)^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))*sinh(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*(3
3*d*cosh(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*cos
h(d*x + c))*sinh(d*x + c) + d)

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giac [B]  time = 0.19, size = 310, normalized size = 2.11 \[ \frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (16 \, a^{3} + 24 \, a^{2} b + 18 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (72 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 54 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 15 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 576 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 576 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 160 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 1152 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 1440 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 528 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(16*a^3 + 24*a^2*b + 18*a*b^2 + 5*b^3) + 4*(72
*a^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 54*a*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 15*b^3*(e^(d*x + c) - e^(-d*
x - c))^5 + 576*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 + 576*a*b^2*(e^(d*x + c) - e^(-d*x - c))^3 + 160*b^3*(e^(
d*x + c) - e^(-d*x - c))^3 + 1152*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 1440*a*b^2*(e^(d*x + c) - e^(-d*x - c))
+ 528*b^3*(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 = 193, normalized size = 1.31 \[ \frac {2 a^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a^{2} b \,\mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2 d}+\frac {3 a^{2} b \arctan \left ({\mathrm e}^{d x +c}\right )}{d}+\frac {3 a \,b^{2} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{4 d}+\frac {9 a \,b^{2} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{8 d}+\frac {9 a \,b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{4 d}+\frac {b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{5}}{6 d}+\frac {5 b^{3} \tanh \left (d x +c \right ) \mathrm {sech}\left (d x +c \right )^{3}}{24 d}+\frac {5 b^{3} \mathrm {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{16 d}+\frac {5 b^{3} \arctan \left ({\mathrm e}^{d x +c}\right )}{8 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a^3*arctan(exp(d*x+c))+3/2/d*a^2*b*sech(d*x+c)*tanh(d*x+c)+3/d*a^2*b*arctan(exp(d*x+c))+3/4/d*a*b^2*tanh(d
*x+c)*sech(d*x+c)^3+9/8/d*a*b^2*sech(d*x+c)*tanh(d*x+c)+9/4/d*a*b^2*arctan(exp(d*x+c))+1/6/d*b^3*tanh(d*x+c)*s
ech(d*x+c)^5+5/24/d*b^3*tanh(d*x+c)*sech(d*x+c)^3+5/16/d*b^3*sech(d*x+c)*tanh(d*x+c)+5/8/d*b^3*arctan(exp(d*x+
c))

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maxima [B]  time = 0.41, size = 365, normalized size = 2.48 \[ -\frac {1}{24} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {3}{4} \, a b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} - 3 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} + 1\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {a^{3} \arctan \left (\sinh \left (d x + c\right )\right )}{d} \]

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]

-1/24*b^3*(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))) - 3/4*a*b^2*(3*arct
an(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))) - 3*a^2*b*(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))) + a^3*arctan(sinh
(d*x + c))/d

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mupad [B]  time = 1.47, size = 535, normalized size = 3.64 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (16\,a^3\,\sqrt {d^2}+5\,b^3\,\sqrt {d^2}+18\,a\,b^2\,\sqrt {d^2}+24\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {256\,a^6+768\,a^5\,b+1152\,a^4\,b^2+1024\,a^3\,b^3+564\,a^2\,b^4+180\,a\,b^5+25\,b^6}}\right )\,\sqrt {256\,a^6+768\,a^5\,b+1152\,a^4\,b^2+1024\,a^3\,b^3+564\,a^2\,b^4+180\,a\,b^5+25\,b^6}}{8\,\sqrt {d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (54\,a\,b^2-b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {80\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (2\,a\,b^2-3\,b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (24\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (-72\,a^2\,b+18\,a\,b^2+5\,b^3\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan((exp(d*x)*exp(c)*(16*a^3*(d^2)^(1/2) + 5*b^3*(d^2)^(1/2) + 18*a*b^2*(d^2)^(1/2) + 24*a^2*b*(d^2)^(1/2)))
/(d*(180*a*b^5 + 768*a^5*b + 256*a^6 + 25*b^6 + 564*a^2*b^4 + 1024*a^3*b^3 + 1152*a^4*b^2)^(1/2)))*(180*a*b^5
+ 768*a^5*b + 256*a^6 + 25*b^6 + 564*a^2*b^4 + 1024*a^3*b^3 + 1152*a^4*b^2)^(1/2))/(8*(d^2)^(1/2)) - (exp(c +
d*x)*(54*a*b^2 - b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (80*b^3*exp(c
+ d*x))/(3*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c +
10*d*x) + 1)) + (6*exp(c + d*x)*(2*a*b^2 - 3*b^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)) - (32*b^3*exp(c + d*x))/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*ex
p(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)) + (exp(c + d*x)*(18*a*b
^2 + 24*a^2*b + 5*b^3))/(8*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(18*a*b^2 - 72*a^2*b + 5*b^3))/(12*d*(2*e
xp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))

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

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

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