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

Optimal. Leaf size=108 \[ -\frac {b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac {3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {(a+b)^3 \tanh (c+d x)}{d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {4146, 373} \[ -\frac {b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac {3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac {(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {(a+b)^3 \tanh (c+d x)}{d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b)^3*Tanh[c + d*x])/d - ((a + b)^2*(a + 4*b)*Tanh[c + d*x]^3)/(3*d) + (3*b*(a + b)*(a + 2*b)*Tanh[c + d*
x]^5)/(5*d) - (b^2*(3*a + 4*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 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 \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left ((a+b)^3-(a+b)^2 (a+4 b) x^2+3 b (a+b) (a+2 b) x^4-b^2 (3 a+4 b) x^6+b^3 x^8\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [B]  time = 1.76, size = 348, normalized size = 3.22 \[ \frac {\text {sech}(c) \text {sech}^9(c+d x) \left (6825 a^3 \sinh (2 c+3 d x)-1995 a^3 \sinh (4 c+3 d x)+3465 a^3 \sinh (4 c+5 d x)-315 a^3 \sinh (6 c+5 d x)+945 a^3 \sinh (6 c+7 d x)+105 a^3 \sinh (8 c+9 d x)-315 a \left (17 a^2+36 a b+24 b^2\right ) \sinh (2 c+d x)+18648 a^2 b \sinh (2 c+3 d x)-2520 a^2 b \sinh (4 c+3 d x)+9072 a^2 b \sinh (4 c+5 d x)+2268 a^2 b \sinh (6 c+7 d x)+252 a^2 b \sinh (8 c+9 d x)+63 \left (125 a^3+324 a^2 b+312 a b^2+128 b^3\right ) \sinh (d x)+18144 a b^2 \sinh (2 c+3 d x)+7776 a b^2 \sinh (4 c+5 d x)+1944 a b^2 \sinh (6 c+7 d x)+216 a b^2 \sinh (8 c+9 d x)+5376 b^3 \sinh (2 c+3 d x)+2304 b^3 \sinh (4 c+5 d x)+576 b^3 \sinh (6 c+7 d x)+64 b^3 \sinh (8 c+9 d x)\right )}{40320 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[c]*Sech[c + d*x]^9*(63*(125*a^3 + 324*a^2*b + 312*a*b^2 + 128*b^3)*Sinh[d*x] - 315*a*(17*a^2 + 36*a*b +
24*b^2)*Sinh[2*c + d*x] + 6825*a^3*Sinh[2*c + 3*d*x] + 18648*a^2*b*Sinh[2*c + 3*d*x] + 18144*a*b^2*Sinh[2*c +
3*d*x] + 5376*b^3*Sinh[2*c + 3*d*x] - 1995*a^3*Sinh[4*c + 3*d*x] - 2520*a^2*b*Sinh[4*c + 3*d*x] + 3465*a^3*Sin
h[4*c + 5*d*x] + 9072*a^2*b*Sinh[4*c + 5*d*x] + 7776*a*b^2*Sinh[4*c + 5*d*x] + 2304*b^3*Sinh[4*c + 5*d*x] - 31
5*a^3*Sinh[6*c + 5*d*x] + 945*a^3*Sinh[6*c + 7*d*x] + 2268*a^2*b*Sinh[6*c + 7*d*x] + 1944*a*b^2*Sinh[6*c + 7*d
*x] + 576*b^3*Sinh[6*c + 7*d*x] + 105*a^3*Sinh[8*c + 9*d*x] + 252*a^2*b*Sinh[8*c + 9*d*x] + 216*a*b^2*Sinh[8*c
+ 9*d*x] + 64*b^3*Sinh[8*c + 9*d*x]))/(40320*d)

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fricas [B]  time = 0.42, size = 1190, normalized size = 11.02 \[ \text {result too large to display} \]

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]

-8/315*(2*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)^7 + 14*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^
3)*cosh(d*x + c)*sinh(d*x + c)^6 + (105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b^3)*sinh(d*x + c)^7 + 6*(245*a^3 + 3
99*a^2*b + 162*a*b^2 + 48*b^3)*cosh(d*x + c)^5 + 3*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3 + 7*(105*a^3 - 126
*a^2*b - 108*a*b^2 - 32*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)
*cosh(d*x + c)^3 + 3*(245*a^3 + 399*a^2*b + 162*a*b^2 + 48*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 18*(245*a^3 +
567*a^2*b + 426*a*b^2 + 64*b^3)*cosh(d*x + c)^3 + (35*(105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b^3)*cosh(d*x + c
)^4 + 945*a^3 + 1134*a^2*b - 108*a*b^2 - 1152*b^3 + 30*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3)*cosh(d*x + c)
^2)*sinh(d*x + c)^3 + 6*(7*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)^5 + 10*(245*a^3 + 399*a^2*b
+ 162*a*b^2 + 48*b^3)*cosh(d*x + c)^3 + 9*(245*a^3 + 567*a^2*b + 426*a*b^2 + 64*b^3)*cosh(d*x + c))*sinh(d*x +
c)^2 + 210*(35*a^3 + 93*a^2*b + 90*a*b^2 + 32*b^3)*cosh(d*x + c) + (7*(105*a^3 - 126*a^2*b - 108*a*b^2 - 32*b
^3)*cosh(d*x + c)^6 + 15*(175*a^3 + 42*a^2*b - 324*a*b^2 - 96*b^3)*cosh(d*x + c)^4 + 525*a^3 + 882*a^2*b + 756
*a*b^2 + 1344*b^3 + 27*(105*a^3 + 126*a^2*b - 12*a*b^2 - 128*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x
+ c)^11 + 11*d*cosh(d*x + c)*sinh(d*x + c)^10 + d*sinh(d*x + c)^11 + 9*d*cosh(d*x + c)^9 + (55*d*cosh(d*x + c)
^2 + 9*d)*sinh(d*x + c)^9 + 3*(55*d*cosh(d*x + c)^3 + 27*d*cosh(d*x + c))*sinh(d*x + c)^8 + 37*d*cosh(d*x + c)
^7 + (330*d*cosh(d*x + c)^4 + 324*d*cosh(d*x + c)^2 + 35*d)*sinh(d*x + c)^7 + 7*(66*d*cosh(d*x + c)^5 + 108*d*
cosh(d*x + c)^3 + 37*d*cosh(d*x + c))*sinh(d*x + c)^6 + 93*d*cosh(d*x + c)^5 + 3*(154*d*cosh(d*x + c)^6 + 378*
d*cosh(d*x + c)^4 + 245*d*cosh(d*x + c)^2 + 25*d)*sinh(d*x + c)^5 + (330*d*cosh(d*x + c)^7 + 1134*d*cosh(d*x +
c)^5 + 1295*d*cosh(d*x + c)^3 + 465*d*cosh(d*x + c))*sinh(d*x + c)^4 + 162*d*cosh(d*x + c)^3 + (165*d*cosh(d*
x + c)^8 + 756*d*cosh(d*x + c)^6 + 1225*d*cosh(d*x + c)^4 + 750*d*cosh(d*x + c)^2 + 90*d)*sinh(d*x + c)^3 + (5
5*d*cosh(d*x + c)^9 + 324*d*cosh(d*x + c)^7 + 777*d*cosh(d*x + c)^5 + 930*d*cosh(d*x + c)^3 + 486*d*cosh(d*x +
c))*sinh(d*x + c)^2 + 210*d*cosh(d*x + c) + (11*d*cosh(d*x + c)^10 + 81*d*cosh(d*x + c)^8 + 245*d*cosh(d*x +
c)^6 + 375*d*cosh(d*x + c)^4 + 270*d*cosh(d*x + c)^2 + 42*d)*sinh(d*x + c))

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giac [B]  time = 0.18, size = 360, normalized size = 3.33 \[ -\frac {4 \, {\left (315 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 1995 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 2520 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 5355 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 7560 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 7875 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 20412 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 19656 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8064 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6825 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 18648 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 18144 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5376 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3465 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9072 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 7776 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2304 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 945 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2268 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 1944 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 576 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 252 \, a^{2} b + 216 \, a b^{2} + 64 \, b^{3}\right )}}{315 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \]

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]

-4/315*(315*a^3*e^(14*d*x + 14*c) + 1995*a^3*e^(12*d*x + 12*c) + 2520*a^2*b*e^(12*d*x + 12*c) + 5355*a^3*e^(10
*d*x + 10*c) + 11340*a^2*b*e^(10*d*x + 10*c) + 7560*a*b^2*e^(10*d*x + 10*c) + 7875*a^3*e^(8*d*x + 8*c) + 20412
*a^2*b*e^(8*d*x + 8*c) + 19656*a*b^2*e^(8*d*x + 8*c) + 8064*b^3*e^(8*d*x + 8*c) + 6825*a^3*e^(6*d*x + 6*c) + 1
8648*a^2*b*e^(6*d*x + 6*c) + 18144*a*b^2*e^(6*d*x + 6*c) + 5376*b^3*e^(6*d*x + 6*c) + 3465*a^3*e^(4*d*x + 4*c)
+ 9072*a^2*b*e^(4*d*x + 4*c) + 7776*a*b^2*e^(4*d*x + 4*c) + 2304*b^3*e^(4*d*x + 4*c) + 945*a^3*e^(2*d*x + 2*c
) + 2268*a^2*b*e^(2*d*x + 2*c) + 1944*a*b^2*e^(2*d*x + 2*c) + 576*b^3*e^(2*d*x + 2*c) + 105*a^3 + 252*a^2*b +
216*a*b^2 + 64*b^3)/(d*(e^(2*d*x + 2*c) + 1)^9)

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maple [A]  time = 0.52, size = 158, normalized size = 1.46 \[ \frac {a^{3} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\mathrm {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \mathrm {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {16}{35}+\frac {\mathrm {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \mathrm {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \mathrm {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {128}{315}+\frac {\mathrm {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \mathrm {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \mathrm {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \mathrm {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d} \]

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/d*(a^3*(2/3+1/3*sech(d*x+c)^2)*tanh(d*x+c)+3*a^2*b*(8/15+1/5*sech(d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c)+3
*a*b^2*(16/35+1/7*sech(d*x+c)^6+6/35*sech(d*x+c)^4+8/35*sech(d*x+c)^2)*tanh(d*x+c)+b^3*(128/315+1/9*sech(d*x+c
)^8+8/63*sech(d*x+c)^6+16/105*sech(d*x+c)^4+64/315*sech(d*x+c)^2)*tanh(d*x+c))

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maxima [B]  time = 0.34, size = 1245, normalized size = 11.53 \[ \text {result too large to display} \]

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="maxima")

[Out]

256/315*b^3*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8
*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) +
e^(-18*d*x - 18*c) + 1)) + 36*e^(-4*d*x - 4*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6
*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16
*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 84*e^(-6*d*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 8
4*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 1
4*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 126*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-
4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 3
6*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 1/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d
*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e
^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1))) + 96/35*a*b^2*(7*e^(-2*d*x - 2*c)/(d*(7*e
^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*
e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 21*e^(-4*d*x - 4*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) + 1)) + 35*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8
*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)) + 1/(d*(7*e^(-2*d*x - 2*
c) + 21*e^(-4*d*x - 4*c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x -
12*c) + e^(-14*d*x - 14*c) + 1))) + 16/5*a^2*b*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c
) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x
- 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) + 1/(d*(5*e
^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1))) +
4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)) + 1/(d*(3*e^
(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1)))

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mupad [B]  time = 1.47, size = 1333, normalized size = 12.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((16*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(315*d) + (4*a^3*exp(6*c + 6*d*x))/(9*d) + (4*a*exp(2*c + 2*d*x
)*(16*a*b + 5*a^2 + 16*b^2))/(21*d) + (8*a^2*exp(4*c + 4*d*x)*(a + 2*b))/(7*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) - ((32*exp(8*c + 8*d*x)*(24*a
*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(9*d) + (8*a^3*exp(2*c + 2*d*x))/(9*d) + (8*a^3*exp(14*c + 14*d*x))/(9*d) +
(8*a*exp(6*c + 6*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(3*d) + (8*a*exp(10*c + 10*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(
3*d) + (16*a^2*exp(4*c + 4*d*x)*(a + 2*b))/(3*d) + (16*a^2*exp(12*c + 12*d*x)*(a + 2*b))/(3*d))/(9*exp(2*c + 2
*d*x) + 36*exp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) + 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) + 84*exp(12*
c + 12*d*x) + 36*exp(14*c + 14*d*x) + 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) + 1) - ((4*a^2*(a + 2*b))/(21*
d) + (2*a^3*exp(2*c + 2*d*x))/(9*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - ((a*(1
6*a*b + 5*a^2 + 16*b^2))/(21*d) + (16*exp(2*c + 2*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(63*d) + (5*a^3
*exp(8*c + 8*d*x))/(9*d) + (10*a*exp(4*c + 4*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(21*d) + (40*a^2*exp(6*c + 6*d*x)
*(a + 2*b))/(21*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) - (a^3/(9*d) + (16*exp(6*c + 6*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3
+ 16*b^3))/(9*d) + (7*a^3*exp(12*c + 12*d*x))/(9*d) + (a*exp(4*c + 4*d*x)*(16*a*b + 5*a^2 + 16*b^2))/d + (5*a*
exp(8*c + 8*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(3*d) + (4*a^2*exp(2*c + 2*d*x)*(a + 2*b))/(3*d) + (4*a^2*exp(10*c
+ 10*d*x)*(a + 2*b))/d)/(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x)
+ 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1) - ((a*(16*a*
b + 5*a^2 + 16*b^2))/(21*d) + (a^3*exp(4*c + 4*d*x))/(3*d) + (4*a^2*exp(2*c + 2*d*x)*(a + 2*b))/(7*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) - ((4*a^2*(a + 2*b))/(21*d) + (
16*exp(4*c + 4*d*x)*(24*a*b^2 + 18*a^2*b + 5*a^3 + 16*b^3))/(21*d) + (2*a^3*exp(10*c + 10*d*x))/(3*d) + (2*a*e
xp(2*c + 2*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(7*d) + (20*a*exp(6*c + 6*d*x)*(16*a*b + 5*a^2 + 16*b^2))/(21*d) +
(20*a^2*exp(8*c + 8*d*x)*(a + 2*b))/(7*d))/(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 3
5*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - a^3/(9*d*(2*exp(
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 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{4}{\left (c + d x \right )}\, dx \]

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

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