### 3.104 $$\int \text{sech}^4(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx$$

Optimal. Leaf size=102 $-\frac{a^2 (a-3 b) \tanh ^3(c+d x)}{3 d}+\frac{a^3 \tanh (c+d x)}{d}-\frac{b^2 (3 a-b) \tanh ^7(c+d x)}{7 d}-\frac{3 a b (a-b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}$

[Out]

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

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Rubi [A]  time = 0.0889386, antiderivative size = 102, normalized size of antiderivative = 1., 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 = {3675, 373} $-\frac{a^2 (a-3 b) \tanh ^3(c+d x)}{3 d}+\frac{a^3 \tanh (c+d x)}{d}-\frac{b^2 (3 a-b) \tanh ^7(c+d x)}{7 d}-\frac{3 a b (a-b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

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]

Rubi steps

\begin{align*} \int \text{sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a+b x^2\right )^3 \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^3-a^2 (a-3 b) x^2-3 a (a-b) b x^4-(3 a-b) b^2 x^6-b^3 x^8\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{a^3 \tanh (c+d x)}{d}-\frac{a^2 (a-3 b) \tanh ^3(c+d x)}{3 d}-\frac{3 a (a-b) b \tanh ^5(c+d x)}{5 d}-\frac{(3 a-b) b^2 \tanh ^7(c+d x)}{7 d}-\frac{b^3 \tanh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [B]  time = 0.825721, size = 218, normalized size = 2.14 $\frac{\tanh (c+d x) \text{sech}^8(c+d x) \left (10 \left (-63 a^2 b+903 a^3-27 a b^2+107 b^3\right ) \cosh (2 (c+d x))+8 \left (126 a^2 b+525 a^3-81 a b^2-50 b^3\right ) \cosh (4 (c+d x))+630 a^2 b \cosh (6 (c+d x))+63 a^2 b \cosh (8 (c+d x))-1071 a^2 b+1050 a^3 \cosh (6 (c+d x))+105 a^3 \cosh (8 (c+d x))+5775 a^3+270 a b^2 \cosh (6 (c+d x))+27 a b^2 \cosh (8 (c+d x))+621 a b^2+50 b^3 \cosh (6 (c+d x))+5 b^3 \cosh (8 (c+d x))-725 b^3\right )}{20160 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((5775*a^3 - 1071*a^2*b + 621*a*b^2 - 725*b^3 + 10*(903*a^3 - 63*a^2*b - 27*a*b^2 + 107*b^3)*Cosh[2*(c + d*x)]
+ 8*(525*a^3 + 126*a^2*b - 81*a*b^2 - 50*b^3)*Cosh[4*(c + d*x)] + 1050*a^3*Cosh[6*(c + d*x)] + 630*a^2*b*Cosh
[6*(c + d*x)] + 270*a*b^2*Cosh[6*(c + d*x)] + 50*b^3*Cosh[6*(c + d*x)] + 105*a^3*Cosh[8*(c + d*x)] + 63*a^2*b*
Cosh[8*(c + d*x)] + 27*a*b^2*Cosh[8*(c + d*x)] + 5*b^3*Cosh[8*(c + d*x)])*Sech[c + d*x]^8*Tanh[c + d*x])/(2016
0*d)

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Maple [B]  time = 0.124, size = 269, normalized size = 2.6 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{2}{3}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{3}} \right ) \tanh \left ( dx+c \right ) +3\,{a}^{2}b \left ( -1/4\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{5}}}+1/4\, \left ({\frac{8}{15}}+1/5\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}+{\frac{4\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{15}} \right ) \tanh \left ( dx+c \right ) \right ) +3\,a{b}^{2} \left ( -1/4\,{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}-1/8\,{\frac{\sinh \left ( dx+c \right ) }{ \left ( \cosh \left ( dx+c \right ) \right ) ^{7}}}+1/8\, \left ({\frac{16}{35}}+1/7\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}+{\frac{6\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{35}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{35}} \right ) \tanh \left ( dx+c \right ) \right ) +{b}^{3} \left ( -{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{4\, \left ( \cosh \left ( dx+c \right ) \right ) ^{9}}}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24\, \left ( \cosh \left ( dx+c \right ) \right ) ^{9}}}-{\frac{5\,\sinh \left ( dx+c \right ) }{64\, \left ( \cosh \left ( dx+c \right ) \right ) ^{9}}}+{\frac{5\,\tanh \left ( dx+c \right ) }{64} \left ({\frac{128}{315}}+{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{8}}{9}}+{\frac{8\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{6}}{63}}+{\frac{16\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{315}} \right ) } \right ) \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(d*x+c)^4*(a+b*tanh(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*(-1/4*sinh(d*x+c)/cosh(d*x+c)^5+1/4*(8/15+1/5*sech(d*x+c)
^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+3*a*b^2*(-1/4*sinh(d*x+c)^3/cosh(d*x+c)^7-1/8*sinh(d*x+c)/cosh(d*x+c)^7+1/
8*(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*(-1/4*sinh(d*x+c)^5/cosh(d*
x+c)^9-5/24*sinh(d*x+c)^3/cosh(d*x+c)^9-5/64*sinh(d*x+c)/cosh(d*x+c)^9+5/64*(128/315+1/9*sech(d*x+c)^8+8/63*se
ch(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 = 1.17249, size = 2493, normalized size = 24.44 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/63*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)) - 27*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)) + 189*e^(-6*d*x - 6*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)) - 189*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) + 36*
e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 315*e^(-10*d*x - 10*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)) - 105*e^(-12*d*x - 12*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)) + 63*
e^(-14*d*x - 14*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)) + 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) + 12
6*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))) + 12/35*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)) - 14*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 - 14*c) + 1)) + 70*e^(-6*d*x - 6*c)/(d*(7*e^(-2*d*x - 2*c) + 2
1*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)) - 35*e^(-8*d*x - 8*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)) + 35*e^(
-10*d*x - 10*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))) + 4/5*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)) - 5*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)) + 15*e^(-6*d*x - 6*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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Fricas [B]  time = 2.01396, size = 3218, normalized size = 31.55 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/315*(2*(105*a^3 + 252*a^2*b + 243*a*b^2 + 80*b^3)*cosh(d*x + c)^7 + 14*(105*a^3 + 252*a^2*b + 243*a*b^2 + 8
0*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (105*a^3 + 441*a^2*b + 459*a*b^2 + 155*b^3)*sinh(d*x + c)^7 + 6*(245*a^
3 + 336*a^2*b + 99*a*b^2 - 40*b^3)*cosh(d*x + c)^5 + 3*(175*a^3 + 483*a^2*b + 117*a*b^2 - 95*b^3 + 7*(105*a^3
+ 441*a^2*b + 459*a*b^2 + 155*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 10*(7*(105*a^3 + 252*a^2*b + 243*a*b^2 +
80*b^3)*cosh(d*x + c)^3 + 3*(245*a^3 + 336*a^2*b + 99*a*b^2 - 40*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 18*(24
5*a^3 + 168*a^2*b + 27*a*b^2 + 40*b^3)*cosh(d*x + c)^3 + (35*(105*a^3 + 441*a^2*b + 459*a*b^2 + 155*b^3)*cosh(
d*x + c)^4 + 945*a^3 + 1701*a^2*b + 459*a*b^2 + 855*b^3 + 30*(175*a^3 + 483*a^2*b + 117*a*b^2 - 95*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^3 + 6*(7*(105*a^3 + 252*a^2*b + 243*a*b^2 + 80*b^3)*cosh(d*x + c)^5 + 10*(245*a^3 + 3
36*a^2*b + 99*a*b^2 - 40*b^3)*cosh(d*x + c)^3 + 9*(245*a^3 + 168*a^2*b + 27*a*b^2 + 40*b^3)*cosh(d*x + c))*sin
h(d*x + c)^2 + 210*(35*a^3 + 12*a^2*b + 9*a*b^2)*cosh(d*x + c) + (7*(105*a^3 + 441*a^2*b + 459*a*b^2 + 155*b^3
)*cosh(d*x + c)^6 + 15*(175*a^3 + 483*a^2*b + 117*a*b^2 - 95*b^3)*cosh(d*x + c)^4 + 525*a^3 + 693*a^2*b + 567*
a*b^2 - 945*b^3 + 27*(105*a^3 + 189*a^2*b + 51*a*b^2 + 95*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*cos
h(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*c
osh(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 + (55*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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname{sech}^{4}{\left (c + d x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.82701, size = 603, normalized size = 5.91 \begin{align*} -\frac{4 \,{\left (315 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 945 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 945 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 315 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 1995 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 3465 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 525 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 5355 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 4725 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 945 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 7875 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3213 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 2457 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 945 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6825 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1827 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1323 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3465 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1323 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 135 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 945 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 567 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 243 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 63 \, a^{2} b + 27 \, a b^{2} + 5 \, b^{3}\right )}}{315 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/315*(315*a^3*e^(14*d*x + 14*c) + 945*a^2*b*e^(14*d*x + 14*c) + 945*a*b^2*e^(14*d*x + 14*c) + 315*b^3*e^(14*
d*x + 14*c) + 1995*a^3*e^(12*d*x + 12*c) + 3465*a^2*b*e^(12*d*x + 12*c) + 945*a*b^2*e^(12*d*x + 12*c) - 525*b^
3*e^(12*d*x + 12*c) + 5355*a^3*e^(10*d*x + 10*c) + 4725*a^2*b*e^(10*d*x + 10*c) + 945*a*b^2*e^(10*d*x + 10*c)
+ 1575*b^3*e^(10*d*x + 10*c) + 7875*a^3*e^(8*d*x + 8*c) + 3213*a^2*b*e^(8*d*x + 8*c) + 2457*a*b^2*e^(8*d*x + 8
*c) - 945*b^3*e^(8*d*x + 8*c) + 6825*a^3*e^(6*d*x + 6*c) + 1827*a^2*b*e^(6*d*x + 6*c) + 1323*a*b^2*e^(6*d*x +
6*c) + 945*b^3*e^(6*d*x + 6*c) + 3465*a^3*e^(4*d*x + 4*c) + 1323*a^2*b*e^(4*d*x + 4*c) + 27*a*b^2*e^(4*d*x + 4
*c) - 135*b^3*e^(4*d*x + 4*c) + 945*a^3*e^(2*d*x + 2*c) + 567*a^2*b*e^(2*d*x + 2*c) + 243*a*b^2*e^(2*d*x + 2*c
) + 45*b^3*e^(2*d*x + 2*c) + 105*a^3 + 63*a^2*b + 27*a*b^2 + 5*b^3)/(d*(e^(2*d*x + 2*c) + 1)^9)