\(\int \text {sech}^4(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\) [104]

Optimal result

Integrand size = 23, antiderivative size = 102 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}-\frac {a^2 (a-3 b) \tanh ^3(c+d x)}{3 d}-\frac {3 a (a-b) b \tanh ^5(c+d x)}{5 d}-\frac {(3 a-b) b^2 \tanh ^7(c+d x)}{7 d}-\frac {b^3 \tanh ^9(c+d x)}{9 d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(218\) vs. \(2(102)=204\).

Time = 2.06 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.14 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {\left (5775 a^3-1071 a^2 b+621 a b^2-725 b^3+10 \left (903 a^3-63 a^2 b-27 a b^2+107 b^3\right ) \cosh (2 (c+d x))+8 \left (525 a^3+126 a^2 b-81 a b^2-50 b^3\right ) \cosh (4 (c+d x))+1050 a^3 \cosh (6 (c+d x))+630 a^2 b \cosh (6 (c+d x))+270 a b^2 \cosh (6 (c+d x))+50 b^3 \cosh (6 (c+d x))+105 a^3 \cosh (8 (c+d x))+63 a^2 b \cosh (8 (c+d x))+27 a b^2 \cosh (8 (c+d x))+5 b^3 \cosh (8 (c+d x))\right ) \text {sech}^8(c+d x) \tanh (c+d x)}{20160 d} \] Input:

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

Output:

((5775*a^3 - 1071*a^2*b + 621*a*b^2 - 725*b^3 + 10*(903*a^3 - 63*a^2*b - 2 
7*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])/(20160*d)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4158, 290, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \sec (i c+i d x)^4 \left (a-b \tan (i c+i d x)^2\right )^3dx\)

\(\Big \downarrow \) 4158

\(\displaystyle \frac {\int \left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )^3d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 290

\(\displaystyle \frac {\int \left (-b^3 \tanh ^8(c+d x)-(3 a-b) b^2 \tanh ^6(c+d x)-3 a (a-b) b \tanh ^4(c+d x)-a^2 (a-3 b) \tanh ^2(c+d x)+a^3\right )d\tanh (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 \tanh (c+d x)-\frac {1}{3} a^2 (a-3 b) \tanh ^3(c+d x)-\frac {1}{7} b^2 (3 a-b) \tanh ^7(c+d x)-\frac {3}{5} a b (a-b) \tanh ^5(c+d x)-\frac {1}{9} b^3 \tanh ^9(c+d x)}{d}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 290

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> I 
nt[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d 
}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4158

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]}, Sim 
p[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] && I 
ntegerQ[m/2] && (IntegersQ[n, p] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] 
 || EqQ[n^2, 16])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(268\) vs. \(2(94)=188\).

Time = 152.41 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.64

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{3}}{4 \cosh \left (d x +c \right )^{7}}-\frac {\sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{7}}+\frac {\left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{4 \cosh \left (d x +c \right )^{9}}-\frac {5 \sinh \left (d x +c \right )^{3}}{24 \cosh \left (d x +c \right )^{9}}-\frac {5 \sinh \left (d x +c \right )}{64 \cosh \left (d x +c \right )^{9}}+\frac {5 \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{64}\right )}{d}\)	\(269\)
default	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{3}}{4 \cosh \left (d x +c \right )^{7}}-\frac {\sinh \left (d x +c \right )}{8 \cosh \left (d x +c \right )^{7}}+\frac {\left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{4 \cosh \left (d x +c \right )^{9}}-\frac {5 \sinh \left (d x +c \right )^{3}}{24 \cosh \left (d x +c \right )^{9}}-\frac {5 \sinh \left (d x +c \right )}{64 \cosh \left (d x +c \right )^{9}}+\frac {5 \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{64}\right )}{d}\)	\(269\)
risch	\(-\frac {4 \left (-945 b^{3} {\mathrm e}^{8 d x +8 c}+945 b^{3} {\mathrm e}^{6 d x +6 c}-135 b^{3} {\mathrm e}^{4 d x +4 c}+945 \,{\mathrm e}^{2 d x +2 c} a^{3}+27 b^{2} a +105 a^{3}+63 a^{2} b +3465 a^{2} b \,{\mathrm e}^{12 d x +12 c}+4725 a^{2} b \,{\mathrm e}^{10 d x +10 c}+945 a \,b^{2} {\mathrm e}^{10 d x +10 c}+3213 a^{2} b \,{\mathrm e}^{8 d x +8 c}+2457 a \,b^{2} {\mathrm e}^{8 d x +8 c}+1827 a^{2} b \,{\mathrm e}^{6 d x +6 c}+1323 a \,b^{2} {\mathrm e}^{6 d x +6 c}+1323 a^{2} b \,{\mathrm e}^{4 d x +4 c}+27 a \,b^{2} {\mathrm e}^{4 d x +4 c}+567 a^{2} b \,{\mathrm e}^{2 d x +2 c}+243 a \,b^{2} {\mathrm e}^{2 d x +2 c}+5 b^{3}+6825 a^{3} {\mathrm e}^{6 d x +6 c}+1575 b^{3} {\mathrm e}^{10 d x +10 c}+7875 a^{3} {\mathrm e}^{8 d x +8 c}-525 b^{3} {\mathrm e}^{12 d x +12 c}+45 b^{3} {\mathrm e}^{2 d x +2 c}+315 a^{3} {\mathrm e}^{14 d x +14 c}+945 a \,b^{2} {\mathrm e}^{12 d x +12 c}+945 a^{2} b \,{\mathrm e}^{14 d x +14 c}+945 a \,b^{2} {\mathrm e}^{14 d x +14 c}+315 b^{3} {\mathrm e}^{14 d x +14 c}+1995 a^{3} {\mathrm e}^{12 d x +12 c}+3465 \,{\mathrm e}^{4 d x +4 c} a^{3}+5355 \,{\mathrm e}^{10 d x +10 c} a^{3}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\)	\(448\)

Input:

int(sech(d*x+c)^4*(a+tanh(d*x+c)^2*b)^3,x,method=_RETURNVERBOSE)
 

Output:

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)/cos 
h(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* 
b^2*a*(-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*sech(d 
*x+c)^6+16/105*sech(d*x+c)^4+64/315*sech(d*x+c)^2)*tanh(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (94) = 188\).

Time = 0.09 (sec) , antiderivative size = 1185, normalized size of antiderivative = 11.62 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-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 + 80*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*(245*a^3 + 168*a^2*b + 27*a*b^2 + 40*b^3)*cos 
h(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 + 336*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))*sinh(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)^...
 

Sympy [F]

\[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \operatorname {sech}^{4}{\left (c + d x \right )}\, dx \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1847 vs. \(2 (94) = 188\).

Time = 0.07 (sec) , antiderivative size = 1847, normalized size of antiderivative = 18.11 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

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^(-1 
8*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*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (94) = 188\).

Time = 0.23 (sec) , antiderivative size = 447, normalized size of antiderivative = 4.38 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {4 \, {\left (315 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 945 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 945 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 315 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 1995 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 3465 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 525 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 5355 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 4725 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 945 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 7875 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 3213 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 2457 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 945 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6825 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1827 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 1323 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 945 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3465 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1323 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 27 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 135 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 945 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 567 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 243 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 63 \, a^{2} b + 27 \, a b^{2} + 5 \, b^{3}\right )}}{315 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \] Input:

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

Output:

-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)
 

Mupad [B] (verification not implemented)

Time = 2.44 (sec) , antiderivative size = 1424, normalized size of antiderivative = 13.96 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

- ((4*(a + b)^2*(a - b))/(21*d) + (2*exp(2*c + 2*d*x)*(a + b)^3)/(9*d))/(3 
*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1) - ((5*exp(8 
*c + 8*d*x)*(a + b)^3)/(9*d) - (a*b^2 + a^2*b - 5*a^3 - 5*b^3)/(21*d) - (1 
0*exp(4*c + 4*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(21*d) + (16*exp(2*c + 
 2*d*x)*(3*a*b^2 - 3*a^2*b + 5*a^3 - 5*b^3))/(63*d) + (40*exp(6*c + 6*d*x) 
*(a + b)^2*(a - 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) - ((4*(a + b)^2*(a - b))/(21*d) + (2*exp(10*c + 10*d*x)*(a 
+ b)^3)/(3*d) - (2*exp(2*c + 2*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(7*d) 
 - (20*exp(6*c + 6*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(21*d) + (16*exp( 
4*c + 4*d*x)*(3*a*b^2 - 3*a^2*b + 5*a^3 - 5*b^3))/(21*d) + (20*exp(8*c + 8 
*d*x)*(a + b)^2*(a - b))/(7*d))/(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) 
+ 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x) + 7*ex 
p(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - ((16*(3*a*b^2 - 3*a^2*b + 5*a 
^3 - 5*b^3))/(315*d) + (4*exp(6*c + 6*d*x)*(a + b)^3)/(9*d) - (4*exp(2*c + 
 2*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(21*d) + (8*exp(4*c + 4*d*x)*(a + 
 b)^2*(a - 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) - ((8*exp(2*c + 
 2*d*x)*(a + b)^3)/(9*d) + (8*exp(14*c + 14*d*x)*(a + b)^3)/(9*d) - (8*exp 
(6*c + 6*d*x)*(a*b^2 + a^2*b - 5*a^3 - 5*b^3))/(3*d) - (8*exp(10*c + 10...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 572, normalized size of antiderivative = 5.61 \[ \int \text {sech}^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-\frac {12 e^{4 d x +4 c} a \,b^{2}}{35}-\frac {4 b^{3}}{63}-12 e^{14 d x +14 c} a^{2} b -12 e^{14 d x +14 c} a \,b^{2}-60 e^{10 d x +10 c} a^{2} b -12 e^{10 d x +10 c} a \,b^{2}-\frac {204 e^{8 d x +8 c} a^{2} b}{5}-\frac {156 e^{8 d x +8 c} a \,b^{2}}{5}-\frac {116 e^{6 d x +6 c} a^{2} b}{5}-\frac {84 e^{6 d x +6 c} a \,b^{2}}{5}-\frac {84 e^{4 d x +4 c} a^{2} b}{5}-\frac {12 a \,b^{2}}{35}+\frac {12 e^{4 d x +4 c} b^{3}}{7}-\frac {108 e^{2 d x +2 c} a \,b^{2}}{35}-\frac {4 a^{3}}{3}-\frac {4 e^{2 d x +2 c} b^{3}}{7}-44 e^{12 d x +12 c} a^{2} b -\frac {76 e^{12 d x +12 c} a^{3}}{3}+\frac {20 e^{12 d x +12 c} b^{3}}{3}-\frac {36 e^{2 d x +2 c} a^{2} b}{5}-12 e^{12 d x +12 c} a \,b^{2}-\frac {4 a^{2} b}{5}-4 e^{14 d x +14 c} a^{3}-4 e^{14 d x +14 c} b^{3}-68 e^{10 d x +10 c} a^{3}-20 e^{10 d x +10 c} b^{3}-100 e^{8 d x +8 c} a^{3}+12 e^{8 d x +8 c} b^{3}-\frac {260 e^{6 d x +6 c} a^{3}}{3}-12 e^{6 d x +6 c} b^{3}-44 e^{4 d x +4 c} a^{3}-12 e^{2 d x +2 c} a^{3}}{d \left (e^{18 d x +18 c}+9 e^{16 d x +16 c}+36 e^{14 d x +14 c}+84 e^{12 d x +12 c}+126 e^{10 d x +10 c}+126 e^{8 d x +8 c}+84 e^{6 d x +6 c}+36 e^{4 d x +4 c}+9 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(4*( - 315*e**(14*c + 14*d*x)*a**3 - 945*e**(14*c + 14*d*x)*a**2*b - 945*e 
**(14*c + 14*d*x)*a*b**2 - 315*e**(14*c + 14*d*x)*b**3 - 1995*e**(12*c + 1 
2*d*x)*a**3 - 3465*e**(12*c + 12*d*x)*a**2*b - 945*e**(12*c + 12*d*x)*a*b* 
*2 + 525*e**(12*c + 12*d*x)*b**3 - 5355*e**(10*c + 10*d*x)*a**3 - 4725*e** 
(10*c + 10*d*x)*a**2*b - 945*e**(10*c + 10*d*x)*a*b**2 - 1575*e**(10*c + 1 
0*d*x)*b**3 - 7875*e**(8*c + 8*d*x)*a**3 - 3213*e**(8*c + 8*d*x)*a**2*b - 
2457*e**(8*c + 8*d*x)*a*b**2 + 945*e**(8*c + 8*d*x)*b**3 - 6825*e**(6*c + 
6*d*x)*a**3 - 1827*e**(6*c + 6*d*x)*a**2*b - 1323*e**(6*c + 6*d*x)*a*b**2 
- 945*e**(6*c + 6*d*x)*b**3 - 3465*e**(4*c + 4*d*x)*a**3 - 1323*e**(4*c + 
4*d*x)*a**2*b - 27*e**(4*c + 4*d*x)*a*b**2 + 135*e**(4*c + 4*d*x)*b**3 - 9 
45*e**(2*c + 2*d*x)*a**3 - 567*e**(2*c + 2*d*x)*a**2*b - 243*e**(2*c + 2*d 
*x)*a*b**2 - 45*e**(2*c + 2*d*x)*b**3 - 105*a**3 - 63*a**2*b - 27*a*b**2 - 
 5*b**3))/(315*d*(e**(18*c + 18*d*x) + 9*e**(16*c + 16*d*x) + 36*e**(14*c 
+ 14*d*x) + 84*e**(12*c + 12*d*x) + 126*e**(10*c + 10*d*x) + 126*e**(8*c + 
 8*d*x) + 84*e**(6*c + 6*d*x) + 36*e**(4*c + 4*d*x) + 9*e**(2*c + 2*d*x) + 
 1))