\(\int \text {sech}^4(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [72]

Optimal result

Integrand size = 23, antiderivative size = 108 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {(a+b)^3 \tanh (c+d x)}{d}-\frac {(a+b)^2 (a+4 b) \tanh ^3(c+d x)}{3 d}+\frac {3 b (a+b) (a+2 b) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+4 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \] Output:

(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
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(238\) vs. \(2(108)=216\).

Time = 5.11 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.20 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {a^3 \tanh (c+d x)}{d}+\frac {3 a^2 b \tanh (c+d x)}{d}+\frac {3 a b^2 \tanh (c+d x)}{d}+\frac {b^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}-\frac {2 a^2 b \tanh ^3(c+d x)}{d}-\frac {3 a b^2 \tanh ^3(c+d x)}{d}-\frac {4 b^3 \tanh ^3(c+d x)}{3 d}+\frac {3 a^2 b \tanh ^5(c+d x)}{5 d}+\frac {9 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {6 b^3 \tanh ^5(c+d x)}{5 d}-\frac {3 a b^2 \tanh ^7(c+d x)}{7 d}-\frac {4 b^3 \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \] Input:

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

Output:

(a^3*Tanh[c + d*x])/d + (3*a^2*b*Tanh[c + d*x])/d + (3*a*b^2*Tanh[c + d*x] 
)/d + (b^3*Tanh[c + d*x])/d - (a^3*Tanh[c + d*x]^3)/(3*d) - (2*a^2*b*Tanh[ 
c + d*x]^3)/d - (3*a*b^2*Tanh[c + d*x]^3)/d - (4*b^3*Tanh[c + d*x]^3)/(3*d 
) + (3*a^2*b*Tanh[c + d*x]^5)/(5*d) + (9*a*b^2*Tanh[c + d*x]^5)/(5*d) + (6 
*b^3*Tanh[c + d*x]^5)/(5*d) - (3*a*b^2*Tanh[c + d*x]^7)/(7*d) - (4*b^3*Tan 
h[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4634, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4634

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

\(\Big \downarrow \) 290

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {1}{7} b^2 (3 a+4 b) \tanh ^7(c+d x)+\frac {3}{5} b (a+b) (a+2 b) \tanh ^5(c+d x)-\frac {1}{3} (a+b)^2 (a+4 b) \tanh ^3(c+d x)+(a+b)^3 \tanh (c+d x)+\frac {1}{9} b^3 \tanh ^9(c+d x)}{d}\)

Input:

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

Output:

((a + b)^3*Tanh[c + d*x] - ((a + b)^2*(a + 4*b)*Tanh[c + d*x]^3)/3 + (3*b* 
(a + b)*(a + 2*b)*Tanh[c + d*x]^5)/5 - (b^2*(3*a + 4*b)*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 4634

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) 
)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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]
 

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}\)	\(158\)
default	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+3 a^{2} b \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )+3 a \,b^{2} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )+b^{3} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}\)	\(158\)
parts	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{d}+\frac {b^{3} \left (\frac {128}{315}+\frac {\operatorname {sech}\left (d x +c \right )^{8}}{9}+\frac {8 \operatorname {sech}\left (d x +c \right )^{6}}{63}+\frac {16 \operatorname {sech}\left (d x +c \right )^{4}}{105}+\frac {64 \operatorname {sech}\left (d x +c \right )^{2}}{315}\right ) \tanh \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {16}{35}+\frac {\operatorname {sech}\left (d x +c \right )^{6}}{7}+\frac {6 \operatorname {sech}\left (d x +c \right )^{4}}{35}+\frac {8 \operatorname {sech}\left (d x +c \right )^{2}}{35}\right ) \tanh \left (d x +c \right )}{d}+\frac {3 a^{2} b \left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{d}\)	\(166\)
parallelrisch	\(\frac {\left (9660 a^{3}+32256 a^{2} b +36288 a \,b^{2}+10752 b^{3}\right ) \sinh \left (3 d x +3 c \right )+\left (6300 a^{3}+18144 a^{2} b +15552 a \,b^{2}+4608 b^{3}\right ) \sinh \left (5 d x +5 c \right )+\left (1890 a^{3}+4536 a^{2} b +3888 a \,b^{2}+1152 b^{3}\right ) \sinh \left (7 d x +7 c \right )+\left (210 a^{3}+504 a^{2} b +432 a \,b^{2}+128 b^{3}\right ) \sinh \left (9 d x +9 c \right )+5040 \sinh \left (d x +c \right ) \left (a^{2}+\frac {8}{5} a b +\frac {8}{5} b^{2}\right ) \left (a +2 b \right )}{315 d \left (\cosh \left (9 d x +9 c \right )+9 \cosh \left (7 d x +7 c \right )+36 \cosh \left (5 d x +5 c \right )+84 \cosh \left (3 d x +3 c \right )+126 \cosh \left (d x +c \right )\right )}\)	\(218\)
risch	\(-\frac {4 \left (252 a^{2} b +216 a \,b^{2}+315 \,{\mathrm e}^{14 d x +14 c} a^{3}+105 a^{3}+576 b^{3} {\mathrm e}^{2 d x +2 c}+945 a^{3} {\mathrm e}^{2 d x +2 c}+2304 b^{3} {\mathrm e}^{4 d x +4 c}+5376 b^{3} {\mathrm e}^{6 d x +6 c}+3465 a^{3} {\mathrm e}^{4 d x +4 c}+6825 a^{3} {\mathrm e}^{6 d x +6 c}+64 b^{3}+18648 a^{2} b \,{\mathrm e}^{6 d x +6 c}+18144 a \,b^{2} {\mathrm e}^{6 d x +6 c}+9072 a^{2} b \,{\mathrm e}^{4 d x +4 c}+7776 a \,b^{2} {\mathrm e}^{4 d x +4 c}+19656 a \,b^{2} {\mathrm e}^{8 d x +8 c}+5355 a^{3} {\mathrm e}^{10 d x +10 c}+1995 a^{3} {\mathrm e}^{12 d x +12 c}+7875 a^{3} {\mathrm e}^{8 d x +8 c}+11340 a^{2} b \,{\mathrm e}^{10 d x +10 c}+7560 a \,b^{2} {\mathrm e}^{10 d x +10 c}+8064 b^{3} {\mathrm e}^{8 d x +8 c}+20412 a^{2} b \,{\mathrm e}^{8 d x +8 c}+2520 a^{2} b \,{\mathrm e}^{12 d x +12 c}+1944 a \,b^{2} {\mathrm e}^{2 d x +2 c}+2268 a^{2} b \,{\mathrm e}^{2 d x +2 c}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\)	\(361\)

Input:

int(sech(d*x+c)^4*(a+b*sech(d*x+c)^2)^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*(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*s 
ech(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 
))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (100) = 200\).

Time = 0.23 (sec) , antiderivative size = 1190, normalized size of antiderivative = 11.02 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-8/315*(2*(105*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)^7 + 14*(1 
05*a^3 + 63*a^2*b + 54*a*b^2 + 16*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (10 
5*a^3 - 126*a^2*b - 108*a*b^2 - 32*b^3)*sinh(d*x + c)^7 + 6*(245*a^3 + 399 
*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)*cos 
h(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 + ...
 

Sympy [F]

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

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

Output:

Integral((a + b*sech(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. 1245 vs. \(2 (100) = 200\).

Time = 0.05 (sec) , antiderivative size = 1245, normalized size of antiderivative = 11.53 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

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) + 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)) + 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) + 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) + 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) ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (100) = 200\).

Time = 0.14 (sec) , antiderivative size = 360, normalized size of antiderivative = 3.33 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {4 \, {\left (315 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 1995 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 2520 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 5355 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 7560 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 7875 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 20412 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 19656 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8064 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6825 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 18648 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 18144 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5376 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3465 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 9072 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 7776 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2304 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 945 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2268 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 1944 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 576 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 252 \, a^{2} b + 216 \, a b^{2} + 64 \, b^{3}\right )}}{315 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}} \] Input:

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

Output:

-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) + 18648*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) + 9 
072*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)
 

Mupad [B] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 1333, normalized size of antiderivative = 12.34 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

- ((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*e 
xp(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) + ex 
p(6*c + 6*d*x) + 1) - ((a*(16*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 + 1 
0*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))/(...
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 479, normalized size of antiderivative = 4.44 \[ \int \text {sech}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {-\frac {3456 e^{4 d x +4 c} a \,b^{2}}{35}-\frac {256 b^{3}}{315}-144 e^{10 d x +10 c} a^{2} b -96 e^{10 d x +10 c} a \,b^{2}-\frac {1296 e^{8 d x +8 c} a^{2} b}{5}-\frac {1248 e^{8 d x +8 c} a \,b^{2}}{5}-\frac {1184 e^{6 d x +6 c} a^{2} b}{5}-\frac {1152 e^{6 d x +6 c} a \,b^{2}}{5}-\frac {576 e^{4 d x +4 c} a^{2} b}{5}-\frac {96 a \,b^{2}}{35}-\frac {1024 e^{4 d x +4 c} b^{3}}{35}-4 e^{14 d x +14 c} a^{3}-68 e^{10 d x +10 c} a^{3}-\frac {864 e^{2 d x +2 c} a \,b^{2}}{35}-\frac {4 a^{3}}{3}-\frac {256 e^{2 d x +2 c} b^{3}}{35}-32 e^{12 d x +12 c} a^{2} b -\frac {76 e^{12 d x +12 c} a^{3}}{3}-\frac {144 e^{2 d x +2 c} a^{2} b}{5}-\frac {16 a^{2} b}{5}-100 e^{8 d x +8 c} a^{3}-\frac {512 e^{8 d x +8 c} b^{3}}{5}-\frac {260 e^{6 d x +6 c} a^{3}}{3}-\frac {1024 e^{6 d x +6 c} b^{3}}{15}-44 e^{4 d x +4 c} a^{3}-12 e^{2 d x +2 c} a^{3}}{d \left (e^{18 d x +18 c}+9 e^{16 d x +16 c}+36 e^{14 d x +14 c}+84 e^{12 d x +12 c}+126 e^{10 d x +10 c}+126 e^{8 d x +8 c}+84 e^{6 d x +6 c}+36 e^{4 d x +4 c}+9 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(4*( - 315*e**(14*c + 14*d*x)*a**3 - 1995*e**(12*c + 12*d*x)*a**3 - 2520*e 
**(12*c + 12*d*x)*a**2*b - 5355*e**(10*c + 10*d*x)*a**3 - 11340*e**(10*c + 
 10*d*x)*a**2*b - 7560*e**(10*c + 10*d*x)*a*b**2 - 7875*e**(8*c + 8*d*x)*a 
**3 - 20412*e**(8*c + 8*d*x)*a**2*b - 19656*e**(8*c + 8*d*x)*a*b**2 - 8064 
*e**(8*c + 8*d*x)*b**3 - 6825*e**(6*c + 6*d*x)*a**3 - 18648*e**(6*c + 6*d* 
x)*a**2*b - 18144*e**(6*c + 6*d*x)*a*b**2 - 5376*e**(6*c + 6*d*x)*b**3 - 3 
465*e**(4*c + 4*d*x)*a**3 - 9072*e**(4*c + 4*d*x)*a**2*b - 7776*e**(4*c + 
4*d*x)*a*b**2 - 2304*e**(4*c + 4*d*x)*b**3 - 945*e**(2*c + 2*d*x)*a**3 - 2 
268*e**(2*c + 2*d*x)*a**2*b - 1944*e**(2*c + 2*d*x)*a*b**2 - 576*e**(2*c + 
 2*d*x)*b**3 - 105*a**3 - 252*a**2*b - 216*a*b**2 - 64*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))