\(\int \text {sech}^7(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [278]

Optimal result

Integrand size = 23, antiderivative size = 135 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {(a+b) \left (5 a^2-2 a b+5 b^2\right ) \arctan (\sinh (c+d x))}{16 d}+\frac {(a-b) \left (5 a^2+8 a b+11 b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {(a-b)^2 (5 a+13 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {(a-b)^3 \text {sech}^5(c+d x) \tanh (c+d x)}{6 d} \] Output:

1/16*(a+b)*(5*a^2-2*a*b+5*b^2)*arctan(sinh(d*x+c))/d+1/16*(a-b)*(5*a^2+8*a 
*b+11*b^2)*sech(d*x+c)*tanh(d*x+c)/d+1/24*(a-b)^2*(5*a+13*b)*sech(d*x+c)^3 
*tanh(d*x+c)/d+1/6*(a-b)^3*sech(d*x+c)^5*tanh(d*x+c)/d
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 10.97 (sec) , antiderivative size = 1192, normalized size of antiderivative = 8.83 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

Integrate[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]
 

Output:

(Csch[c + d*x]^5*(-117228825*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] - 1092656 
25*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 274542345*a^2*b*A 
rcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 - 17069535*a^3*ArcTanh[Sqrt 
[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 - 260465625*a^2*b*ArcTanh[Sqrt[-Sinh[c 
 + d*x]^2]]*Sinh[c + d*x]^4 - 215549775*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^ 
2]]*Sinh[c + d*x]^4 + 142065*a^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + 
d*x]^6 - 41427855*a^2*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 
207173295*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 58009455 
*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 - 210735*a^2*b*ArcTan 
h[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^8 - 33756345*a*b^2*ArcTanh[Sqrt[-S 
inh[c + d*x]^2]]*Sinh[c + d*x]^8 - 56109375*b^3*ArcTanh[Sqrt[-Sinh[c + d*x 
]^2]]*Sinh[c + d*x]^8 - 174825*a*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[ 
c + d*x]^10 - 9261945*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^10 
 - 48825*b^3*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^12 + 117228825* 
a^3*Sqrt[-Sinh[c + d*x]^2] + 4093425*a^3*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d* 
x]^2] + 168951510*a^2*b*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 215549775 
*a*b^2*Sinh[c + d*x]^4*Sqrt[-Sinh[c + d*x]^2] + 9514449*a^2*b*Sinh[c + d*x 
]^6*Sqrt[-Sinh[c + d*x]^2] + 135323370*a*b^2*Sinh[c + d*x]^6*Sqrt[-Sinh[c 
+ d*x]^2] + 58009455*b^3*Sinh[c + d*x]^6*Sqrt[-Sinh[c + d*x]^2] + 7808535* 
a*b^2*Sinh[c + d*x]^8*Sqrt[-Sinh[c + d*x]^2] + 36772890*b^3*Sinh[c + d*...
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.26, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 3669, 315, 401, 25, 298, 216}

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.

\(\displaystyle \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-b \sin (i c+i d x)^2\right )^3}{\cos (i c+i d x)^7}dx\)

\(\Big \downarrow \) 3669

\(\displaystyle \frac {\int \frac {\left (b \sinh ^2(c+d x)+a\right )^3}{\left (\sinh ^2(c+d x)+1\right )^4}d\sinh (c+d x)}{d}\)

\(\Big \downarrow \) 315

\(\displaystyle \frac {\frac {1}{6} \int \frac {\left (b \sinh ^2(c+d x)+a\right ) \left (b (a+5 b) \sinh ^2(c+d x)+a (5 a+b)\right )}{\left (\sinh ^2(c+d x)+1\right )^3}d\sinh (c+d x)+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\)

\(\Big \downarrow \) 401

\(\displaystyle \frac {\frac {1}{6} \left (\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}-\frac {1}{4} \int -\frac {b \left (5 a^2+4 b a+15 b^2\right ) \sinh ^2(c+d x)+a \left (15 a^2+4 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {b \left (5 a^2+4 b a+15 b^2\right ) \sinh ^2(c+d x)+a \left (15 a^2+4 b a+5 b^2\right )}{\left (\sinh ^2(c+d x)+1\right )^2}d\sinh (c+d x)+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} (a+b) \left (5 a^2-2 a b+5 b^2\right ) \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {3}{2} (a+b) \left (5 a^2-2 a b+5 b^2\right ) \arctan (\sinh (c+d x))+\frac {(a-b) \left (15 a^2+14 a b+15 b^2\right ) \sinh (c+d x)}{2 \left (\sinh ^2(c+d x)+1\right )}\right )+\frac {5 \left (a^2-b^2\right ) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )}{4 \left (\sinh ^2(c+d x)+1\right )^2}\right )+\frac {(a-b) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 \left (\sinh ^2(c+d x)+1\right )^3}}{d}\)

Input:

Int[Sech[c + d*x]^7*(a + b*Sinh[c + d*x]^2)^3,x]
 

Output:

(((a - b)*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(6*(1 + Sinh[c + d*x]^2 
)^3) + ((5*(a^2 - b^2)*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2))/(4*(1 + Sinh 
[c + d*x]^2)^2) + ((3*(a + b)*(5*a^2 - 2*a*b + 5*b^2)*ArcTan[Sinh[c + d*x] 
])/2 + ((a - b)*(15*a^2 + 14*a*b + 15*b^2)*Sinh[c + d*x])/(2*(1 + Sinh[c + 
 d*x]^2)))/4)/6)/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 401

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 0]
 

rule 3042

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

rule 3669

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ 
(p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f   S 
ubst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x] 
/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(127)=254\).

Time = 0.27 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.33

\[\frac {a^{3} \left (\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{5 \cosh \left (d x +c \right )^{6}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )}{5}+\frac {\arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{6}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{6}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{6}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{5}}{6}+\frac {5 \operatorname {sech}\left (d x +c \right )^{3}}{24}+\frac {5 \,\operatorname {sech}\left (d x +c \right )}{16}\right ) \tanh \left (d x +c \right )+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{8}\right )}{d}\]

Input:

int(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x)
 

Output:

1/d*(a^3*((1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x 
+c)+5/8*arctan(exp(d*x+c)))+3*a^2*b*(-1/5*sinh(d*x+c)/cosh(d*x+c)^6+1/5*(1 
/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+1/8*arct 
an(exp(d*x+c)))+3*b^2*a*(-1/3*sinh(d*x+c)^3/cosh(d*x+c)^6-1/5*sinh(d*x+c)/ 
cosh(d*x+c)^6+1/5*(1/6*sech(d*x+c)^5+5/24*sech(d*x+c)^3+5/16*sech(d*x+c))* 
tanh(d*x+c)+1/8*arctan(exp(d*x+c)))+b^3*(-sinh(d*x+c)^5/cosh(d*x+c)^6-5/3* 
sinh(d*x+c)^3/cosh(d*x+c)^6-sinh(d*x+c)/cosh(d*x+c)^6+(1/6*sech(d*x+c)^5+5 
/24*sech(d*x+c)^3+5/16*sech(d*x+c))*tanh(d*x+c)+5/8*arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3675 vs. \(2 (127) = 254\).

Time = 0.12 (sec) , antiderivative size = 3675, normalized size of antiderivative = 27.22 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:

integrate(sech(d*x+c)**7*(a+b*sinh(d*x+c)**2)**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 646 vs. \(2 (127) = 254\).

Time = 0.13 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.79 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

-1/24*b^3*(15*arctan(e^(-d*x - c))/d + (33*e^(-d*x - c) - 5*e^(-3*d*x - 3* 
c) + 90*e^(-5*d*x - 5*c) - 90*e^(-7*d*x - 7*c) + 5*e^(-9*d*x - 9*c) - 33*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))) - 1/24*a^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))) - 1/8*a^2*b*(3*arctan(e^(-d*x - c))/d - (3*e^(- 
d*x - c) + 17*e^(-3*d*x - 3*c) - 114*e^(-5*d*x - 5*c) + 114*e^(-7*d*x - 7* 
c) - 17*e^(-9*d*x - 9*c) - 3*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))) - 1/8*a*b^2*(3*arctan(e^(-d*x - c 
))/d - (3*e^(-d*x - c) - 47*e^(-3*d*x - 3*c) + 78*e^(-5*d*x - 5*c) - 78*e^ 
(-7*d*x - 7*c) + 47*e^(-9*d*x - 9*c) - 3*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)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (127) = 254\).

Time = 0.19 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.84 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (5 \, a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (15 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 9 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} - 33 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 96 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 96 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 160 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 144 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 240 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \] Input:

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")
 

Output:

1/96*(3*(pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(5*a^3 + 3 
*a^2*b + 3*a*b^2 + 5*b^3) + 4*(15*a^3*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a 
^2*b*(e^(d*x + c) - e^(-d*x - c))^5 + 9*a*b^2*(e^(d*x + c) - e^(-d*x - c)) 
^5 - 33*b^3*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^3*(e^(d*x + c) - e^(-d* 
x - c))^3 + 96*a^2*b*(e^(d*x + c) - e^(-d*x - c))^3 - 96*a*b^2*(e^(d*x + c 
) - e^(-d*x - c))^3 - 160*b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 528*a^3*(e^ 
(d*x + c) - e^(-d*x - c)) - 144*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 144*a 
*b^2*(e^(d*x + c) - e^(-d*x - c)) - 240*b^3*(e^(d*x + c) - e^(-d*x - c)))/ 
((e^(d*x + c) - e^(-d*x - c))^2 + 4)^3)/d
 

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 601, normalized size of antiderivative = 4.45 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^3\,\sqrt {d^2}+5\,b^3\,\sqrt {d^2}+3\,a\,b^2\,\sqrt {d^2}+3\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}\right )\,\sqrt {25\,a^6+30\,a^5\,b+39\,a^4\,b^2+68\,a^3\,b^3+39\,a^2\,b^4+30\,a\,b^5+25\,b^6}}{8\,\sqrt {d^2}}-\frac {6\,{\mathrm {e}}^{c+d\,x}\,\left (3\,a^3-11\,a^2\,b+13\,a\,b^2-5\,b^3\right )}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-57\,a^2\,b+111\,a\,b^2-55\,b^3\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b+3\,a\,b^2-11\,b^3\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^3+3\,a^2\,b-93\,a\,b^2+85\,b^3\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {80\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {32\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )} \] Input:

int((a + b*sinh(c + d*x)^2)^3/cosh(c + d*x)^7,x)
 

Output:

(atan((exp(d*x)*exp(c)*(5*a^3*(d^2)^(1/2) + 5*b^3*(d^2)^(1/2) + 3*a*b^2*(d 
^2)^(1/2) + 3*a^2*b*(d^2)^(1/2)))/(d*(30*a*b^5 + 30*a^5*b + 25*a^6 + 25*b^ 
6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2)))*(30*a*b^5 + 30*a^5*b + 2 
5*a^6 + 25*b^6 + 39*a^2*b^4 + 68*a^3*b^3 + 39*a^4*b^2)^(1/2))/(8*(d^2)^(1/ 
2)) - (6*exp(c + d*x)*(13*a*b^2 - 11*a^2*b + 3*a^3 - 5*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 
)) + (exp(c + d*x)*(111*a*b^2 - 57*a^2*b + a^3 - 55*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)) + (exp(c + d*x)*(3* 
a*b^2 + 3*a^2*b + 5*a^3 - 11*b^3))/(8*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + 
 d*x)*(3*a^2*b - 93*a*b^2 + 5*a^3 + 85*b^3))/(12*d*(2*exp(2*c + 2*d*x) + e 
xp(4*c + 4*d*x) + 1)) + (80*exp(c + d*x)*(3*a*b^2 - 3*a^2*b + a^3 - b^3))/ 
(3*d*(5*exp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*e 
xp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1)) - (32*exp(c + d*x)*(3*a*b^2 - 3 
*a^2*b + a^3 - b^3))/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*e 
xp(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))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1058, normalized size of antiderivative = 7.84 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:

int(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^3,x)
 

Output:

(15*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a**3 + 9*e**(12*c + 12*d*x)*atan 
(e**(c + d*x))*a**2*b + 9*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a*b**2 + 1 
5*e**(12*c + 12*d*x)*atan(e**(c + d*x))*b**3 + 90*e**(10*c + 10*d*x)*atan( 
e**(c + d*x))*a**3 + 54*e**(10*c + 10*d*x)*atan(e**(c + d*x))*a**2*b + 54* 
e**(10*c + 10*d*x)*atan(e**(c + d*x))*a*b**2 + 90*e**(10*c + 10*d*x)*atan( 
e**(c + d*x))*b**3 + 225*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**3 + 135*e* 
*(8*c + 8*d*x)*atan(e**(c + d*x))*a**2*b + 135*e**(8*c + 8*d*x)*atan(e**(c 
 + d*x))*a*b**2 + 225*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b**3 + 300*e**(6 
*c + 6*d*x)*atan(e**(c + d*x))*a**3 + 180*e**(6*c + 6*d*x)*atan(e**(c + d* 
x))*a**2*b + 180*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a*b**2 + 300*e**(6*c 
+ 6*d*x)*atan(e**(c + d*x))*b**3 + 225*e**(4*c + 4*d*x)*atan(e**(c + d*x)) 
*a**3 + 135*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + 135*e**(4*c + 4*d 
*x)*atan(e**(c + d*x))*a*b**2 + 225*e**(4*c + 4*d*x)*atan(e**(c + d*x))*b* 
*3 + 90*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**3 + 54*e**(2*c + 2*d*x)*ata 
n(e**(c + d*x))*a**2*b + 54*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**2 + 9 
0*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**3 + 15*atan(e**(c + d*x))*a**3 + 
9*atan(e**(c + d*x))*a**2*b + 9*atan(e**(c + d*x))*a*b**2 + 15*atan(e**(c 
+ d*x))*b**3 + 15*e**(11*c + 11*d*x)*a**3 + 9*e**(11*c + 11*d*x)*a**2*b + 
9*e**(11*c + 11*d*x)*a*b**2 - 33*e**(11*c + 11*d*x)*b**3 + 85*e**(9*c + 9* 
d*x)*a**3 + 51*e**(9*c + 9*d*x)*a**2*b - 141*e**(9*c + 9*d*x)*a*b**2 + ...