\(\int \text {sech}^7(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\) [267]

Optimal result

Integrand size = 23, antiderivative size = 121 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\left (5 a^2+2 a b+b^2\right ) \arctan (\sinh (c+d x))}{16 d}+\frac {\left (5 a^2+2 a b+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{16 d}+\frac {(a-b) (5 a+7 b) \text {sech}^3(c+d x) \tanh (c+d x)}{24 d}+\frac {(a-b)^2 \text {sech}^5(c+d x) \tanh (c+d x)}{6 d} \] Output:

1/16*(5*a^2+2*a*b+b^2)*arctan(sinh(d*x+c))/d+1/16*(5*a^2+2*a*b+b^2)*sech(d 
*x+c)*tanh(d*x+c)/d+1/24*(a-b)*(5*a+7*b)*sech(d*x+c)^3*tanh(d*x+c)/d+1/6*( 
a-b)^2*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 = 8.91 (sec) , antiderivative size = 715, normalized size of antiderivative = 5.91 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\text {csch}^3(c+d x) \left (65625 a^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right )+36855 a^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^2(c+d x)+91875 a b \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^2(c+d x)+1680 a^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)+54180 a b \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)+32970 b^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^4(c+d x)+1365 a b \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)+19845 b^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^6(c+d x)+525 b^2 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \sinh ^8(c+d x)-65625 a^2 \sqrt {-\sinh ^2(c+d x)}+14980 a^2 \left (-\sinh ^2(c+d x)\right )^{3/2}+91875 a b \left (-\sinh ^2(c+d x)\right )^{3/2}+8855 b^2 \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+16 a^2 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+32 a b \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^6(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}+16 b^2 \, _6F_5\left (\frac {3}{2},2,2,2,2,2;1,1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2}-23555 a b \left (-\sinh ^2(c+d x)\right )^{5/2}-32970 b^2 \left (-\sinh ^2(c+d x)\right )^{5/2}+32 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2} \left (5 a^2+9 a b \sinh ^2(c+d x)+4 b^2 \sinh ^4(c+d x)\right )+4 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};-\sinh ^2(c+d x)\right ) \sinh ^4(c+d x) \left (-\sinh ^2(c+d x)\right )^{3/2} \left (155 a^2+242 a b \sinh ^2(c+d x)+95 b^2 \sinh ^4(c+d x)\right )\right )}{2520 d \sqrt {-\sinh ^2(c+d x)}} \] Input:

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

Output:

(Csch[c + d*x]^3*(65625*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] + 36855*a^2*Ar 
cTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^2 + 91875*a*b*ArcTanh[Sqrt[-Si 
nh[c + d*x]^2]]*Sinh[c + d*x]^2 + 1680*a^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]] 
*Sinh[c + d*x]^4 + 54180*a*b*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x] 
^4 + 32970*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^4 + 1365*a*b* 
ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 19845*b^2*ArcTanh[Sqrt[- 
Sinh[c + d*x]^2]]*Sinh[c + d*x]^6 + 525*b^2*ArcTanh[Sqrt[-Sinh[c + d*x]^2] 
]*Sinh[c + d*x]^8 - 65625*a^2*Sqrt[-Sinh[c + d*x]^2] + 14980*a^2*(-Sinh[c 
+ d*x]^2)^(3/2) + 91875*a*b*(-Sinh[c + d*x]^2)^(3/2) + 8855*b^2*Sinh[c + d 
*x]^4*(-Sinh[c + d*x]^2)^(3/2) + 16*a^2*HypergeometricPFQ[{3/2, 2, 2, 2, 2 
, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x] 
^2)^(3/2) + 32*a*b*HypergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/ 
2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^6*(-Sinh[c + d*x]^2)^(3/2) + 16*b^2*Hy 
pergeometricPFQ[{3/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 9/2}, -Sinh[c + d*x]^2] 
*Sinh[c + d*x]^8*(-Sinh[c + d*x]^2)^(3/2) - 23555*a*b*(-Sinh[c + d*x]^2)^( 
5/2) - 32970*b^2*(-Sinh[c + d*x]^2)^(5/2) + 32*HypergeometricPFQ[{3/2, 2, 
2, 2, 2}, {1, 1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^4*(-Sinh[c + d*x 
]^2)^(3/2)*(5*a^2 + 9*a*b*Sinh[c + d*x]^2 + 4*b^2*Sinh[c + d*x]^4) + 4*Hyp 
ergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, -Sinh[c + d*x]^2]*Sinh[c + d*x 
]^4*(-Sinh[c + d*x]^2)^(3/2)*(155*a^2 + 242*a*b*Sinh[c + d*x]^2 + 95*b^...
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 3669, 315, 298, 215, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3669

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 298

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 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 [A] (verified)

Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.72

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

Input:

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

Output:

1/d*(a^2*((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)))+2*a*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*arctan 
(exp(d*x+c)))+b^2*(-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))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2824 vs. \(2 (113) = 226\).

Time = 0.11 (sec) , antiderivative size = 2824, normalized size of antiderivative = 23.34 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(3*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^11 + 33*(5*a^2 + 2*a*b + b^2)* 
cosh(d*x + c)*sinh(d*x + c)^10 + 3*(5*a^2 + 2*a*b + b^2)*sinh(d*x + c)^11 
+ (85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^9 + (165*(5*a^2 + 2*a*b + b^2)* 
cosh(d*x + c)^2 + 85*a^2 + 34*a*b - 47*b^2)*sinh(d*x + c)^9 + 9*(55*(5*a^2 
 + 2*a*b + b^2)*cosh(d*x + c)^3 + (85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c) 
)*sinh(d*x + c)^8 + 6*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^7 + 6*(165* 
(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^4 + 6*(85*a^2 + 34*a*b - 47*b^2)*cosh( 
d*x + c)^2 + 33*a^2 - 38*a*b + 13*b^2)*sinh(d*x + c)^7 + 42*(33*(5*a^2 + 2 
*a*b + b^2)*cosh(d*x + c)^5 + 2*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^3 
 + (33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 - 6*(33*a^2 - 
 38*a*b + 13*b^2)*cosh(d*x + c)^5 + 6*(231*(5*a^2 + 2*a*b + b^2)*cosh(d*x 
+ c)^6 + 21*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^4 + 21*(33*a^2 - 38*a 
*b + 13*b^2)*cosh(d*x + c)^2 - 33*a^2 + 38*a*b - 13*b^2)*sinh(d*x + c)^5 + 
 6*(165*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 21*(85*a^2 + 34*a*b - 47*b 
^2)*cosh(d*x + c)^5 + 35*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^3 - 5*(3 
3*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - (85*a^2 + 34*a*b 
 - 47*b^2)*cosh(d*x + c)^3 + (495*(5*a^2 + 2*a*b + b^2)*cosh(d*x + c)^8 + 
84*(85*a^2 + 34*a*b - 47*b^2)*cosh(d*x + c)^6 + 210*(33*a^2 - 38*a*b + 13* 
b^2)*cosh(d*x + c)^4 - 60*(33*a^2 - 38*a*b + 13*b^2)*cosh(d*x + c)^2 - 85* 
a^2 - 34*a*b + 47*b^2)*sinh(d*x + c)^3 + 3*(55*(5*a^2 + 2*a*b + b^2)*co...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (113) = 226\).

Time = 0.13 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.99 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=-\frac {1}{24} \, a^{2} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} + 85 \, e^{\left (-3 \, d x - 3 \, c\right )} + 198 \, e^{\left (-5 \, d x - 5 \, c\right )} - 198 \, e^{\left (-7 \, d x - 7 \, c\right )} - 85 \, e^{\left (-9 \, d x - 9 \, c\right )} - 15 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{12} \, a b {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} + 17 \, e^{\left (-3 \, d x - 3 \, c\right )} - 114 \, e^{\left (-5 \, d x - 5 \, c\right )} + 114 \, e^{\left (-7 \, d x - 7 \, c\right )} - 17 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} - \frac {1}{24} \, b^{2} {\left (\frac {3 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {3 \, e^{\left (-d x - c\right )} - 47 \, e^{\left (-3 \, d x - 3 \, c\right )} + 78 \, e^{\left (-5 \, d x - 5 \, c\right )} - 78 \, e^{\left (-7 \, d x - 7 \, c\right )} + 47 \, e^{\left (-9 \, d x - 9 \, c\right )} - 3 \, e^{\left (-11 \, d x - 11 \, c\right )}}{d {\left (6 \, e^{\left (-2 \, d x - 2 \, c\right )} + 15 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20 \, e^{\left (-6 \, d x - 6 \, c\right )} + 15 \, e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, e^{\left (-10 \, d x - 10 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} + 1\right )}}\right )} \] Input:

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

Output:

-1/24*a^2*(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))) - 1/12*a*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))) - 1/24*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. 291 vs. \(2 (113) = 226\).

Time = 0.16 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.40 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (5 \, a^{2} + 2 \, a b + b^{2}\right )} + \frac {4 \, {\left (15 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 6 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{5} + 160 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 64 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} - 32 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 528 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 96 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 48 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{3}}}{96 \, d} \] Input:

integrate(sech(d*x+c)^7*(a+b*sinh(d*x+c)^2)^2,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^2 + 2 
*a*b + b^2) + 4*(15*a^2*(e^(d*x + c) - e^(-d*x - c))^5 + 6*a*b*(e^(d*x + c 
) - e^(-d*x - c))^5 + 3*b^2*(e^(d*x + c) - e^(-d*x - c))^5 + 160*a^2*(e^(d 
*x + c) - e^(-d*x - c))^3 + 64*a*b*(e^(d*x + c) - e^(-d*x - c))^3 - 32*b^2 
*(e^(d*x + c) - e^(-d*x - c))^3 + 528*a^2*(e^(d*x + c) - e^(-d*x - c)) - 9 
6*a*b*(e^(d*x + c) - e^(-d*x - c)) - 48*b^2*(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 = 1.96 (sec) , antiderivative size = 582, normalized size of antiderivative = 4.81 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,a^2\,\sqrt {d^2}+b^2\,\sqrt {d^2}+2\,a\,b\,\sqrt {d^2}\right )}{d\,\sqrt {25\,a^4+20\,a^3\,b+14\,a^2\,b^2+4\,a\,b^3+b^4}}\right )\,\sqrt {25\,a^4+20\,a^3\,b+14\,a^2\,b^2+4\,a\,b^3+b^4}}{8\,\sqrt {d^2}}-\frac {\frac {2\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d}+\frac {2\,b^2\,{\mathrm {e}}^{9\,c+9\,d\,x}}{3\,d}+\frac {4\,{\mathrm {e}}^{5\,c+5\,d\,x}\,\left (8\,a^2-8\,a\,b+3\,b^2\right )}{3\,d}+\frac {8\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (2\,a-b\right )}{3\,d}+\frac {8\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}\,\left (2\,a-b\right )}{3\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (11\,a^2-26\,a\,b+15\,b^2\right )}{3\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^2+2\,a\,b+b^2\right )}{8\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {16\,{\mathrm {e}}^{c+d\,x}\,\left (a^2-2\,a\,b+b^2\right )}{3\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2-22\,a\,b+21\,b^2\right )}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (5\,a^2+2\,a\,b-23\,b^2\right )}{12\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:

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

Output:

(atan((exp(d*x)*exp(c)*(5*a^2*(d^2)^(1/2) + b^2*(d^2)^(1/2) + 2*a*b*(d^2)^ 
(1/2)))/(d*(4*a*b^3 + 20*a^3*b + 25*a^4 + b^4 + 14*a^2*b^2)^(1/2)))*(4*a*b 
^3 + 20*a^3*b + 25*a^4 + b^4 + 14*a^2*b^2)^(1/2))/(8*(d^2)^(1/2)) - ((2*b^ 
2*exp(c + d*x))/(3*d) + (2*b^2*exp(9*c + 9*d*x))/(3*d) + (4*exp(5*c + 5*d* 
x)*(8*a^2 - 8*a*b + 3*b^2))/(3*d) + (8*b*exp(3*c + 3*d*x)*(2*a - b))/(3*d) 
 + (8*b*exp(7*c + 7*d*x)*(2*a - b))/(3*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) - (2*exp(c + d*x)*(11*a^2 - 26*a*b + 15*b^2) 
)/(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)*(2*a*b + 5*a^2 + b^2))/(8*d*(exp(2*c + 
 2*d*x) + 1)) + (16*exp(c + d*x)*(a^2 - 2*a*b + b^2))/(3*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) + ex 
p(10*c + 10*d*x) + 1)) + (exp(c + d*x)*(a^2 - 22*a*b + 21*b^2))/(3*d*(3*ex 
p(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (exp(c + d* 
x)*(2*a*b + 5*a^2 - 23*b^2))/(12*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) 
+ 1))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 781, normalized size of antiderivative = 6.45 \[ \int \text {sech}^7(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx=\frac {90 e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) a b +120 e^{6 d x +6 c} \mathit {atan} \left (e^{d x +c}\right ) a b +90 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a b +36 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a b +34 e^{9 d x +9 c} a b +225 e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+45 e^{8 d x +8 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+300 e^{6 d x +6 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+60 e^{6 d x +6 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+225 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+45 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+90 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+18 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+6 \mathit {atan} \left (e^{d x +c}\right ) a b +6 e^{12 d x +12 c} \mathit {atan} \left (e^{d x +c}\right ) a b +36 e^{10 d x +10 c} \mathit {atan} \left (e^{d x +c}\right ) a b +15 e^{12 d x +12 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+3 e^{12 d x +12 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}+90 e^{10 d x +10 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2}+18 e^{10 d x +10 c} \mathit {atan} \left (e^{d x +c}\right ) b^{2}-34 e^{3 d x +3 c} a b -6 e^{d x +c} a b +15 \mathit {atan} \left (e^{d x +c}\right ) a^{2}+3 \mathit {atan} \left (e^{d x +c}\right ) b^{2}+78 e^{7 d x +7 c} b^{2}-78 e^{5 d x +5 c} b^{2}+47 e^{3 d x +3 c} b^{2}-3 e^{d x +c} b^{2}-198 e^{5 d x +5 c} a^{2}+15 e^{11 d x +11 c} a^{2}+85 e^{9 d x +9 c} a^{2}+198 e^{7 d x +7 c} a^{2}+6 e^{11 d x +11 c} a b -228 e^{7 d x +7 c} a b +228 e^{5 d x +5 c} a b +3 e^{11 d x +11 c} b^{2}-47 e^{9 d x +9 c} b^{2}-85 e^{3 d x +3 c} a^{2}-15 e^{d x +c} a^{2}}{24 d \left (e^{12 d x +12 c}+6 e^{10 d x +10 c}+15 e^{8 d x +8 c}+20 e^{6 d x +6 c}+15 e^{4 d x +4 c}+6 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(15*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a**2 + 6*e**(12*c + 12*d*x)*atan 
(e**(c + d*x))*a*b + 3*e**(12*c + 12*d*x)*atan(e**(c + d*x))*b**2 + 90*e** 
(10*c + 10*d*x)*atan(e**(c + d*x))*a**2 + 36*e**(10*c + 10*d*x)*atan(e**(c 
 + d*x))*a*b + 18*e**(10*c + 10*d*x)*atan(e**(c + d*x))*b**2 + 225*e**(8*c 
 + 8*d*x)*atan(e**(c + d*x))*a**2 + 90*e**(8*c + 8*d*x)*atan(e**(c + d*x)) 
*a*b + 45*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b**2 + 300*e**(6*c + 6*d*x)* 
atan(e**(c + d*x))*a**2 + 120*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a*b + 60 
*e**(6*c + 6*d*x)*atan(e**(c + d*x))*b**2 + 225*e**(4*c + 4*d*x)*atan(e**( 
c + d*x))*a**2 + 90*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a*b + 45*e**(4*c + 
 4*d*x)*atan(e**(c + d*x))*b**2 + 90*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a 
**2 + 36*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b + 18*e**(2*c + 2*d*x)*ata 
n(e**(c + d*x))*b**2 + 15*atan(e**(c + d*x))*a**2 + 6*atan(e**(c + d*x))*a 
*b + 3*atan(e**(c + d*x))*b**2 + 15*e**(11*c + 11*d*x)*a**2 + 6*e**(11*c + 
 11*d*x)*a*b + 3*e**(11*c + 11*d*x)*b**2 + 85*e**(9*c + 9*d*x)*a**2 + 34*e 
**(9*c + 9*d*x)*a*b - 47*e**(9*c + 9*d*x)*b**2 + 198*e**(7*c + 7*d*x)*a**2 
 - 228*e**(7*c + 7*d*x)*a*b + 78*e**(7*c + 7*d*x)*b**2 - 198*e**(5*c + 5*d 
*x)*a**2 + 228*e**(5*c + 5*d*x)*a*b - 78*e**(5*c + 5*d*x)*b**2 - 85*e**(3* 
c + 3*d*x)*a**2 - 34*e**(3*c + 3*d*x)*a*b + 47*e**(3*c + 3*d*x)*b**2 - 15* 
e**(c + d*x)*a**2 - 6*e**(c + d*x)*a*b - 3*e**(c + d*x)*b**2)/(24*d*(e**(1 
2*c + 12*d*x) + 6*e**(10*c + 10*d*x) + 15*e**(8*c + 8*d*x) + 20*e**(6*c...