\(\int \cosh ^3(c+d x) (a+b \text {sech}^2(c+d x))^3 \, dx\) [66]

Optimal result

Integrand size = 23, antiderivative size = 81 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {b^2 (6 a+b) \arctan (\sinh (c+d x))}{2 d}+\frac {a^2 (a+3 b) \sinh (c+d x)}{d}+\frac {a^3 \sinh ^3(c+d x)}{3 d}+\frac {b^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Output:

1/2*b^2*(6*a+b)*arctan(sinh(d*x+c))/d+a^2*(a+3*b)*sinh(d*x+c)/d+1/3*a^3*si 
nh(d*x+c)^3/d+1/2*b^3*sech(d*x+c)*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 = 6.64 (sec) , antiderivative size = 483, normalized size of antiderivative = 5.96 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\coth ^3(c+d x) \text {csch}^2(c+d x) (a \cosh (c+d x)+b \text {sech}(c+d x))^3 \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b+a \sinh ^2(c+d x)\right )^3-\frac {315 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (b^3 \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)-47 \sinh ^6(c+d x)\right )+3 a^2 b \cosh ^4(c+d x) \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+a^3 \cosh ^6(c+d x) \left (2401+1875 \sinh ^2(c+d x)+243 \sinh ^4(c+d x)+\sinh ^6(c+d x)\right )+3 a b^2 \left (2401+4276 \sinh ^2(c+d x)+2118 \sinh ^4(c+d x)+148 \sinh ^6(c+d x)+\sinh ^8(c+d x)\right )\right )}{\sqrt {-\sinh ^2(c+d x)}}+21 \left (b^3 \left (36015+16120 \sinh ^2(c+d x)+1473 \sinh ^4(c+d x)\right )+3 a b^2 \left (36015+52135 \sinh ^2(c+d x)+17593 \sinh ^4(c+d x)+753 \sinh ^6(c+d x)\right )+3 a^2 b \left (36015+88150 \sinh ^2(c+d x)+69728 \sinh ^4(c+d x)+19786 \sinh ^6(c+d x)+753 \sinh ^8(c+d x)\right )+a^3 \left (36015+124165 \sinh ^2(c+d x)+157878 \sinh ^4(c+d x)+89514 \sinh ^6(c+d x)+19579 \sinh ^8(c+d x)+753 \sinh ^{10}(c+d x)\right )\right )\right )}{3780 d (a+2 b+a \cosh (2 c+2 d x))^3} \] Input:

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

Output:

(Coth[c + d*x]^3*Csch[c + d*x]^2*(a*Cosh[c + d*x] + b*Sech[c + d*x])^3*(-2 
56*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2}, -Sinh[c + d*x]^2] 
*Sinh[c + d*x]^8*(a + b + a*Sinh[c + d*x]^2)^3 - (315*ArcTanh[Sqrt[-Sinh[c 
 + d*x]^2]]*(b^3*(2401 + 1875*Sinh[c + d*x]^2 + 243*Sinh[c + d*x]^4 - 47*S 
inh[c + d*x]^6) + 3*a^2*b*Cosh[c + d*x]^4*(2401 + 1875*Sinh[c + d*x]^2 + 2 
43*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + a^3*Cosh[c + d*x]^6*(2401 + 1875*S 
inh[c + d*x]^2 + 243*Sinh[c + d*x]^4 + Sinh[c + d*x]^6) + 3*a*b^2*(2401 + 
4276*Sinh[c + d*x]^2 + 2118*Sinh[c + d*x]^4 + 148*Sinh[c + d*x]^6 + Sinh[c 
 + d*x]^8)))/Sqrt[-Sinh[c + d*x]^2] + 21*(b^3*(36015 + 16120*Sinh[c + d*x] 
^2 + 1473*Sinh[c + d*x]^4) + 3*a*b^2*(36015 + 52135*Sinh[c + d*x]^2 + 1759 
3*Sinh[c + d*x]^4 + 753*Sinh[c + d*x]^6) + 3*a^2*b*(36015 + 88150*Sinh[c + 
 d*x]^2 + 69728*Sinh[c + d*x]^4 + 19786*Sinh[c + d*x]^6 + 753*Sinh[c + d*x 
]^8) + a^3*(36015 + 124165*Sinh[c + d*x]^2 + 157878*Sinh[c + d*x]^4 + 8951 
4*Sinh[c + d*x]^6 + 19579*Sinh[c + d*x]^8 + 753*Sinh[c + d*x]^10))))/(3780 
*d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4635, 300, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4635

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((b^2*(6*a + b)*ArcTan[Sinh[c + d*x]])/2 + a^2*(a + 3*b)*Sinh[c + d*x] + ( 
a^3*Sinh[c + d*x]^3)/3 + (b^3*Sinh[c + d*x])/(2*(1 + Sinh[c + d*x]^2)))/d
 

Defintions of rubi rules used

rule 300

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int 
[PolynomialDivide[(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] && ILtQ[q, 0] && GeQ[p, -q]
 

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 4635

Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ 
))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f 
 Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m 
+ n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In 
tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
 

Maple [A] (verified)

Time = 1.87 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+3 a^{2} b \sinh \left (d x +c \right )+6 a \,b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(79\)
default	\(\frac {a^{3} \left (\frac {2}{3}+\frac {\cosh \left (d x +c \right )^{2}}{3}\right ) \sinh \left (d x +c \right )+3 a^{2} b \sinh \left (d x +c \right )+6 a \,b^{2} \arctan \left ({\mathrm e}^{d x +c}\right )+b^{3} \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(79\)
parallelrisch	\(\frac {-72 i b^{2} \left (\cosh \left (2 d x +2 c \right )+1\right ) \left (a +\frac {b}{6}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+72 i b^{2} \left (\cosh \left (2 d x +2 c \right )+1\right ) \left (a +\frac {b}{6}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )+\left (11 a^{3}+36 a^{2} b \right ) \sinh \left (3 d x +3 c \right )+a^{3} \sinh \left (5 d x +5 c \right )+10 \sinh \left (d x +c \right ) \left (a^{3}+\frac {18}{5} a^{2} b +\frac {12}{5} b^{3}\right )}{24 d \left (\cosh \left (2 d x +2 c \right )+1\right )}\)	\(148\)
risch	\(\frac {a^{3} {\mathrm e}^{3 d x +3 c}}{24 d}+\frac {3 a^{3} {\mathrm e}^{d x +c}}{8 d}+\frac {3 a^{2} {\mathrm e}^{d x +c} b}{2 d}-\frac {3 a^{3} {\mathrm e}^{-d x -c}}{8 d}-\frac {3 a^{2} {\mathrm e}^{-d x -c} b}{2 d}-\frac {a^{3} {\mathrm e}^{-3 d x -3 c}}{24 d}+\frac {b^{3} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d}+\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d}-\frac {3 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d}\)	\(215\)

Input:

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

Output:

1/d*(a^3*(2/3+1/3*cosh(d*x+c)^2)*sinh(d*x+c)+3*a^2*b*sinh(d*x+c)+6*a*b^2*a 
rctan(exp(d*x+c))+b^3*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (75) = 150\).

Time = 0.35 (sec) , antiderivative size = 1409, normalized size of antiderivative = 17.40 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(a^3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*sinh(d*x + c)^9 + a^3*si 
nh(d*x + c)^10 + (11*a^3 + 36*a^2*b)*cosh(d*x + c)^8 + (45*a^3*cosh(d*x + 
c)^2 + 11*a^3 + 36*a^2*b)*sinh(d*x + c)^8 + 8*(15*a^3*cosh(d*x + c)^3 + (1 
1*a^3 + 36*a^2*b)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(5*a^3 + 18*a^2*b + 1 
2*b^3)*cosh(d*x + c)^6 + 2*(105*a^3*cosh(d*x + c)^4 + 5*a^3 + 18*a^2*b + 1 
2*b^3 + 14*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^2)*sinh(d*x + c)^6 + 4*(63*a^ 
3*cosh(d*x + c)^5 + 14*(11*a^3 + 36*a^2*b)*cosh(d*x + c)^3 + 3*(5*a^3 + 18 
*a^2*b + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 2*(5*a^3 + 18*a^2*b + 12 
*b^3)*cosh(d*x + c)^4 + 2*(105*a^3*cosh(d*x + c)^6 + 35*(11*a^3 + 36*a^2*b 
)*cosh(d*x + c)^4 - 5*a^3 - 18*a^2*b - 12*b^3 + 15*(5*a^3 + 18*a^2*b + 12* 
b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*a^3*cosh(d*x + c)^7 + 7*(11* 
a^3 + 36*a^2*b)*cosh(d*x + c)^5 + 5*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + 
 c)^3 - (5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - a^3 - 
 (11*a^3 + 36*a^2*b)*cosh(d*x + c)^2 + (45*a^3*cosh(d*x + c)^8 + 28*(11*a^ 
3 + 36*a^2*b)*cosh(d*x + c)^6 + 30*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + 
c)^4 - 11*a^3 - 36*a^2*b - 12*(5*a^3 + 18*a^2*b + 12*b^3)*cosh(d*x + c)^2) 
*sinh(d*x + c)^2 + 24*((6*a*b^2 + b^3)*cosh(d*x + c)^7 + 7*(6*a*b^2 + b^3) 
*cosh(d*x + c)*sinh(d*x + c)^6 + (6*a*b^2 + b^3)*sinh(d*x + c)^7 + 2*(6*a* 
b^2 + b^3)*cosh(d*x + c)^5 + (12*a*b^2 + 2*b^3 + 21*(6*a*b^2 + b^3)*cosh(d 
*x + c)^2)*sinh(d*x + c)^5 + 5*(7*(6*a*b^2 + b^3)*cosh(d*x + c)^3 + 2*(...
 

Sympy [F(-1)]

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

integrate(cosh(d*x+c)**3*(a+b*sech(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. 179 vs. \(2 (75) = 150\).

Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.21 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-b^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{24} \, a^{3} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} - \frac {6 \, a b^{2} \arctan \left (e^{\left (-d x - c\right )}\right )}{d} \] Input:

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

Output:

-b^3*(arctan(e^(-d*x - c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^( 
-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 1/24*a^3*(e^(3*d*x + 3*c)/d + 9* 
e^(d*x + c)/d - 9*e^(-d*x - c)/d - e^(-3*d*x - 3*c)/d) + 3/2*a^2*b*(e^(d*x 
 + c)/d - e^(-d*x - c)/d) - 6*a*b^2*arctan(e^(-d*x - c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (75) = 150\).

Time = 0.15 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.01 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 12 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 36 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + \frac {24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (6 \, a b^{2} + b^{3}\right )}}{24 \, d} \] Input:

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

Output:

1/24*(a^3*(e^(d*x + c) - e^(-d*x - c))^3 + 12*a^3*(e^(d*x + c) - e^(-d*x - 
 c)) + 36*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 24*b^3*(e^(d*x + c) - e^(-d 
*x - c))/((e^(d*x + c) - e^(-d*x - c))^2 + 4) + 6*(pi + 2*arctan(1/2*(e^(2 
*d*x + 2*c) - 1)*e^(-d*x - c)))*(6*a*b^2 + b^3))/d
 

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.69 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {d^2}+6\,a\,b^2\,\sqrt {d^2}\right )}{d\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}\right )\,\sqrt {36\,a^2\,b^4+12\,a\,b^5+b^6}}{\sqrt {d^2}}-\frac {a^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {a^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {3\,a^2\,{\mathrm {e}}^{-c-d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {3\,a^2\,{\mathrm {e}}^{c+d\,x}\,\left (a+4\,b\right )}{8\,d}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:

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

Output:

(atan((exp(d*x)*exp(c)*(b^3*(d^2)^(1/2) + 6*a*b^2*(d^2)^(1/2)))/(d*(12*a*b 
^5 + b^6 + 36*a^2*b^4)^(1/2)))*(12*a*b^5 + b^6 + 36*a^2*b^4)^(1/2))/(d^2)^ 
(1/2) - (a^3*exp(- 3*c - 3*d*x))/(24*d) + (a^3*exp(3*c + 3*d*x))/(24*d) - 
(3*a^2*exp(- c - d*x)*(a + 4*b))/(8*d) + (3*a^2*exp(c + d*x)*(a + 4*b))/(8 
*d) + (b^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1)) - (2*b^3*exp(c + d*x)) 
/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 358, normalized size of antiderivative = 4.42 \[ \int \cosh ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {144 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+24 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+288 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+48 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+144 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+24 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+e^{10 d x +10 c} a^{3}+11 e^{8 d x +8 c} a^{3}+36 e^{8 d x +8 c} a^{2} b +10 e^{6 d x +6 c} a^{3}+36 e^{6 d x +6 c} a^{2} b +24 e^{6 d x +6 c} b^{3}-10 e^{4 d x +4 c} a^{3}-36 e^{4 d x +4 c} a^{2} b -24 e^{4 d x +4 c} b^{3}-11 e^{2 d x +2 c} a^{3}-36 e^{2 d x +2 c} a^{2} b -a^{3}}{24 e^{3 d x +3 c} d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(144*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a*b**2 + 24*e**(7*c + 7*d*x)*atan 
(e**(c + d*x))*b**3 + 288*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a*b**2 + 48* 
e**(5*c + 5*d*x)*atan(e**(c + d*x))*b**3 + 144*e**(3*c + 3*d*x)*atan(e**(c 
 + d*x))*a*b**2 + 24*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**3 + e**(10*c + 
 10*d*x)*a**3 + 11*e**(8*c + 8*d*x)*a**3 + 36*e**(8*c + 8*d*x)*a**2*b + 10 
*e**(6*c + 6*d*x)*a**3 + 36*e**(6*c + 6*d*x)*a**2*b + 24*e**(6*c + 6*d*x)* 
b**3 - 10*e**(4*c + 4*d*x)*a**3 - 36*e**(4*c + 4*d*x)*a**2*b - 24*e**(4*c 
+ 4*d*x)*b**3 - 11*e**(2*c + 2*d*x)*a**3 - 36*e**(2*c + 2*d*x)*a**2*b - a* 
*3)/(24*e**(3*c + 3*d*x)*d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))