\(\int \text {sech}^3(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\) [274]

Optimal result

Integrand size = 23, antiderivative size = 91 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {(a-b)^2 (a+5 b) \arctan (\sinh (c+d x))}{2 d}+\frac {(3 a-2 b) b^2 \sinh (c+d x)}{d}+\frac {b^3 \sinh ^3(c+d x)}{3 d}+\frac {(a-b)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 d} \] Output:

1/2*(a-b)^2*(a+5*b)*arctan(sinh(d*x+c))/d+(3*a-2*b)*b^2*sinh(d*x+c)/d+1/3* 
b^3*sinh(d*x+c)^3/d+1/2*(a-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 = 5.76 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.81 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {\text {csch}^5(c+d x) \left (-256 \, _5F_4\left (\frac {3}{2},2,2,2,2;1,1,1,\frac {11}{2};-\sinh ^2(c+d x)\right ) \sinh ^8(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3+21 \left (36015 a^3+5 a^2 (3224 a+21609 b) \sinh ^2(c+d x)+3 a \left (491 a^2+16120 a b+36015 b^2\right ) \sinh ^4(c+d x)+3 b \left (753 a^2+18280 a b+10805 b^2\right ) \sinh ^6(c+d x)+b^2 (2259 a+17320 b) \sinh ^8(c+d x)+753 b^3 \sinh ^{10}(c+d x)\right )-\frac {315 \text {arctanh}\left (\sqrt {-\sinh ^2(c+d x)}\right ) \left (2401 a^3+3 a^2 (625 a+2401 b) \sinh ^2(c+d x)+3 a \left (81 a^2+1875 a b+2401 b^2\right ) \sinh ^4(c+d x)+\left (-47 a^3+585 a^2 b+6057 a b^2+2161 b^3\right ) \sinh ^6(c+d x)+3 b \left (a^2+243 a b+625 b^2\right ) \sinh ^8(c+d x)+3 b^2 (a+81 b) \sinh ^{10}(c+d x)+b^3 \sinh ^{12}(c+d x)\right )}{\sqrt {-\sinh ^2(c+d x)}}\right )}{30240 d} \] Input:

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

Output:

(Csch[c + d*x]^5*(-256*HypergeometricPFQ[{3/2, 2, 2, 2, 2}, {1, 1, 1, 11/2 
}, -Sinh[c + d*x]^2]*Sinh[c + d*x]^8*(a + b*Sinh[c + d*x]^2)^3 + 21*(36015 
*a^3 + 5*a^2*(3224*a + 21609*b)*Sinh[c + d*x]^2 + 3*a*(491*a^2 + 16120*a*b 
 + 36015*b^2)*Sinh[c + d*x]^4 + 3*b*(753*a^2 + 18280*a*b + 10805*b^2)*Sinh 
[c + d*x]^6 + b^2*(2259*a + 17320*b)*Sinh[c + d*x]^8 + 753*b^3*Sinh[c + d* 
x]^10) - (315*ArcTanh[Sqrt[-Sinh[c + d*x]^2]]*(2401*a^3 + 3*a^2*(625*a + 2 
401*b)*Sinh[c + d*x]^2 + 3*a*(81*a^2 + 1875*a*b + 2401*b^2)*Sinh[c + d*x]^ 
4 + (-47*a^3 + 585*a^2*b + 6057*a*b^2 + 2161*b^3)*Sinh[c + d*x]^6 + 3*b*(a 
^2 + 243*a*b + 625*b^2)*Sinh[c + d*x]^8 + 3*b^2*(a + 81*b)*Sinh[c + d*x]^1 
0 + b^3*Sinh[c + d*x]^12))/Sqrt[-Sinh[c + d*x]^2]))/(30240*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 89, 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, 3669, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3669

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

\(\Big \downarrow \) 300

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(((a - b)^2*(a + 5*b)*ArcTan[Sinh[c + d*x]])/2 + (3*a - 2*b)*b^2*Sinh[c + 
d*x] + (b^3*Sinh[c + d*x]^3)/3 + ((a - 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 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. \(218\) vs. \(2(85)=170\).

Time = 0.17 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.41

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

Input:

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

Output:

1/d*(a^3*(1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*a^2*b*(-sinh(d 
*x+c)/cosh(d*x+c)^2+1/2*sech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*b^2* 
a*(sinh(d*x+c)^3/cosh(d*x+c)^2+3*sinh(d*x+c)/cosh(d*x+c)^2-3/2*sech(d*x+c) 
*tanh(d*x+c)-3*arctan(exp(d*x+c)))+b^3*(1/3*sinh(d*x+c)^5/cosh(d*x+c)^2-5/ 
3*sinh(d*x+c)^3/cosh(d*x+c)^2-5*sinh(d*x+c)/cosh(d*x+c)^2+5/2*sech(d*x+c)* 
tanh(d*x+c)+5*arctan(exp(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (85) = 170\).

Time = 0.10 (sec) , antiderivative size = 1679, normalized size of antiderivative = 18.45 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(b^3*cosh(d*x + c)^10 + 10*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + b^3*si 
nh(d*x + c)^10 + (36*a*b^2 - 25*b^3)*cosh(d*x + c)^8 + (45*b^3*cosh(d*x + 
c)^2 + 36*a*b^2 - 25*b^3)*sinh(d*x + c)^8 + 8*(15*b^3*cosh(d*x + c)^3 + (3 
6*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 2*(12*a^3 - 36*a^2*b + 
54*a*b^2 - 25*b^3)*cosh(d*x + c)^6 + 2*(105*b^3*cosh(d*x + c)^4 + 12*a^3 - 
 36*a^2*b + 54*a*b^2 - 25*b^3 + 14*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^2)*si 
nh(d*x + c)^6 + 4*(63*b^3*cosh(d*x + c)^5 + 14*(36*a*b^2 - 25*b^3)*cosh(d* 
x + c)^3 + 3*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c))*sinh(d 
*x + c)^5 - 2*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^4 + 2* 
(105*b^3*cosh(d*x + c)^6 + 35*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^4 - 12*a^3 
 + 36*a^2*b - 54*a*b^2 + 25*b^3 + 15*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^ 
3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(15*b^3*cosh(d*x + c)^7 + 7*(36*a* 
b^2 - 25*b^3)*cosh(d*x + c)^5 + 5*(12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)* 
cosh(d*x + c)^3 - (12*a^3 - 36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c))*s 
inh(d*x + c)^3 - b^3 - (36*a*b^2 - 25*b^3)*cosh(d*x + c)^2 + (45*b^3*cosh( 
d*x + c)^8 + 28*(36*a*b^2 - 25*b^3)*cosh(d*x + c)^6 + 30*(12*a^3 - 36*a^2* 
b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^4 - 36*a*b^2 + 25*b^3 - 12*(12*a^3 - 
36*a^2*b + 54*a*b^2 - 25*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 24*((a^3 
+ 3*a^2*b - 9*a*b^2 + 5*b^3)*cosh(d*x + c)^7 + 7*(a^3 + 3*a^2*b - 9*a*b^2 
+ 5*b^3)*cosh(d*x + c)*sinh(d*x + c)^6 + (a^3 + 3*a^2*b - 9*a*b^2 + 5*b...
 

Sympy [F(-1)]

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

integrate(sech(d*x+c)**3*(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. 357 vs. \(2 (85) = 170\).

Time = 0.12 (sec) , antiderivative size = 357, normalized size of antiderivative = 3.92 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {1}{24} \, b^{3} {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} - 3 \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} - a^{3} {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} \] Input:

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

Output:

1/24*b^3*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 120*arctan(e^(-d*x - c) 
)/d - (25*e^(-2*d*x - 2*c) + 77*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1) 
/(d*(e^(-3*d*x - 3*c) + 2*e^(-5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 3/2*a*b 
^2*(6*arctan(e^(-d*x - c))/d - e^(-d*x - c)/d + (4*e^(-2*d*x - 2*c) - e^(- 
4*d*x - 4*c) + 1)/(d*(e^(-d*x - c) + 2*e^(-3*d*x - 3*c) + e^(-5*d*x - 5*c) 
))) - 3*a^2*b*(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))) - a^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)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (85) = 170\).

Time = 0.17 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.71 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 36 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 24 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 6 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{3} + 3 \, a^{2} b - 9 \, a b^{2} + 5 \, b^{3}\right )} + \frac {24 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 3 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4}}{24 \, d} \] Input:

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

Output:

1/24*(b^3*(e^(d*x + c) - e^(-d*x - c))^3 + 36*a*b^2*(e^(d*x + c) - e^(-d*x 
 - c)) - 24*b^3*(e^(d*x + c) - e^(-d*x - c)) + 6*(pi + 2*arctan(1/2*(e^(2* 
d*x + 2*c) - 1)*e^(-d*x - c)))*(a^3 + 3*a^2*b - 9*a*b^2 + 5*b^3) + 24*(a^3 
*(e^(d*x + c) - e^(-d*x - c)) - 3*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 3*a 
*b^2*(e^(d*x + c) - e^(-d*x - c)) - b^3*(e^(d*x + c) - e^(-d*x - c)))/((e^ 
(d*x + c) - e^(-d*x - c))^2 + 4))/d
 

Mupad [B] (verification not implemented)

Time = 3.25 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.38 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {b^3\,{\mathrm {e}}^{3\,c+3\,d\,x}}{24\,d}-\frac {b^3\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{24\,d}+\frac {3\,b^2\,{\mathrm {e}}^{c+d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {3\,b^2\,{\mathrm {e}}^{-c-d\,x}\,\left (4\,a-3\,b\right )}{8\,d}+\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )-{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (-{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^3+3\,a^2\,b-9\,a\,b^2+5\,b^3\right )+{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}\right )\,{\left (a-b\right )}^2\,\left (a+5\,b\right )\,1{}\mathrm {i}}{2\,d} \] Input:

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

Output:

(b^3*exp(3*c + 3*d*x))/(24*d) - (b^3*exp(- 3*c - 3*d*x))/(24*d) + (log(- ( 
a - b)^2*(a + 5*b)*1i - exp(d*x)*exp(c)*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3)) 
*(a - b)^2*(a + 5*b)*1i)/(2*d) - (log((a - b)^2*(a + 5*b)*1i - exp(d*x)*ex 
p(c)*(3*a^2*b - 9*a*b^2 + a^3 + 5*b^3))*(a - b)^2*(a + 5*b)*1i)/(2*d) + (3 
*b^2*exp(c + d*x)*(4*a - 3*b))/(8*d) + (exp(c + d*x)*(3*a*b^2 - 3*a^2*b + 
a^3 - b^3))/(d*(exp(2*c + 2*d*x) + 1)) - (2*exp(c + d*x)*(3*a*b^2 - 3*a^2* 
b + a^3 - b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (3*b^2*e 
xp(- c - d*x)*(4*a - 3*b))/(8*d)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 531, normalized size of antiderivative = 5.84 \[ \int \text {sech}^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx=\frac {-b^{3}+48 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}+240 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+72 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -216 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+72 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -216 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+144 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b -432 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a \,b^{2}+25 e^{2 d x +2 c} b^{3}+e^{10 d x +10 c} b^{3}+24 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}+120 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+24 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a^{3}+120 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+36 e^{8 d x +8 c} a \,b^{2}-72 e^{6 d x +6 c} a^{2} b +108 e^{6 d x +6 c} a \,b^{2}+72 e^{4 d x +4 c} a^{2} b -108 e^{4 d x +4 c} a \,b^{2}-50 e^{6 d x +6 c} b^{3}+50 e^{4 d x +4 c} b^{3}-25 e^{8 d x +8 c} b^{3}+24 e^{6 d x +6 c} a^{3}-24 e^{4 d x +4 c} a^{3}-36 e^{2 d x +2 c} a \,b^{2}}{24 e^{3 d x +3 c} d \left (e^{4 d x +4 c}+2 e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

(24*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a**3 + 72*e**(7*c + 7*d*x)*atan(e* 
*(c + d*x))*a**2*b - 216*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a*b**2 + 120* 
e**(7*c + 7*d*x)*atan(e**(c + d*x))*b**3 + 48*e**(5*c + 5*d*x)*atan(e**(c 
+ d*x))*a**3 + 144*e**(5*c + 5*d*x)*atan(e**(c + d*x))*a**2*b - 432*e**(5* 
c + 5*d*x)*atan(e**(c + d*x))*a*b**2 + 240*e**(5*c + 5*d*x)*atan(e**(c + d 
*x))*b**3 + 24*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a**3 + 72*e**(3*c + 3*d 
*x)*atan(e**(c + d*x))*a**2*b - 216*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a* 
b**2 + 120*e**(3*c + 3*d*x)*atan(e**(c + d*x))*b**3 + e**(10*c + 10*d*x)*b 
**3 + 36*e**(8*c + 8*d*x)*a*b**2 - 25*e**(8*c + 8*d*x)*b**3 + 24*e**(6*c + 
 6*d*x)*a**3 - 72*e**(6*c + 6*d*x)*a**2*b + 108*e**(6*c + 6*d*x)*a*b**2 - 
50*e**(6*c + 6*d*x)*b**3 - 24*e**(4*c + 4*d*x)*a**3 + 72*e**(4*c + 4*d*x)* 
a**2*b - 108*e**(4*c + 4*d*x)*a*b**2 + 50*e**(4*c + 4*d*x)*b**3 - 36*e**(2 
*c + 2*d*x)*a*b**2 + 25*e**(2*c + 2*d*x)*b**3 - b**3)/(24*e**(3*c + 3*d*x) 
*d*(e**(4*c + 4*d*x) + 2*e**(2*c + 2*d*x) + 1))