\(\int \sinh (c+d x) (a+b \tanh ^3(c+d x))^2 \, dx\) [60]

Optimal result

Integrand size = 21, antiderivative size = 120 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-\frac {3 a b \arctan (\sinh (c+d x))}{d}+\frac {a^2 \cosh (c+d x)}{d}+\frac {b^2 \cosh (c+d x)}{d}+\frac {3 b^2 \text {sech}(c+d x)}{d}-\frac {b^2 \text {sech}^3(c+d x)}{d}+\frac {b^2 \text {sech}^5(c+d x)}{5 d}+\frac {2 a b \sinh (c+d x)}{d}+\frac {a b \text {sech}(c+d x) \tanh (c+d x)}{d} \] Output:

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

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.75 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {5 \left (a^2+b^2\right ) \cosh (c+d x)+b \left (-5 b \text {sech}^3(c+d x)+b \text {sech}^5(c+d x)+10 a \left (-3 \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\sinh (c+d x)\right )+5 \text {sech}(c+d x) (3 b+a \tanh (c+d x))\right )}{5 d} \] Input:

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

Output:

(5*(a^2 + b^2)*Cosh[c + d*x] + b*(-5*b*Sech[c + d*x]^3 + b*Sech[c + d*x]^5 
 + 10*a*(-3*ArcTan[Tanh[(c + d*x)/2]] + Sinh[c + d*x]) + 5*Sech[c + d*x]*( 
3*b + a*Tanh[c + d*x])))/(5*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.64 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 26, 4149, 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.

\(\displaystyle \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4149

\(\displaystyle -i \int \left (i b^2 \sinh (c+d x) \tanh ^6(c+d x)+2 i a b \sinh (c+d x) \tanh ^3(c+d x)+i a^2 \sinh (c+d x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -i \left (\frac {i a^2 \cosh (c+d x)}{d}-\frac {3 i a b \arctan (\sinh (c+d x))}{d}+\frac {3 i a b \sinh (c+d x)}{d}-\frac {i a b \sinh (c+d x) \tanh ^2(c+d x)}{d}+\frac {i b^2 \cosh (c+d x)}{d}+\frac {i b^2 \text {sech}^5(c+d x)}{5 d}-\frac {i b^2 \text {sech}^3(c+d x)}{d}+\frac {3 i b^2 \text {sech}(c+d x)}{d}\right )\)

Input:

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

Output:

(-I)*(((-3*I)*a*b*ArcTan[Sinh[c + d*x]])/d + (I*a^2*Cosh[c + d*x])/d + (I* 
b^2*Cosh[c + d*x])/d + ((3*I)*b^2*Sech[c + d*x])/d - (I*b^2*Sech[c + d*x]^ 
3)/d + ((I/5)*b^2*Sech[c + d*x]^5)/d + ((3*I)*a*b*Sinh[c + d*x])/d - (I*a* 
b*Sinh[c + d*x]*Tanh[c + d*x]^2)/d)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 4149

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a 
 + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && 
 IGtQ[p, 0]
 

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {a^{2} \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )}{d}\)	\(145\)
default	\(\frac {a^{2} \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{6}}{\cosh \left (d x +c \right )^{5}}+\frac {6 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}+\frac {8 \sinh \left (d x +c \right )^{2}}{\cosh \left (d x +c \right )^{5}}+\frac {16}{5 \cosh \left (d x +c \right )^{5}}\right )}{d}\)	\(145\)
risch	\(\frac {{\mathrm e}^{d x +c} a^{2}}{2 d}+\frac {{\mathrm e}^{d x +c} a b}{d}+\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}+\frac {{\mathrm e}^{-d x -c} a^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} a b}{d}+\frac {{\mathrm e}^{-d x -c} b^{2}}{2 d}+\frac {2 b \,{\mathrm e}^{d x +c} \left (5 \,{\mathrm e}^{8 d x +8 c} a +15 \,{\mathrm e}^{8 d x +8 c} b +10 \,{\mathrm e}^{6 d x +6 c} a +40 \,{\mathrm e}^{6 d x +6 c} b +66 b \,{\mathrm e}^{4 d x +4 c}-10 \,{\mathrm e}^{2 d x +2 c} a +40 \,{\mathrm e}^{2 d x +2 c} b -5 a +15 b \right )}{5 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}+\frac {3 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}-\frac {3 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}\)	\(244\)

Input:

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

Output:

1/d*(a^2*cosh(d*x+c)+2*a*b*(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^2*(sinh(d*x+ 
c)^6/cosh(d*x+c)^5+6*sinh(d*x+c)^4/cosh(d*x+c)^5+8*sinh(d*x+c)^2/cosh(d*x+ 
c)^5+16/5/cosh(d*x+c)^5))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2230 vs. \(2 (118) = 236\).

Time = 0.12 (sec) , antiderivative size = 2230, normalized size of antiderivative = 18.58 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/10*(5*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^12 + 60*(a^2 + 2*a*b + b^2)*cosh 
(d*x + c)*sinh(d*x + c)^11 + 5*(a^2 + 2*a*b + b^2)*sinh(d*x + c)^12 + 30*( 
a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^10 + 30*(11*(a^2 + 2*a*b + b^2)*cosh(d* 
x + c)^2 + a^2 + 2*a*b + 3*b^2)*sinh(d*x + c)^10 + 100*(11*(a^2 + 2*a*b + 
b^2)*cosh(d*x + c)^3 + 3*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c))*sinh(d*x + c 
)^9 + 5*(15*a^2 + 18*a*b + 47*b^2)*cosh(d*x + c)^8 + 5*(495*(a^2 + 2*a*b + 
 b^2)*cosh(d*x + c)^4 + 270*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^2 + 15*a^2 
 + 18*a*b + 47*b^2)*sinh(d*x + c)^8 + 40*(99*(a^2 + 2*a*b + b^2)*cosh(d*x 
+ c)^5 + 90*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^3 + (15*a^2 + 18*a*b + 47* 
b^2)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(25*a^2 + 91*b^2)*cosh(d*x + c)^6 
+ 4*(1155*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^6 + 1575*(a^2 + 2*a*b + 3*b^2) 
*cosh(d*x + c)^4 + 35*(15*a^2 + 18*a*b + 47*b^2)*cosh(d*x + c)^2 + 25*a^2 
+ 91*b^2)*sinh(d*x + c)^6 + 8*(495*(a^2 + 2*a*b + b^2)*cosh(d*x + c)^7 + 9 
45*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^5 + 35*(15*a^2 + 18*a*b + 47*b^2)*c 
osh(d*x + c)^3 + 3*(25*a^2 + 91*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 5*(1 
5*a^2 - 18*a*b + 47*b^2)*cosh(d*x + c)^4 + 5*(495*(a^2 + 2*a*b + b^2)*cosh 
(d*x + c)^8 + 1260*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^6 + 70*(15*a^2 + 18 
*a*b + 47*b^2)*cosh(d*x + c)^4 + 12*(25*a^2 + 91*b^2)*cosh(d*x + c)^2 + 15 
*a^2 - 18*a*b + 47*b^2)*sinh(d*x + c)^4 + 20*(55*(a^2 + 2*a*b + b^2)*cosh( 
d*x + c)^9 + 180*(a^2 + 2*a*b + 3*b^2)*cosh(d*x + c)^7 + 14*(15*a^2 + 1...
 

Sympy [F]

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

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

Output:

Integral((a + b*tanh(c + d*x)**3)**2*sinh(c + d*x), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (118) = 236\).

Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.11 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=a b {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {1}{10} \, b^{2} {\left (\frac {5 \, e^{\left (-d x - c\right )}}{d} + \frac {85 \, e^{\left (-2 \, d x - 2 \, c\right )} + 210 \, e^{\left (-4 \, d x - 4 \, c\right )} + 314 \, e^{\left (-6 \, d x - 6 \, c\right )} + 185 \, e^{\left (-8 \, d x - 8 \, c\right )} + 65 \, e^{\left (-10 \, d x - 10 \, c\right )} + 5}{d {\left (e^{\left (-d x - c\right )} + 5 \, e^{\left (-3 \, d x - 3 \, c\right )} + 10 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 5 \, e^{\left (-9 \, d x - 9 \, c\right )} + e^{\left (-11 \, d x - 11 \, c\right )}\right )}}\right )} + \frac {a^{2} \cosh \left (d x + c\right )}{d} \] Input:

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

Output:

a*b*(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 
)))) + 1/10*b^2*(5*e^(-d*x - c)/d + (85*e^(-2*d*x - 2*c) + 210*e^(-4*d*x - 
 4*c) + 314*e^(-6*d*x - 6*c) + 185*e^(-8*d*x - 8*c) + 65*e^(-10*d*x - 10*c 
) + 5)/(d*(e^(-d*x - c) + 5*e^(-3*d*x - 3*c) + 10*e^(-5*d*x - 5*c) + 10*e^ 
(-7*d*x - 7*c) + 5*e^(-9*d*x - 9*c) + e^(-11*d*x - 11*c)))) + a^2*cosh(d*x 
 + c)/d
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.69 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-\frac {60 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) - 5 \, a^{2} e^{\left (d x + c\right )} - 10 \, a b e^{\left (d x + c\right )} - 5 \, b^{2} e^{\left (d x + c\right )} - 5 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-d x - c\right )} - \frac {4 \, {\left (5 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 15 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 10 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 40 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 66 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 10 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 40 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 5 \, a b e^{\left (d x + c\right )} + 15 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{10 \, d} \] Input:

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

Output:

-1/10*(60*a*b*arctan(e^(d*x + c)) - 5*a^2*e^(d*x + c) - 10*a*b*e^(d*x + c) 
 - 5*b^2*e^(d*x + c) - 5*(a^2 - 2*a*b + b^2)*e^(-d*x - c) - 4*(5*a*b*e^(9* 
d*x + 9*c) + 15*b^2*e^(9*d*x + 9*c) + 10*a*b*e^(7*d*x + 7*c) + 40*b^2*e^(7 
*d*x + 7*c) + 66*b^2*e^(5*d*x + 5*c) - 10*a*b*e^(3*d*x + 3*c) + 40*b^2*e^( 
3*d*x + 3*c) - 5*a*b*e^(d*x + c) + 15*b^2*e^(d*x + c))/(e^(2*d*x + 2*c) + 
1)^5)/d
 

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.82 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^2}{2\,d}+\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a-b\right )}^2}{2\,d}-\frac {6\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {d^2}}+\frac {72\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {64\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}+\frac {32\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}+\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (3\,b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (2\,b^2+a\,b\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:

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

Output:

(exp(c + d*x)*(a + b)^2)/(2*d) + (exp(- c - d*x)*(a - b)^2)/(2*d) - (6*ata 
n((a*b*exp(d*x)*exp(c)*(d^2)^(1/2))/(d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/ 
(d^2)^(1/2) + (72*b^2*exp(c + d*x))/(5*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 
 4*d*x) + exp(6*c + 6*d*x) + 1)) - (64*b^2*exp(c + d*x))/(5*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)) 
 + (32*b^2*exp(c + d*x))/(5*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)) + (2*e 
xp(c + d*x)*(a*b + 3*b^2))/(d*(exp(2*c + 2*d*x) + 1)) - (4*exp(c + d*x)*(a 
*b + 2*b^2))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 611, normalized size of antiderivative = 5.09 \[ \int \sinh (c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {-480 e^{d x +c} \mathit {atan} \left (e^{d x +c}\right ) a b -4 e^{3 d x +3 c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right )^{4} b^{2}-4 e^{d x +c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right )^{4} b^{2}-16 e^{3 d x +3 c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{5} b^{2}+80 e^{3 d x +3 c} \sinh \left (d x +c \right ) a b -16 e^{d x +c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{5} b^{2}+80 e^{d x +c} \sinh \left (d x +c \right ) a b -22 e^{3 d x +3 c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right )^{2} b^{2}-22 e^{d x +c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right )^{2} b^{2}-28 e^{3 d x +3 c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{3} b^{2}+44 e^{3 d x +3 c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right ) b^{2}-28 e^{d x +c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{3} b^{2}+44 e^{d x +c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right ) b^{2}-480 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a b +80 e^{3 d x +3 c} \cosh \left (d x +c \right ) a^{2}-44 e^{3 d x +3 c} \cosh \left (d x +c \right ) b^{2}+80 e^{d x +c} \cosh \left (d x +c \right ) a^{2}-44 e^{d x +c} \cosh \left (d x +c \right ) b^{2}+450 e^{2 d x +2 c} b^{2}+75 b^{2}-80 e^{3 d x +3 c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right ) a b -80 e^{d x +c} \cosh \left (d x +c \right ) \tanh \left (d x +c \right ) a b -80 e^{3 d x +3 c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{2} a b -80 e^{d x +c} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )^{2} a b +120 e^{4 d x +4 c} a b -120 a b +75 e^{4 d x +4 c} b^{2}}{80 e^{d x +c} d \left (e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

( - 480*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a*b - 480*e**(c + d*x)*atan(e* 
*(c + d*x))*a*b - 4*e**(3*c + 3*d*x)*cosh(c + d*x)*tanh(c + d*x)**4*b**2 - 
 22*e**(3*c + 3*d*x)*cosh(c + d*x)*tanh(c + d*x)**2*b**2 - 80*e**(3*c + 3* 
d*x)*cosh(c + d*x)*tanh(c + d*x)*a*b + 80*e**(3*c + 3*d*x)*cosh(c + d*x)*a 
**2 - 44*e**(3*c + 3*d*x)*cosh(c + d*x)*b**2 - 4*e**(c + d*x)*cosh(c + d*x 
)*tanh(c + d*x)**4*b**2 - 22*e**(c + d*x)*cosh(c + d*x)*tanh(c + d*x)**2*b 
**2 - 80*e**(c + d*x)*cosh(c + d*x)*tanh(c + d*x)*a*b + 80*e**(c + d*x)*co 
sh(c + d*x)*a**2 - 44*e**(c + d*x)*cosh(c + d*x)*b**2 + 120*e**(4*c + 4*d* 
x)*a*b + 75*e**(4*c + 4*d*x)*b**2 - 16*e**(3*c + 3*d*x)*sinh(c + d*x)*tanh 
(c + d*x)**5*b**2 - 28*e**(3*c + 3*d*x)*sinh(c + d*x)*tanh(c + d*x)**3*b** 
2 - 80*e**(3*c + 3*d*x)*sinh(c + d*x)*tanh(c + d*x)**2*a*b + 44*e**(3*c + 
3*d*x)*sinh(c + d*x)*tanh(c + d*x)*b**2 + 80*e**(3*c + 3*d*x)*sinh(c + d*x 
)*a*b + 450*e**(2*c + 2*d*x)*b**2 - 16*e**(c + d*x)*sinh(c + d*x)*tanh(c + 
 d*x)**5*b**2 - 28*e**(c + d*x)*sinh(c + d*x)*tanh(c + d*x)**3*b**2 - 80*e 
**(c + d*x)*sinh(c + d*x)*tanh(c + d*x)**2*a*b + 44*e**(c + d*x)*sinh(c + 
d*x)*tanh(c + d*x)*b**2 + 80*e**(c + d*x)*sinh(c + d*x)*a*b - 120*a*b + 75 
*b**2)/(80*e**(c + d*x)*d*(e**(2*c + 2*d*x) + 1))