\(\int \text {csch}^3(c+d x) (a+b \tanh ^2(c+d x))^2 \, dx\) [15]

Optimal result

Integrand size = 23, antiderivative size = 74 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {a (a-4 b) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a^2 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {2 a b \text {sech}(c+d x)}{d}-\frac {b^2 \text {sech}^3(c+d x)}{3 d} \] Output:

1/2*a*(a-4*b)*arctanh(cosh(d*x+c))/d-1/2*a^2*coth(d*x+c)*csch(d*x+c)/d+2*a 
*b*sech(d*x+c)/d-1/3*b^2*sech(d*x+c)^3/d
 

Mathematica [A] (verified)

Time = 3.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.72 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {3 a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-12 a^2 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+48 a b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+12 a^2 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-48 a b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+3 a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-48 a b \text {sech}(c+d x)+8 b^2 \text {sech}^3(c+d x)}{24 d} \] Input:

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

Output:

-1/24*(3*a^2*Csch[(c + d*x)/2]^2 - 12*a^2*Log[Cosh[(c + d*x)/2]] + 48*a*b* 
Log[Cosh[(c + d*x)/2]] + 12*a^2*Log[Sinh[(c + d*x)/2]] - 48*a*b*Log[Sinh[( 
c + d*x)/2]] + 3*a^2*Sech[(c + d*x)/2]^2 - 48*a*b*Sech[c + d*x] + 8*b^2*Se 
ch[c + d*x]^3)/d
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4147, 366, 363, 262, 219}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4147

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

\(\Big \downarrow \) 366

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {a^2 \text {sech}^3(c+d x)}{2 \left (1-\text {sech}^2(c+d x)\right )}+\frac {1}{2} \left (\frac {2}{3} b^2 \text {sech}^3(c+d x)-a (a-4 b) (\text {arctanh}(\text {sech}(c+d x))-\text {sech}(c+d x))\right )}{d}\)

Input:

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

Output:

-(((a^2*Sech[c + d*x]^3)/(2*(1 - Sech[c + d*x]^2)) + (-(a*(a - 4*b)*(ArcTa 
nh[Sech[c + d*x]] - Sech[c + d*x])) + (2*b^2*Sech[c + d*x]^3)/3)/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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

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

rule 366

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

rule 3042

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

rule 4147

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

Maple [A] (verified)

Time = 8.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )-\frac {b^{2}}{3 \cosh \left (d x +c \right )^{3}}}{d}\)	\(67\)
default	\(\frac {a^{2} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+2 a b \left (\frac {1}{\cosh \left (d x +c \right )}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )-\frac {b^{2}}{3 \cosh \left (d x +c \right )^{3}}}{d}\)	\(67\)
risch	\(-\frac {{\mathrm e}^{d x +c} \left (3 \,{\mathrm e}^{8 d x +8 c} a^{2}-12 \,{\mathrm e}^{8 d x +8 c} a b +12 \,{\mathrm e}^{6 d x +6 c} a^{2}+8 \,{\mathrm e}^{6 d x +6 c} b^{2}+18 \,{\mathrm e}^{4 d x +4 c} a^{2}+24 \,{\mathrm e}^{4 d x +4 c} a b -16 \,{\mathrm e}^{4 d x +4 c} b^{2}+12 \,{\mathrm e}^{2 d x +2 c} a^{2}+8 b^{2} {\mathrm e}^{2 d x +2 c}+3 a^{2}-12 a b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {2 a b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}-\frac {a^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {2 a b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}\)	\(239\)

Input:

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

Output:

1/d*(a^2*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+2*a*b*(1/cosh( 
d*x+c)-2*arctanh(exp(d*x+c)))-1/3*b^2/cosh(d*x+c)^3)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2462 vs. \(2 (68) = 136\).

Time = 0.10 (sec) , antiderivative size = 2462, normalized size of antiderivative = 33.27 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(6*(a^2 - 4*a*b)*cosh(d*x + c)^9 + 54*(a^2 - 4*a*b)*cosh(d*x + c)*sin 
h(d*x + c)^8 + 6*(a^2 - 4*a*b)*sinh(d*x + c)^9 + 8*(3*a^2 + 2*b^2)*cosh(d* 
x + c)^7 + 8*(27*(a^2 - 4*a*b)*cosh(d*x + c)^2 + 3*a^2 + 2*b^2)*sinh(d*x + 
 c)^7 + 56*(9*(a^2 - 4*a*b)*cosh(d*x + c)^3 + (3*a^2 + 2*b^2)*cosh(d*x + c 
))*sinh(d*x + c)^6 + 4*(9*a^2 + 12*a*b - 8*b^2)*cosh(d*x + c)^5 + 4*(189*( 
a^2 - 4*a*b)*cosh(d*x + c)^4 + 42*(3*a^2 + 2*b^2)*cosh(d*x + c)^2 + 9*a^2 
+ 12*a*b - 8*b^2)*sinh(d*x + c)^5 + 4*(189*(a^2 - 4*a*b)*cosh(d*x + c)^5 + 
 70*(3*a^2 + 2*b^2)*cosh(d*x + c)^3 + 5*(9*a^2 + 12*a*b - 8*b^2)*cosh(d*x 
+ c))*sinh(d*x + c)^4 + 8*(3*a^2 + 2*b^2)*cosh(d*x + c)^3 + 8*(63*(a^2 - 4 
*a*b)*cosh(d*x + c)^6 + 35*(3*a^2 + 2*b^2)*cosh(d*x + c)^4 + 5*(9*a^2 + 12 
*a*b - 8*b^2)*cosh(d*x + c)^2 + 3*a^2 + 2*b^2)*sinh(d*x + c)^3 + 8*(27*(a^ 
2 - 4*a*b)*cosh(d*x + c)^7 + 21*(3*a^2 + 2*b^2)*cosh(d*x + c)^5 + 5*(9*a^2 
 + 12*a*b - 8*b^2)*cosh(d*x + c)^3 + 3*(3*a^2 + 2*b^2)*cosh(d*x + c))*sinh 
(d*x + c)^2 + 6*(a^2 - 4*a*b)*cosh(d*x + c) - 3*((a^2 - 4*a*b)*cosh(d*x + 
c)^10 + 10*(a^2 - 4*a*b)*cosh(d*x + c)*sinh(d*x + c)^9 + (a^2 - 4*a*b)*sin 
h(d*x + c)^10 + (a^2 - 4*a*b)*cosh(d*x + c)^8 + (45*(a^2 - 4*a*b)*cosh(d*x 
 + c)^2 + a^2 - 4*a*b)*sinh(d*x + c)^8 + 8*(15*(a^2 - 4*a*b)*cosh(d*x + c) 
^3 + (a^2 - 4*a*b)*cosh(d*x + c))*sinh(d*x + c)^7 - 2*(a^2 - 4*a*b)*cosh(d 
*x + c)^6 + 2*(105*(a^2 - 4*a*b)*cosh(d*x + c)^4 + 14*(a^2 - 4*a*b)*cosh(d 
*x + c)^2 - a^2 + 4*a*b)*sinh(d*x + c)^6 + 4*(63*(a^2 - 4*a*b)*cosh(d*x...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (68) = 136\).

Time = 0.04 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.45 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {1}{2} \, a^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - 2 \, a b {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} - \frac {8 \, b^{2}}{3 \, d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}} \] Input:

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

Output:

1/2*a^2*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(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))) - 
2*a*b*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d - 2*e^(-d*x - c)/ 
(d*(e^(-2*d*x - 2*c) + 1))) - 8/3*b^2/(d*(e^(d*x + c) + e^(-d*x - c))^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (68) = 136\).

Time = 0.17 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.07 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {\frac {12 \, a^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4} - 3 \, {\left (a^{2} - 4 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) + 3 \, {\left (a^{2} - 4 \, a b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {16 \, {\left (3 \, a b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 2 \, b^{2}\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3}}}{12 \, d} \] Input:

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

Output:

-1/12*(12*a^2*(e^(d*x + c) + e^(-d*x - c))/((e^(d*x + c) + e^(-d*x - c))^2 
 - 4) - 3*(a^2 - 4*a*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) + 3*(a^2 - 4*a 
*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 16*(3*a*b*(e^(d*x + c) + e^(-d*x 
 - c))^2 - 2*b^2)/(e^(d*x + c) + e^(-d*x - c))^3)/d
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 261, normalized size of antiderivative = 3.53 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a^2\,\sqrt {-d^2}-4\,a\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^4-8\,a^3\,b+16\,a^2\,b^2}}\right )\,\sqrt {a^4-8\,a^3\,b+16\,a^2\,b^2}}{\sqrt {-d^2}}+\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^2\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}+\frac {4\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:

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

Output:

(atan((exp(d*x)*exp(c)*(a^2*(-d^2)^(1/2) - 4*a*b*(-d^2)^(1/2)))/(d*(a^4 - 
8*a^3*b + 16*a^2*b^2)^(1/2)))*(a^4 - 8*a^3*b + 16*a^2*b^2)^(1/2))/(-d^2)^( 
1/2) + (8*b^2*exp(c + d*x))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) 
+ exp(6*c + 6*d*x) + 1)) - (a^2*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - 
 (2*a^2*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8 
*b^2*exp(c + d*x))/(3*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) + (4* 
a*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1))
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 768, normalized size of antiderivative = 10.38 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {-3 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+24 e^{9 d x +9 c} a b -3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+12 \,\mathrm {log}\left (e^{d x +c}-1\right ) a b -12 \,\mathrm {log}\left (e^{d x +c}+1\right ) a b +6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}-6 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-48 e^{5 d x +5 c} a b +6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}-6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{2}+24 e^{d x +c} a b +12 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -12 e^{10 d x +10 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +12 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -12 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b -24 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b +24 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b -24 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b +24 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b +12 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) a b -12 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) a b -6 e^{9 d x +9 c} a^{2}-24 e^{7 d x +7 c} a^{2}-16 e^{7 d x +7 c} b^{2}-36 e^{5 d x +5 c} a^{2}-24 e^{3 d x +3 c} a^{2}+32 e^{5 d x +5 c} b^{2}-16 e^{3 d x +3 c} b^{2}-3 \,\mathrm {log}\left (e^{d x +c}-1\right ) a^{2}+3 \,\mathrm {log}\left (e^{d x +c}+1\right ) a^{2}-6 e^{d x +c} a^{2}}{6 d \left (e^{10 d x +10 c}+e^{8 d x +8 c}-2 e^{6 d x +6 c}-2 e^{4 d x +4 c}+e^{2 d x +2 c}+1\right )} \] Input:

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

Output:

( - 3*e**(10*c + 10*d*x)*log(e**(c + d*x) - 1)*a**2 + 12*e**(10*c + 10*d*x 
)*log(e**(c + d*x) - 1)*a*b + 3*e**(10*c + 10*d*x)*log(e**(c + d*x) + 1)*a 
**2 - 12*e**(10*c + 10*d*x)*log(e**(c + d*x) + 1)*a*b - 6*e**(9*c + 9*d*x) 
*a**2 + 24*e**(9*c + 9*d*x)*a*b - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1) 
*a**2 + 12*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a*b + 3*e**(8*c + 8*d*x) 
*log(e**(c + d*x) + 1)*a**2 - 12*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a* 
b - 24*e**(7*c + 7*d*x)*a**2 - 16*e**(7*c + 7*d*x)*b**2 + 6*e**(6*c + 6*d* 
x)*log(e**(c + d*x) - 1)*a**2 - 24*e**(6*c + 6*d*x)*log(e**(c + d*x) - 1)* 
a*b - 6*e**(6*c + 6*d*x)*log(e**(c + d*x) + 1)*a**2 + 24*e**(6*c + 6*d*x)* 
log(e**(c + d*x) + 1)*a*b - 36*e**(5*c + 5*d*x)*a**2 - 48*e**(5*c + 5*d*x) 
*a*b + 32*e**(5*c + 5*d*x)*b**2 + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1) 
*a**2 - 24*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a*b - 6*e**(4*c + 4*d*x) 
*log(e**(c + d*x) + 1)*a**2 + 24*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a* 
b - 24*e**(3*c + 3*d*x)*a**2 - 16*e**(3*c + 3*d*x)*b**2 - 3*e**(2*c + 2*d* 
x)*log(e**(c + d*x) - 1)*a**2 + 12*e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)* 
a*b + 3*e**(2*c + 2*d*x)*log(e**(c + d*x) + 1)*a**2 - 12*e**(2*c + 2*d*x)* 
log(e**(c + d*x) + 1)*a*b - 6*e**(c + d*x)*a**2 + 24*e**(c + d*x)*a*b - 3* 
log(e**(c + d*x) - 1)*a**2 + 12*log(e**(c + d*x) - 1)*a*b + 3*log(e**(c + 
d*x) + 1)*a**2 - 12*log(e**(c + d*x) + 1)*a*b)/(6*d*(e**(10*c + 10*d*x) + 
e**(8*c + 8*d*x) - 2*e**(6*c + 6*d*x) - 2*e**(4*c + 4*d*x) + e**(2*c + ...