\(\int \frac {\text {csch}^3(c+d x)}{(a+b \tanh ^2(c+d x))^3} \, dx\) [47]

Optimal result

Integrand size = 23, antiderivative size = 196 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\frac {(a+6 b) \text {arctanh}(\cosh (c+d x))}{2 a^4 d}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d \left (a+b-b \text {sech}^2(c+d x)\right )^2}-\frac {3 b \text {sech}(c+d x)}{4 a^2 d \left (a+b-b \text {sech}^2(c+d x)\right )^2}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{8 a^3 (a+b) d \left (a+b-b \text {sech}^2(c+d x)\right )} \] Output:

1/2*(a+6*b)*arctanh(cosh(d*x+c))/a^4/d-1/8*b^(1/2)*(15*a^2+40*a*b+24*b^2)* 
arctanh(b^(1/2)*sech(d*x+c)/(a+b)^(1/2))/a^4/(a+b)^(3/2)/d-1/2*coth(d*x+c) 
*csch(d*x+c)/a/d/(a+b-b*sech(d*x+c)^2)^2-3/4*b*sech(d*x+c)/a^2/d/(a+b-b*se 
ch(d*x+c)^2)^2-1/8*b*(11*a+12*b)*sech(d*x+c)/a^3/(a+b)/d/(a+b-b*sech(d*x+c 
)^2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.17 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.46 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=-\frac {\frac {i \sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \arctan \left (\frac {-i \sqrt {a+b}-\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{3/2}}+\frac {i \sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \arctan \left (\frac {-i \sqrt {a+b}+\sqrt {a} \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b}}\right )}{(a+b)^{3/2}}+\frac {8 a^2 b^2 \cosh (c+d x)}{(a+b) (a-b+(a+b) \cosh (2 (c+d x)))^2}+\frac {2 a b (9 a+8 b) \cosh (c+d x)}{(a+b) (a-b+(a+b) \cosh (2 (c+d x)))}+a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 (a+6 b) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 (a+6 b) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d} \] Input:

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

Output:

-1/8*((I*Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] - Sqr 
t[a]*Tanh[(c + d*x)/2])/Sqrt[b]])/(a + b)^(3/2) + (I*Sqrt[b]*(15*a^2 + 40* 
a*b + 24*b^2)*ArcTan[((-I)*Sqrt[a + b] + Sqrt[a]*Tanh[(c + d*x)/2])/Sqrt[b 
]])/(a + b)^(3/2) + (8*a^2*b^2*Cosh[c + d*x])/((a + b)*(a - b + (a + b)*Co 
sh[2*(c + d*x)])^2) + (2*a*b*(9*a + 8*b)*Cosh[c + d*x])/((a + b)*(a - b + 
(a + b)*Cosh[2*(c + d*x)])) + a*Csch[(c + d*x)/2]^2 - 4*(a + 6*b)*Log[Cosh 
[(c + d*x)/2]] + 4*(a + 6*b)*Log[Sinh[(c + d*x)/2]] + a*Sech[(c + d*x)/2]^ 
2)/(a^4*d)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 26, 4147, 373, 402, 27, 402, 25, 397, 219, 221}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4147

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

\(\Big \downarrow \) 373

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

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {-\frac {\int -\frac {2 (a+b) \left (9 b \text {sech}^2(c+d x)+2 a+3 b\right )}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{4 a (a+b)}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\int \frac {9 b \text {sech}^2(c+d x)+2 a+3 b}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )^2}d\text {sech}(c+d x)}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 402

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {-\frac {\int -\frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\int \frac {4 a^2+17 b a+12 b^2+b (11 a+12 b) \text {sech}^2(c+d x)}{\left (1-\text {sech}^2(c+d x)\right ) \left (-b \text {sech}^2(c+d x)+a+b\right )}d\text {sech}(c+d x)}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 397

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \int \frac {1}{1-\text {sech}^2(c+d x)}d\text {sech}(c+d x)}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {b \left (15 a^2+40 a b+24 b^2\right ) \int \frac {1}{-b \text {sech}^2(c+d x)+a+b}d\text {sech}(c+d x)}{a}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\text {sech}(c+d x)}{2 a \left (1-\text {sech}^2(c+d x)\right ) \left (a-b \text {sech}^2(c+d x)+b\right )^2}-\frac {\frac {\frac {\frac {4 (a+b) (a+6 b) \text {arctanh}(\text {sech}(c+d x))}{a}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \text {sech}(c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b (11 a+12 b) \text {sech}(c+d x)}{2 a (a+b) \left (a-b \text {sech}^2(c+d x)+b\right )}}{2 a}-\frac {3 b \text {sech}(c+d x)}{2 a \left (a-b \text {sech}^2(c+d x)+b\right )^2}}{2 a}}{d}\)

Input:

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

Output:

-((Sech[c + d*x]/(2*a*(1 - Sech[c + d*x]^2)*(a + b - b*Sech[c + d*x]^2)^2) 
 - ((-3*b*Sech[c + d*x])/(2*a*(a + b - b*Sech[c + d*x]^2)^2) + (((4*(a + b 
)*(a + 6*b)*ArcTanh[Sech[c + d*x]])/a - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2 
)*ArcTanh[(Sqrt[b]*Sech[c + d*x])/Sqrt[a + b]])/(a*Sqrt[a + b]))/(2*a*(a + 
 b)) - (b*(11*a + 12*b)*Sech[c + d*x])/(2*a*(a + b)*(a + b - b*Sech[c + d* 
x]^2)))/(2*a))/(2*a))/d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 221

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

rule 373

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

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && 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 = 16.25 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.55

method	result	size
derivativedivides	\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{3}}-\frac {1}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+32 a b +24 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a +b \right )}-\frac {\left (27 a^{3}+102 a^{2} b +152 b^{2} a +80 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a +b \right )}-\frac {a \left (27 a^{2}+80 a b +56 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a +b \right )}-\frac {a^{2} \left (9 a +10 b \right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{4}}}{d}\)	\(304\)
default	\(\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{3}}-\frac {1}{8 a^{3} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a -12 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{4}}+\frac {2 b \left (\frac {-\frac {\left (9 a^{2}+32 a b +24 b^{2}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8 \left (a +b \right )}-\frac {\left (27 a^{3}+102 a^{2} b +152 b^{2} a +80 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8 \left (a +b \right )}-\frac {a \left (27 a^{2}+80 a b +56 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a +b \right )}-\frac {a^{2} \left (9 a +10 b \right )}{8 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +4 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+a \right )^{2}}-\frac {\left (15 a^{2}+40 a b +24 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 a +4 b}{4 \sqrt {a b +b^{2}}}\right )}{16 \left (a +b \right ) \sqrt {a b +b^{2}}}\right )}{a^{4}}}{d}\)	\(304\)
risch	\(-\frac {\left (4 \,{\mathrm e}^{10 d x +10 c} a^{3}+21 a^{2} b \,{\mathrm e}^{10 d x +10 c}+29 a \,b^{2} {\mathrm e}^{10 d x +10 c}+12 b^{3} {\mathrm e}^{10 d x +10 c}+20 a^{3} {\mathrm e}^{8 d x +8 c}+37 a^{2} b \,{\mathrm e}^{8 d x +8 c}-15 a \,b^{2} {\mathrm e}^{8 d x +8 c}-36 b^{3} {\mathrm e}^{8 d x +8 c}+40 a^{3} {\mathrm e}^{6 d x +6 c}+6 a^{2} b \,{\mathrm e}^{6 d x +6 c}-14 a \,b^{2} {\mathrm e}^{6 d x +6 c}+24 b^{3} {\mathrm e}^{6 d x +6 c}+40 \,{\mathrm e}^{4 d x +4 c} a^{3}+6 a^{2} b \,{\mathrm e}^{4 d x +4 c}-14 a \,b^{2} {\mathrm e}^{4 d x +4 c}+24 b^{3} {\mathrm e}^{4 d x +4 c}+20 \,{\mathrm e}^{2 d x +2 c} a^{3}+37 a^{2} b \,{\mathrm e}^{2 d x +2 c}-15 a \,b^{2} {\mathrm e}^{2 d x +2 c}-36 b^{3} {\mathrm e}^{2 d x +2 c}+4 a^{3}+21 a^{2} b +29 b^{2} a +12 b^{3}\right ) {\mathrm e}^{d x +c}}{4 \left (a +b \right ) \left ({\mathrm e}^{4 d x +4 c} a +b \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a -2 \,{\mathrm e}^{2 d x +2 c} b +a +b \right )^{2} d \,a^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 a^{3} d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{d \,a^{4}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{d \,a^{4}}+\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{2} d \,a^{2}}+\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{2 \left (a +b \right )^{2} d \,a^{3}}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{2} d \,a^{4}}-\frac {15 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right )}{16 \left (a +b \right )^{2} d \,a^{2}}-\frac {5 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b}{2 \left (a +b \right )^{2} d \,a^{3}}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}\, {\mathrm e}^{d x +c}}{a +b}+1\right ) b^{2}}{2 \left (a +b \right )^{2} d \,a^{4}}\)	\(788\)

Input:

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

Output:

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a^3-1/8/a^3/tanh(1/2*d*x+1/2*c)^2+1/4/a^4*( 
-2*a-12*b)*ln(tanh(1/2*d*x+1/2*c))+2*b/a^4*((-1/8*(9*a^2+32*a*b+24*b^2)*a/ 
(a+b)*tanh(1/2*d*x+1/2*c)^6-1/8*(27*a^3+102*a^2*b+152*a*b^2+80*b^3)/(a+b)* 
tanh(1/2*d*x+1/2*c)^4-1/8*a*(27*a^2+80*a*b+56*b^2)/(a+b)*tanh(1/2*d*x+1/2* 
c)^2-1/8*a^2*(9*a+10*b)/(a+b))/(tanh(1/2*d*x+1/2*c)^4*a+2*tanh(1/2*d*x+1/2 
*c)^2*a+4*b*tanh(1/2*d*x+1/2*c)^2+a)^2-1/16*(15*a^2+40*a*b+24*b^2)/(a+b)/( 
a*b+b^2)^(1/2)*arctanh(1/4*(2*tanh(1/2*d*x+1/2*c)^2*a+2*a+4*b)/(a*b+b^2)^( 
1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11576 vs. \(2 (187) = 374\).

Time = 0.40 (sec) , antiderivative size = 21301, normalized size of antiderivative = 108.68 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/4*((4*a^3*e^(11*c) + 21*a^2*b*e^(11*c) + 29*a*b^2*e^(11*c) + 12*b^3*e^( 
11*c))*e^(11*d*x) + (20*a^3*e^(9*c) + 37*a^2*b*e^(9*c) - 15*a*b^2*e^(9*c) 
- 36*b^3*e^(9*c))*e^(9*d*x) + 2*(20*a^3*e^(7*c) + 3*a^2*b*e^(7*c) - 7*a*b^ 
2*e^(7*c) + 12*b^3*e^(7*c))*e^(7*d*x) + 2*(20*a^3*e^(5*c) + 3*a^2*b*e^(5*c 
) - 7*a*b^2*e^(5*c) + 12*b^3*e^(5*c))*e^(5*d*x) + (20*a^3*e^(3*c) + 37*a^2 
*b*e^(3*c) - 15*a*b^2*e^(3*c) - 36*b^3*e^(3*c))*e^(3*d*x) + (4*a^3*e^c + 2 
1*a^2*b*e^c + 29*a*b^2*e^c + 12*b^3*e^c)*e^(d*x))/(a^6*d + 3*a^5*b*d + 3*a 
^4*b^2*d + a^3*b^3*d + (a^6*d*e^(12*c) + 3*a^5*b*d*e^(12*c) + 3*a^4*b^2*d* 
e^(12*c) + a^3*b^3*d*e^(12*c))*e^(12*d*x) + 2*(a^6*d*e^(10*c) - a^5*b*d*e^ 
(10*c) - 5*a^4*b^2*d*e^(10*c) - 3*a^3*b^3*d*e^(10*c))*e^(10*d*x) - (a^6*d* 
e^(8*c) + 3*a^5*b*d*e^(8*c) - 13*a^4*b^2*d*e^(8*c) - 15*a^3*b^3*d*e^(8*c)) 
*e^(8*d*x) - 4*(a^6*d*e^(6*c) - a^5*b*d*e^(6*c) + 3*a^4*b^2*d*e^(6*c) + 5* 
a^3*b^3*d*e^(6*c))*e^(6*d*x) - (a^6*d*e^(4*c) + 3*a^5*b*d*e^(4*c) - 13*a^4 
*b^2*d*e^(4*c) - 15*a^3*b^3*d*e^(4*c))*e^(4*d*x) + 2*(a^6*d*e^(2*c) - a^5* 
b*d*e^(2*c) - 5*a^4*b^2*d*e^(2*c) - 3*a^3*b^3*d*e^(2*c))*e^(2*d*x)) + 1/2* 
(a + 6*b)*log((e^(d*x + c) + 1)*e^(-c))/(a^4*d) - 1/2*(a + 6*b)*log((e^(d* 
x + c) - 1)*e^(-c))/(a^4*d) + 8*integrate(1/32*((15*a^2*b*e^(3*c) + 40*a*b 
^2*e^(3*c) + 24*b^3*e^(3*c))*e^(3*d*x) - (15*a^2*b*e^c + 40*a*b^2*e^c + 24 
*b^3*e^c)*e^(d*x))/(a^6 + 2*a^5*b + a^4*b^2 + (a^6*e^(4*c) + 2*a^5*b*e^(4* 
c) + a^4*b^2*e^(4*c))*e^(4*d*x) + 2*(a^6*e^(2*c) - a^4*b^2*e^(2*c))*e^(...
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \] Input:

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^3),x)
 

Output:

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^2)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 6723, normalized size of antiderivative = 34.30 \[ \int \frac {\text {csch}^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b 
) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**4 + 70*e**(12*c + 12*d*x)*sqr 
t(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c 
+ d*x)*sqrt(b))*a**3*b + 119*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(e* 
*(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a**2*b* 
*2 + 88*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a 
 + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))*a*b**3 + 24*e**(12*c + 12*d* 
x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2* 
e**(c + d*x)*sqrt(b))*b**4 - 15*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log 
(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a**4 
 - 70*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + 
 b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*a**3*b - 119*e**(12*c + 12*d*x 
)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e 
**(c + d*x)*sqrt(b))*a**2*b**2 - 88*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b) 
*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))* 
a*b**3 - 24*e**(12*c + 12*d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sq 
rt(a + b) + sqrt(a + b) + 2*e**(c + d*x)*sqrt(b))*b**4 + 30*e**(10*c + 10* 
d*x)*sqrt(b)*sqrt(a + b)*log(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 
2*e**(c + d*x)*sqrt(b))*a**4 + 20*e**(10*c + 10*d*x)*sqrt(b)*sqrt(a + b)*l 
og(e**(2*c + 2*d*x)*sqrt(a + b) + sqrt(a + b) - 2*e**(c + d*x)*sqrt(b))...