\(\int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx\) [76]

Optimal result

Integrand size = 12, antiderivative size = 163 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=\frac {x}{a^3}+\frac {b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right )^{5/2} d}-\frac {b^2 \coth (c+d x)}{2 a \left (a^2+b^2\right ) d (a+b \text {csch}(c+d x))^2}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{2 a^2 \left (a^2+b^2\right )^2 d (a+b \text {csch}(c+d x))} \] Output:

x/a^3+b*(6*a^4+5*a^2*b^2+2*b^4)*arctanh((a-b*tanh(1/2*d*x+1/2*c))/(a^2+b^2 
)^(1/2))/a^3/(a^2+b^2)^(5/2)/d-1/2*b^2*coth(d*x+c)/a/(a^2+b^2)/d/(a+b*csch 
(d*x+c))^2-1/2*b^2*(5*a^2+2*b^2)*coth(d*x+c)/a^2/(a^2+b^2)^2/d/(a+b*csch(d 
*x+c))
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.31 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=\frac {\text {csch}^2(c+d x) (b+a \sinh (c+d x)) \left (\frac {a b^3 \coth (c+d x)}{a^2+b^2}-\frac {3 a b^2 \left (2 a^2+b^2\right ) \coth (c+d x) (b+a \sinh (c+d x))}{\left (a^2+b^2\right )^2}+2 (c+d x) \text {csch}(c+d x) (b+a \sinh (c+d x))^2-\frac {2 b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right ) \text {csch}(c+d x) (b+a \sinh (c+d x))^2}{\left (-a^2-b^2\right )^{5/2}}\right )}{2 a^3 d (a+b \text {csch}(c+d x))^3} \] Input:

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

Output:

(Csch[c + d*x]^2*(b + a*Sinh[c + d*x])*((a*b^3*Coth[c + d*x])/(a^2 + b^2) 
- (3*a*b^2*(2*a^2 + b^2)*Coth[c + d*x]*(b + a*Sinh[c + d*x]))/(a^2 + b^2)^ 
2 + 2*(c + d*x)*Csch[c + d*x]*(b + a*Sinh[c + d*x])^2 - (2*b*(6*a^4 + 5*a^ 
2*b^2 + 2*b^4)*ArcTan[(a - b*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]]*Csch[c + 
 d*x]*(b + a*Sinh[c + d*x])^2)/(-a^2 - b^2)^(5/2)))/(2*a^3*d*(a + b*Csch[c 
 + d*x])^3)
 

Rubi [A] (warning: unable to verify)

Time = 1.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.24, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {3042, 4272, 25, 3042, 4548, 25, 3042, 4407, 26, 3042, 26, 4318, 3042, 3139, 1083, 217}

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.

\(\displaystyle \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+i b \csc (i c+i d x))^3}dx\)

\(\Big \downarrow \) 4272

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {\int \frac {-b^2 \csc (i c+i d x)^2-2 i a b \csc (i c+i d x)+2 \left (a^2+b^2\right )}{(a+i b \csc (i c+i d x))^2}dx}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {-\frac {\int -\frac {2 \left (a^2+b^2\right )^2-a b \left (4 a^2+b^2\right ) \text {csch}(c+d x)}{a+b \text {csch}(c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}}{2 a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 \left (a^2+b^2\right )^2-a b \left (4 a^2+b^2\right ) \text {csch}(c+d x)}{a+b \text {csch}(c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}}{2 a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\int \frac {2 \left (a^2+b^2\right )^2-i a b \left (4 a^2+b^2\right ) \csc (i c+i d x)}{a+i b \csc (i c+i d x)}dx}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4407

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {i b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int -\frac {i \text {csch}(c+d x)}{a+b \text {csch}(c+d x)}dx}{a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {\text {csch}(c+d x)}{a+b \text {csch}(c+d x)}dx}{a}}{a \left (a^2+b^2\right )}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}}{2 a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {i \csc (i c+i d x)}{a+i b \csc (i c+i d x)}dx}{a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {i b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {\csc (i c+i d x)}{a+i b \csc (i c+i d x)}dx}{a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4318

\(\displaystyle \frac {\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {\left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {1}{\frac {a \sinh (c+d x)}{b}+1}dx}{a}}{a \left (a^2+b^2\right )}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}}{2 a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {\left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {1}{1-\frac {i a \sin (i c+i d x)}{b}}dx}{a}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3139

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}+\frac {2 i \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {1}{-\tanh ^2\left (\frac {1}{2} (c+d x)\right )+\frac {2 a \tanh \left (\frac {1}{2} (c+d x)\right )}{b}+1}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a d}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}+\frac {-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}+\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {4 i \left (6 a^4+5 a^2 b^2+2 b^4\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (\frac {a^2}{b^2}+1\right )}d\left (2 i \tanh \left (\frac {1}{2} (c+d x)\right )-\frac {2 i a}{b}\right )}{a d}}{a \left (a^2+b^2\right )}}{2 a \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\frac {2 x \left (a^2+b^2\right )^2}{a}-\frac {2 b \left (6 a^4+5 a^2 b^2+2 b^4\right ) \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {b^2 \left (5 a^2+2 b^2\right ) \coth (c+d x)}{a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))}}{2 a \left (a^2+b^2\right )}-\frac {b^2 \coth (c+d x)}{2 a d \left (a^2+b^2\right ) (a+b \text {csch}(c+d x))^2}\)

Input:

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

Output:

-1/2*(b^2*Coth[c + d*x])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*x])^2) + (((2* 
(a^2 + b^2)^2*x)/a - (2*b*(6*a^4 + 5*a^2*b^2 + 2*b^4)*ArcTanh[(b*Tanh[(c + 
 d*x)/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d))/(a*(a^2 + b^2)) - ( 
b^2*(5*a^2 + 2*b^2)*Coth[c + d*x])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*x])) 
)/(2*a*(a^2 + b^2))
 

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 217

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

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0]
 

rule 4548

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - 
a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 
 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^( 
m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x 
] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, 
 b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(327\) vs. \(2(154)=308\).

Time = 0.45 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {b \,a^{2} \left (4 a^{2}+b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {4 a \left (10 a^{4}-a^{2} b^{2}-2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}+\frac {4 a^{2} b \left (16 a^{2}+7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \,b^{2} \left (5 a^{2}+2 b^{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}}{\left (-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+b \right )^{2}}-\frac {2 \left (6 a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{4}+8 a^{2} b^{2}+4 b^{4}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}}{d}\)	\(328\)
default	\(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{3}}-\frac {2 b \left (\frac {-\frac {b \,a^{2} \left (4 a^{2}+b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {4 a \left (10 a^{4}-a^{2} b^{2}-2 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}+\frac {4 a^{2} b \left (16 a^{2}+7 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}+\frac {4 a \,b^{2} \left (5 a^{2}+2 b^{2}\right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}}{\left (-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+b \right )^{2}}-\frac {2 \left (6 a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (4 a^{4}+8 a^{2} b^{2}+4 b^{4}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}}{d}\)	\(328\)
risch	\(\frac {x}{a^{3}}-\frac {b^{2} \left (-7 a^{3} b \,{\mathrm e}^{3 d x +3 c}-4 a \,b^{3} {\mathrm e}^{3 d x +3 c}+6 a^{4} {\mathrm e}^{2 d x +2 c}-9 \,{\mathrm e}^{2 d x +2 c} a^{2} b^{2}-6 b^{4} {\mathrm e}^{2 d x +2 c}+17 a^{3} b \,{\mathrm e}^{d x +c}+8 b^{3} {\mathrm e}^{d x +c} a -6 a^{4}-3 a^{2} b^{2}\right )}{a^{3} \left (a^{2}+b^{2}\right )^{2} d \left (a \,{\mathrm e}^{2 d x +2 c}+2 b \,{\mathrm e}^{d x +c}-a \right )^{2}}+\frac {3 b a \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} d}+\frac {5 b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}} d a}+\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} d \,a^{3}}-\frac {3 b a \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} d}-\frac {5 b^{3} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}} d a}-\frac {b^{5} \ln \left ({\mathrm e}^{d x +c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} b -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{a \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}} d \,a^{3}}\)	\(619\)

Input:

int(1/(a+csch(d*x+c)*b)^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/a^3*ln(tanh(1/2*d*x+1/2*c)+1)-2/a^ 
3*b*(4*(-1/8*b*a^2*(4*a^2+b^2)/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^3+1 
/8*a*(10*a^4-a^2*b^2-2*b^4)/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)^2+1/8* 
a^2*b*(16*a^2+7*b^2)/(a^4+2*a^2*b^2+b^4)*tanh(1/2*d*x+1/2*c)+1/8*a*b^2*(5* 
a^2+2*b^2)/(a^4+2*a^2*b^2+b^4))/(-tanh(1/2*d*x+1/2*c)^2*b+2*a*tanh(1/2*d*x 
+1/2*c)+b)^2-2*(6*a^4+5*a^2*b^2+2*b^4)/(4*a^4+8*a^2*b^2+4*b^4)/(a^2+b^2)^( 
1/2)*arctanh(1/2*(-2*b*tanh(1/2*d*x+1/2*c)+2*a)/(a^2+b^2)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2094 vs. \(2 (156) = 312\).

Time = 0.16 (sec) , antiderivative size = 2094, normalized size of antiderivative = 12.85 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*csch(d*x+c))^3,x, algorithm="fricas")
 

Output:

1/2*(12*a^6*b^2 + 18*a^4*b^4 + 6*a^2*b^6 + 2*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 
+ a^2*b^6)*d*x*cosh(d*x + c)^4 + 2*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6) 
*d*x*sinh(d*x + c)^4 + 2*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 4*(a^7*b + 3* 
a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*cosh(d*x + c)^3 + 2*(7*a^5*b^3 + 11*a^3* 
b^5 + 4*a*b^7 + 4*(a^8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c 
) + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*sinh(d*x + c)^3 + 2*(a^ 
8 + 3*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x - 2*(6*a^6*b^2 - 3*a^4*b^4 - 15*a 
^2*b^6 - 6*b^8 + 2*(a^8 + a^6*b^2 - 3*a^4*b^4 - 5*a^2*b^6 - 2*b^8)*d*x)*co 
sh(d*x + c)^2 - 2*(6*a^6*b^2 - 3*a^4*b^4 - 15*a^2*b^6 - 6*b^8 - 6*(a^8 + 3 
*a^6*b^2 + 3*a^4*b^4 + a^2*b^6)*d*x*cosh(d*x + c)^2 + 2*(a^8 + a^6*b^2 - 3 
*a^4*b^4 - 5*a^2*b^6 - 2*b^8)*d*x - 3*(7*a^5*b^3 + 11*a^3*b^5 + 4*a*b^7 + 
4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*d*x)*cosh(d*x + c))*sinh(d*x + c 
)^2 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5) 
*cosh(d*x + c)^4 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*sinh(d*x + c)^4 + 4*( 
6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c)^3 + 4*(6*a^5*b^2 + 5*a^3*b^ 
4 + 2*a*b^6 + (6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5)*cosh(d*x + c))*sinh(d*x + 
c)^3 - 2*(6*a^6*b - 7*a^4*b^3 - 8*a^2*b^5 - 4*b^7)*cosh(d*x + c)^2 - 2*(6* 
a^6*b - 7*a^4*b^3 - 8*a^2*b^5 - 4*b^7 - 3*(6*a^6*b + 5*a^4*b^3 + 2*a^2*b^5 
)*cosh(d*x + c)^2 - 6*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c))*sin 
h(d*x + c)^2 - 4*(6*a^5*b^2 + 5*a^3*b^4 + 2*a*b^6)*cosh(d*x + c) - 4*(6...
 

Sympy [F]

\[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=\int \frac {1}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right )^{3}}\, dx \] Input:

integrate(1/(a+b*csch(d*x+c))**3,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (156) = 312\).

Time = 0.13 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.29 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=-\frac {{\left (6 \, a^{4} b + 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \log \left (\frac {a e^{\left (-d x - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {6 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + {\left (17 \, a^{3} b^{3} + 8 \, a b^{5}\right )} e^{\left (-d x - c\right )} - 3 \, {\left (2 \, a^{4} b^{2} - 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - {\left (7 \, a^{3} b^{3} + 4 \, a b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4} + 4 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} e^{\left (-d x - c\right )} - 2 \, {\left (a^{9} - 3 \, a^{5} b^{4} - 2 \, a^{3} b^{6}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 4 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} e^{\left (-3 \, d x - 3 \, c\right )} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {d x + c}{a^{3} d} \] Input:

integrate(1/(a+b*csch(d*x+c))^3,x, algorithm="maxima")
 

Output:

-1/2*(6*a^4*b + 5*a^2*b^3 + 2*b^5)*log((a*e^(-d*x - c) - b - sqrt(a^2 + b^ 
2))/(a*e^(-d*x - c) - b + sqrt(a^2 + b^2)))/((a^7 + 2*a^5*b^2 + a^3*b^4)*s 
qrt(a^2 + b^2)*d) - (6*a^4*b^2 + 3*a^2*b^4 + (17*a^3*b^3 + 8*a*b^5)*e^(-d* 
x - c) - 3*(2*a^4*b^2 - 3*a^2*b^4 - 2*b^6)*e^(-2*d*x - 2*c) - (7*a^3*b^3 + 
 4*a*b^5)*e^(-3*d*x - 3*c))/((a^9 + 2*a^7*b^2 + a^5*b^4 + 4*(a^8*b + 2*a^6 
*b^3 + a^4*b^5)*e^(-d*x - c) - 2*(a^9 - 3*a^5*b^4 - 2*a^3*b^6)*e^(-2*d*x - 
 2*c) - 4*(a^8*b + 2*a^6*b^3 + a^4*b^5)*e^(-3*d*x - 3*c) + (a^9 + 2*a^7*b^ 
2 + a^5*b^4)*e^(-4*d*x - 4*c))*d) + (d*x + c)/(a^3*d)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.80 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx=-\frac {\frac {{\left (6 \, a^{4} b + 5 \, a^{2} b^{3} + 2 \, b^{5}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (7 \, a^{3} b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 4 \, a b^{5} e^{\left (3 \, d x + 3 \, c\right )} - 6 \, a^{4} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 9 \, a^{2} b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b^{6} e^{\left (2 \, d x + 2 \, c\right )} - 17 \, a^{3} b^{3} e^{\left (d x + c\right )} - 8 \, a b^{5} e^{\left (d x + c\right )} + 6 \, a^{4} b^{2} + 3 \, a^{2} b^{4}\right )}}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} {\left (a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, b e^{\left (d x + c\right )} - a\right )}^{2}} - \frac {2 \, {\left (d x + c\right )}}{a^{3}}}{2 \, d} \] Input:

integrate(1/(a+b*csch(d*x+c))^3,x, algorithm="giac")
 

Output:

-1/2*((6*a^4*b + 5*a^2*b^3 + 2*b^5)*log(abs(2*a*e^(d*x + c) + 2*b - 2*sqrt 
(a^2 + b^2))/abs(2*a*e^(d*x + c) + 2*b + 2*sqrt(a^2 + b^2)))/((a^7 + 2*a^5 
*b^2 + a^3*b^4)*sqrt(a^2 + b^2)) - 2*(7*a^3*b^3*e^(3*d*x + 3*c) + 4*a*b^5* 
e^(3*d*x + 3*c) - 6*a^4*b^2*e^(2*d*x + 2*c) + 9*a^2*b^4*e^(2*d*x + 2*c) + 
6*b^6*e^(2*d*x + 2*c) - 17*a^3*b^3*e^(d*x + c) - 8*a*b^5*e^(d*x + c) + 6*a 
^4*b^2 + 3*a^2*b^4)/((a^7 + 2*a^5*b^2 + a^3*b^4)*(a*e^(2*d*x + 2*c) + 2*b* 
e^(d*x + c) - a)^2) - 2*(d*x + c)/a^3)/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 1647, normalized size of antiderivative = 10.10 \[ \int \frac {1}{(a+b \text {csch}(c+d x))^3} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*csch(d*x+c))^3,x)
 

Output:

( - 24*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sq 
rt(a**2 + b**2))*a**6*b*i - 20*e**(4*c + 4*d*x)*sqrt(a**2 + b**2)*atan((e* 
*(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**4*b**3*i - 8*e**(4*c + 4*d*x)* 
sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**2*b* 
*5*i - 96*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i) 
/sqrt(a**2 + b**2))*a**5*b**2*i - 80*e**(3*c + 3*d*x)*sqrt(a**2 + b**2)*at 
an((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**3*b**4*i - 32*e**(3*c + 
3*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))* 
a*b**6*i + 48*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + 
b*i)/sqrt(a**2 + b**2))*a**6*b*i - 56*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*a 
tan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**4*b**3*i - 64*e**(2*c + 
 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2)) 
*a**2*b**5*i - 32*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a* 
i + b*i)/sqrt(a**2 + b**2))*b**7*i + 96*e**(c + d*x)*sqrt(a**2 + b**2)*ata 
n((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**5*b**2*i + 80*e**(c + d*x 
)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/sqrt(a**2 + b**2))*a**3* 
b**4*i + 32*e**(c + d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + b*i)/s 
qrt(a**2 + b**2))*a*b**6*i - 24*sqrt(a**2 + b**2)*atan((e**(c + d*x)*a*i + 
 b*i)/sqrt(a**2 + b**2))*a**6*b*i - 20*sqrt(a**2 + b**2)*atan((e**(c + d*x 
)*a*i + b*i)/sqrt(a**2 + b**2))*a**4*b**3*i - 8*sqrt(a**2 + b**2)*atan(...