\(\int \frac {x}{(a+b \text {csch}(c+d x^2))^2} \, dx\) [31]

Optimal result

Integrand size = 16, antiderivative size = 113 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {x^2}{2 a^2}+\frac {b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac {b^2 \coth \left (c+d x^2\right )}{2 a \left (a^2+b^2\right ) d \left (a+b \text {csch}\left (c+d x^2\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.42 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {\text {csch}\left (c+d x^2\right ) \left (-\frac {a b^2 \coth \left (c+d x^2\right )}{a^2+b^2}+\left (c+d x^2\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right )+\frac {2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )}{\sqrt {-a^2-b^2}}\right ) \left (a+b \text {csch}\left (c+d x^2\right )\right )}{\left (-a^2-b^2\right )^{3/2}}\right ) \left (b+a \sinh \left (c+d x^2\right )\right )}{2 a^2 d \left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \] Input:

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

Output:

(Csch[c + d*x^2]*(-((a*b^2*Coth[c + d*x^2])/(a^2 + b^2)) + (c + d*x^2)*(a 
+ b*Csch[c + d*x^2]) + (2*b*(2*a^2 + b^2)*ArcTan[(a - b*Tanh[(c + d*x^2)/2 
])/Sqrt[-a^2 - b^2]]*(a + b*Csch[c + d*x^2]))/(-a^2 - b^2)^(3/2))*(b + a*S 
inh[c + d*x^2]))/(2*a^2*d*(a + b*Csch[c + d*x^2])^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.77 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {5960, 3042, 4272, 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.

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

\(\Big \downarrow \) 5960

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4272

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4407

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

((((a^2 + b^2)*x^2)/a - (2*b*(2*a^2 + b^2)*ArcTanh[(b*Tanh[(c + d*x^2)/2]) 
/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d))/(a*(a^2 + b^2)) - (b^2*Coth[ 
c + d*x^2])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*x^2])))/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 5960

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.67

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (\frac {\frac {a^{2} \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}}{2 d}\)	\(189\)
default	\(\frac {\frac {\ln \left (1+\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\ln \left (\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {2 b \left (\frac {\frac {a^{2} \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{-\frac {\tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )^{2} b}{2}+a \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+\frac {b}{2}}-\frac {2 \left (2 a^{2}+b^{2}\right ) \operatorname {arctanh}\left (\frac {-2 b \tanh \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+2 a}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (2 a^{2}+2 b^{2}\right ) \sqrt {a^{2}+b^{2}}}\right )}{a^{2}}}{2 d}\)	\(189\)
risch	\(\frac {x^{2}}{2 a^{2}}-\frac {b^{2} \left (-b \,{\mathrm e}^{d \,x^{2}+c}+a \right )}{a^{2} \left (a^{2}+b^{2}\right ) d \left (a \,{\mathrm e}^{2 d \,x^{2}+2 c}+2 b \,{\mathrm e}^{d \,x^{2}+c}-a \right )}+\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}+\frac {b^{3} \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b +a^{4}+2 a^{2} b^{2}+b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} d}-\frac {b^{3} \ln \left ({\mathrm e}^{d \,x^{2}+c}+\frac {\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b -a^{4}-2 a^{2} b^{2}-b^{4}}{a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}} d \,a^{2}}\)	\(346\)

Input:

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

Output:

1/2/d*(1/a^2*ln(1+tanh(1/2*d*x^2+1/2*c))-1/a^2*ln(tanh(1/2*d*x^2+1/2*c)-1) 
-2*b/a^2*((1/2*a^2/(a^2+b^2)*tanh(1/2*d*x^2+1/2*c)+1/2*b*a/(a^2+b^2))/(-1/ 
2*tanh(1/2*d*x^2+1/2*c)^2*b+a*tanh(1/2*d*x^2+1/2*c)+1/2*b)-2*(2*a^2+b^2)/( 
2*a^2+2*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(-2*b*tanh(1/2*d*x^2+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. 711 vs. \(2 (106) = 212\).

Time = 0.09 (sec) , antiderivative size = 711, normalized size of antiderivative = 6.29 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*((a^5 + 2*a^3*b^2 + a*b^4)*d*x^2*cosh(d*x^2 + c)^2 + (a^5 + 2*a^3*b^2 
+ a*b^4)*d*x^2*sinh(d*x^2 + c)^2 - 2*a^3*b^2 - 2*a*b^4 - (a^5 + 2*a^3*b^2 
+ a*b^4)*d*x^2 - (2*a^3*b + a*b^3 - (2*a^3*b + a*b^3)*cosh(d*x^2 + c)^2 - 
(2*a^3*b + a*b^3)*sinh(d*x^2 + c)^2 - 2*(2*a^2*b^2 + b^4)*cosh(d*x^2 + c) 
- 2*(2*a^2*b^2 + b^4 + (2*a^3*b + a*b^3)*cosh(d*x^2 + c))*sinh(d*x^2 + c)) 
*sqrt(a^2 + b^2)*log((a^2*cosh(d*x^2 + c)^2 + a^2*sinh(d*x^2 + c)^2 + 2*a* 
b*cosh(d*x^2 + c) + a^2 + 2*b^2 + 2*(a^2*cosh(d*x^2 + c) + a*b)*sinh(d*x^2 
 + c) + 2*sqrt(a^2 + b^2)*(a*cosh(d*x^2 + c) + a*sinh(d*x^2 + c) + b))/(a* 
cosh(d*x^2 + c)^2 + a*sinh(d*x^2 + c)^2 + 2*b*cosh(d*x^2 + c) + 2*(a*cosh( 
d*x^2 + c) + b)*sinh(d*x^2 + c) - a)) + 2*(a^2*b^3 + b^5 + (a^4*b + 2*a^2* 
b^3 + b^5)*d*x^2)*cosh(d*x^2 + c) + 2*(a^2*b^3 + b^5 + (a^5 + 2*a^3*b^2 + 
a*b^4)*d*x^2*cosh(d*x^2 + c) + (a^4*b + 2*a^2*b^3 + b^5)*d*x^2)*sinh(d*x^2 
 + c))/((a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x^2 + c)^2 + (a^7 + 2*a^5*b^2 
 + a^3*b^4)*d*sinh(d*x^2 + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d*cosh(d 
*x^2 + c) - (a^7 + 2*a^5*b^2 + a^3*b^4)*d + 2*((a^7 + 2*a^5*b^2 + a^3*b^4) 
*d*cosh(d*x^2 + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d)*sinh(d*x^2 + c))
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.77 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {a e^{\left (-d x^{2} - c\right )} - b - \sqrt {a^{2} + b^{2}}}{a e^{\left (-d x^{2} - c\right )} - b + \sqrt {a^{2} + b^{2}}}\right )}{2 \, {\left (a^{4} + a^{2} b^{2}\right )} \sqrt {a^{2} + b^{2}} d} - \frac {b^{3} e^{\left (-d x^{2} - c\right )} + a b^{2}}{{\left (a^{5} + a^{3} b^{2} + 2 \, {\left (a^{4} b + a^{2} b^{3}\right )} e^{\left (-d x^{2} - c\right )} - {\left (a^{5} + a^{3} b^{2}\right )} e^{\left (-2 \, d x^{2} - 2 \, c\right )}\right )} d} + \frac {d x^{2} + c}{2 \, a^{2} d} \] Input:

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

Output:

-1/2*(2*a^2*b + b^3)*log((a*e^(-d*x^2 - c) - b - sqrt(a^2 + b^2))/(a*e^(-d 
*x^2 - c) - b + sqrt(a^2 + b^2)))/((a^4 + a^2*b^2)*sqrt(a^2 + b^2)*d) - (b 
^3*e^(-d*x^2 - c) + a*b^2)/((a^5 + a^3*b^2 + 2*(a^4*b + a^2*b^3)*e^(-d*x^2 
 - c) - (a^5 + a^3*b^2)*e^(-2*d*x^2 - 2*c))*d) + 1/2*(d*x^2 + c)/(a^2*d)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.57 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=-\frac {{\left (2 \, a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a e^{\left (d x^{2} + c\right )} + 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{4} d + a^{2} b^{2} d\right )} \sqrt {a^{2} + b^{2}}} + \frac {b^{3} e^{\left (d x^{2} + c\right )} - a b^{2}}{{\left (a^{4} d + a^{2} b^{2} d\right )} {\left (a e^{\left (2 \, d x^{2} + 2 \, c\right )} + 2 \, b e^{\left (d x^{2} + c\right )} - a\right )}} + \frac {d x^{2} + c}{2 \, a^{2} d} \] Input:

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

Output:

-1/2*(2*a^2*b + b^3)*log(abs(2*a*e^(d*x^2 + c) + 2*b - 2*sqrt(a^2 + b^2))/ 
abs(2*a*e^(d*x^2 + c) + 2*b + 2*sqrt(a^2 + b^2)))/((a^4*d + a^2*b^2*d)*sqr 
t(a^2 + b^2)) + (b^3*e^(d*x^2 + c) - a*b^2)/((a^4*d + a^2*b^2*d)*(a*e^(2*d 
*x^2 + 2*c) + 2*b*e^(d*x^2 + c) - a)) + 1/2*(d*x^2 + c)/(a^2*d)
 

Mupad [B] (verification not implemented)

Time = 3.02 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.57 \[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\frac {x^2}{2\,a^2}-\frac {\frac {b^2}{d\,\left (a^3+a\,b^2\right )}-\frac {b^3\,{\mathrm {e}}^{d\,x^2+c}}{a\,d\,\left (a^3+a\,b^2\right )}}{2\,b\,{\mathrm {e}}^{d\,x^2+c}-a+a\,{\mathrm {e}}^{2\,d\,x^2+2\,c}}-\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2+b^2\right )}{a^3\,\left (a^2+b^2\right )}-\frac {2\,b\,x\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{d\,x^2+c}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{2\,a^2\,d\,{\left (a^2+b^2\right )}^{3/2}}+\frac {b\,\ln \left (\frac {2\,b\,x\,{\mathrm {e}}^{d\,x^2+c}\,\left (2\,a^2+b^2\right )}{a^3\,\left (a^2+b^2\right )}+\frac {2\,b\,x\,\left (2\,a^2+b^2\right )\,\left (a-b\,{\mathrm {e}}^{d\,x^2+c}\right )}{a^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+b^2\right )}{2\,a^2\,d\,{\left (a^2+b^2\right )}^{3/2}} \] Input:

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

Output:

x^2/(2*a^2) - (b^2/(d*(a*b^2 + a^3)) - (b^3*exp(c + d*x^2))/(a*d*(a*b^2 + 
a^3)))/(2*b*exp(c + d*x^2) - a + a*exp(2*c + 2*d*x^2)) - (b*log((2*b*x*exp 
(c + d*x^2)*(2*a^2 + b^2))/(a^3*(a^2 + b^2)) - (2*b*x*(2*a^2 + b^2)*(a - b 
*exp(c + d*x^2)))/(a^3*(a^2 + b^2)^(3/2)))*(2*a^2 + b^2))/(2*a^2*d*(a^2 + 
b^2)^(3/2)) + (b*log((2*b*x*exp(c + d*x^2)*(2*a^2 + b^2))/(a^3*(a^2 + b^2) 
) + (2*b*x*(2*a^2 + b^2)*(a - b*exp(c + d*x^2)))/(a^3*(a^2 + b^2)^(3/2)))* 
(2*a^2 + b^2))/(2*a^2*d*(a^2 + b^2)^(3/2))
 

Reduce [F]

\[ \int \frac {x}{\left (a+b \text {csch}\left (c+d x^2\right )\right )^2} \, dx=\text {too large to display} \] Input:

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

Output:

( - 5*e**(2*c + 2*d*x**2)*sqrt(a**2 + b**2)*atan((e**(c + d*x**2)*a*i + b* 
i)/sqrt(a**2 + b**2))*a**3*b*i - 2*e**(2*c + 2*d*x**2)*sqrt(a**2 + b**2)*a 
tan((e**(c + d*x**2)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i - 10*e**(c + d 
*x**2)*sqrt(a**2 + b**2)*atan((e**(c + d*x**2)*a*i + b*i)/sqrt(a**2 + b**2 
))*a**2*b**2*i - 4*e**(c + d*x**2)*sqrt(a**2 + b**2)*atan((e**(c + d*x**2) 
*a*i + b*i)/sqrt(a**2 + b**2))*b**4*i + 5*sqrt(a**2 + b**2)*atan((e**(c + 
d*x**2)*a*i + b*i)/sqrt(a**2 + b**2))*a**3*b*i + 2*sqrt(a**2 + b**2)*atan( 
(e**(c + d*x**2)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i + 4*e**(2*c + 2*d* 
x**2)*int(x/(e**(4*c + 4*d*x**2)*a**2 + 4*e**(3*c + 3*d*x**2)*a*b - 2*e**( 
2*c + 2*d*x**2)*a**2 + 4*e**(2*c + 2*d*x**2)*b**2 - 4*e**(c + d*x**2)*a*b 
+ a**2),x)*a**7*d + 8*e**(2*c + 2*d*x**2)*int(x/(e**(4*c + 4*d*x**2)*a**2 
+ 4*e**(3*c + 3*d*x**2)*a*b - 2*e**(2*c + 2*d*x**2)*a**2 + 4*e**(2*c + 2*d 
*x**2)*b**2 - 4*e**(c + d*x**2)*a*b + a**2),x)*a**5*b**2*d + 4*e**(2*c + 2 
*d*x**2)*int(x/(e**(4*c + 4*d*x**2)*a**2 + 4*e**(3*c + 3*d*x**2)*a*b - 2*e 
**(2*c + 2*d*x**2)*a**2 + 4*e**(2*c + 2*d*x**2)*b**2 - 4*e**(c + d*x**2)*a 
*b + a**2),x)*a**3*b**4*d + e**(2*c + 2*d*x**2)*log(e**(2*c + 2*d*x**2)*a 
+ 2*e**(c + d*x**2)*b - a)*a**5 + 2*e**(2*c + 2*d*x**2)*log(e**(2*c + 2*d* 
x**2)*a + 2*e**(c + d*x**2)*b - a)*a**3*b**2 + e**(2*c + 2*d*x**2)*log(e** 
(2*c + 2*d*x**2)*a + 2*e**(c + d*x**2)*b - a)*a*b**4 + e**(2*c + 2*d*x**2) 
*a**5 - e**(2*c + 2*d*x**2)*a**3*b**2 - 2*e**(2*c + 2*d*x**2)*a*b**4 + ...