\(\int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x^2)^2} \, dx\) [110]

Optimal result

Integrand size = 18, antiderivative size = 713 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=-\frac {a+b \text {csch}^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \text {csch}^{-1}(c x)}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d-e}}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 d^{3/2} \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 (-d)^{3/2} \sqrt {e}} \] Output:

-1/4*(a+b*arccsch(c*x))/d/((-d)^(1/2)*e^(1/2)-d/x)+1/4*(a+b*arccsch(c*x))/ 
d/((-d)^(1/2)*e^(1/2)+d/x)+1/4*b*arctanh((c^2*d-(-d)^(1/2)*e^(1/2)/x)/c/d^ 
(1/2)/(c^2*d-e)^(1/2)/(1+1/c^2/x^2)^(1/2))/d^(3/2)/(c^2*d-e)^(1/2)+1/4*b*a 
rctanh((c^2*d+(-d)^(1/2)*e^(1/2)/x)/c/d^(1/2)/(c^2*d-e)^(1/2)/(1+1/c^2/x^2 
)^(1/2))/d^(3/2)/(c^2*d-e)^(1/2)-1/4*(a+b*arccsch(c*x))*ln(1-c*(-d)^(1/2)* 
(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2) 
+1/4*(a+b*arccsch(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^( 
1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4*(a+b*arccsch(c*x))*ln(1-c*( 
-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/ 
2)/e^(1/2)+1/4*(a+b*arccsch(c*x))*ln(1+c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^( 
1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)+1/4*b*polylog(2,-c*(- 
d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)-(-c^2*d+e)^(1/2)))/(-d)^(3/2 
)/e^(1/2)-1/4*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2 
)-(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)+1/4*b*polylog(2,-c*(-d)^(1/2)*(1/c 
/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^(1/2)))/(-d)^(3/2)/e^(1/2)-1/4 
*b*polylog(2,c*(-d)^(1/2)*(1/c/x+(1+1/c^2/x^2)^(1/2))/(e^(1/2)+(-c^2*d+e)^ 
(1/2)))/(-d)^(3/2)/e^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.84 (sec) , antiderivative size = 1437, normalized size of antiderivative = 2.02 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) 
 + (b*((2*Sqrt[d]*ArcCsch[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (2*Sqrt[d]* 
ArcCsch[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]/( 
c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch 
[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[e] + ((8*I)*ArcSin[Sqrt[1 - Sqrt[e]/( 
c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch 
[c*x])/4])/Sqrt[-(c^2*d) + e]])/Sqrt[e] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[ 
-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + ((2*I)*ArcCsch[c*x] 
*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/ 
Sqrt[e] - (4*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(-Sq 
rt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (Pi*Lo 
g[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqr 
t[e] - ((2*I)*ArcCsch[c*x]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Ar 
cCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] + (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d]) 
]/Sqrt[2]]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*S 
qrt[d])])/Sqrt[e] + (Pi*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsc 
h[c*x])/(c*Sqrt[d])])/Sqrt[e] - ((2*I)*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + 
Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (4*ArcSin[Sqrt 
[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e 
])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[e] - (Pi*Log[1 + (I*(Sqrt[e] + Sq...
 

Rubi [A] (verified)

Time = 2.44 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6848, 6238, 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.

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

\(\Big \downarrow \) 6848

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

\(\Big \downarrow \) 6238

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{4 d \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{4 d \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{4 (-d)^{3/2} \sqrt {e}}+\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 d^{3/2} \sqrt {c^2 d-e}}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {c^2 d-e}}\right )}{4 d^{3/2} \sqrt {c^2 d-e}}\)

Input:

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

Output:

-1/4*(a + b*ArcSinh[1/(c*x)])/(d*(Sqrt[-d]*Sqrt[e] - d/x)) + (a + b*ArcSin 
h[1/(c*x)])/(4*d*(Sqrt[-d]*Sqrt[e] + d/x)) + (b*ArcTanh[(c^2*d - (Sqrt[-d] 
*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2 
)*Sqrt[c^2*d - e]) + (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]* 
Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*d^(3/2)*Sqrt[c^2*d - e]) - ((a 
 + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - 
Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ((a + b*ArcSinh[1/(c*x)])*L 
og[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4 
*(-d)^(3/2)*Sqrt[e]) - ((a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^Arc 
Sinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + ( 
(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] 
+ Sqrt[-(c^2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[- 
d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sqr 
t[e]) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^ 
2*d) + e])])/(4*(-d)^(3/2)*Sqrt[e]) + (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSin 
h[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(4*(-d)^(3/2)*Sqrt[e]) - (b* 
PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])] 
)/(4*(-d)^(3/2)*Sqrt[e])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6238

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e 
_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, 
 (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[e, c^ 
2*d] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
 

rule 6848

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_)^2)^(p_.), 
 x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSinh[x/c])^n/x^(2*(p + 1) 
)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[p 
]
 

Maple [F]

\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{\left (x^{2} e +d \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

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

Output:

integral((b*arccsch(c*x) + a)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
 

Sympy [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{\left (d + e x^{2}\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:

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

Output:

integrate((b*arccsch(c*x) + a)/(e*x^2 + d)^2, x)
 

Mupad [F(-1)]

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

int((a + b*asinh(1/(c*x)))/(d + e*x^2)^2,x)
 

Output:

int((a + b*asinh(1/(c*x)))/(d + e*x^2)^2, x)
 

Reduce [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{\left (d+e x^2\right )^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}+2 \left (\int \frac {\mathit {acsch} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{3} e +2 \left (\int \frac {\mathit {acsch} \left (c x \right )}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{2} e^{2} x^{2}+a d e x}{2 d^{2} e \left (e \,x^{2}+d \right )} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d + sqrt(e)*sqrt(d)*atan( 
(e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int(acsch(c*x)/(d**2 + 2*d*e*x**2 + 
e**2*x**4),x)*b*d**3*e + 2*int(acsch(c*x)/(d**2 + 2*d*e*x**2 + e**2*x**4), 
x)*b*d**2*e**2*x**2 + a*d*e*x)/(2*d**2*e*(d + e*x**2))