\(\int \frac {x^2 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^2} \, dx\) [109]

Optimal result

Integrand size = 21, antiderivative size = 719 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {csch}^{-1}(c x)}{4 e \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {b \text {arctanh}\left (\frac {c^2 d-\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} \sqrt {c^2 d-e} e}-\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {\sqrt {-d} \sqrt {e}}{x}}{c \sqrt {d} \sqrt {c^2 d-e} \sqrt {1+\frac {1}{c^2 x^2}}}\right )}{4 \sqrt {d} \sqrt {c^2 d-e} e}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.64 (sec) , antiderivative size = 1442, normalized size of antiderivative = 2.01 \[ \int \frac {x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] 
+ b*((2*ArcCsch[c*x])/(I*Sqrt[d] - Sqrt[e]*x) - (2*ArcCsch[c*x])/(I*Sqrt[d 
] + Sqrt[e]*x) + ((8*I)*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcT 
an[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) 
+ e]])/Sqrt[d] + ((8*I)*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcT 
an[((c*Sqrt[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) 
+ e]])/Sqrt[d] - (Pi*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[ 
c*x])/(c*Sqrt[d])])/Sqrt[d] + ((2*I)*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + S 
qrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (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[d] + (Pi*Log[1 + (I*(-Sqrt[e] + Sqrt 
[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] - ((2*I)*ArcCsch[c*x 
]*Log[1 + (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])]) 
/Sqrt[d] + (4*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (I*(-S 
qrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqrt[d] + (Pi*L 
og[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])])/Sqr 
t[d] - ((2*I)*ArcCsch[c*x]*Log[1 - (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^Arc 
Csch[c*x])/(c*Sqrt[d])])/Sqrt[d] - (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*Sqr 
t[d])])/Sqrt[d] - (Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCs...
 

Rubi [A] (verified)

Time = 1.65 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6858, 6208, 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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6208

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

\(\Big \downarrow \) 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 \sqrt {-d} e^{3/2}}-\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 \sqrt {-d} e^{3/2}}+\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 \sqrt {-d} e^{3/2}}-\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 \sqrt {-d} e^{3/2}}+\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{4 e \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{4 e \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 \sqrt {-d} e^{3/2}}+\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 \sqrt {-d} e^{3/2}}-\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 \sqrt {-d} e^{3/2}}+\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 \sqrt {-d} e^{3/2}}-\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 \sqrt {d} e \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 \sqrt {d} e \sqrt {c^2 d-e}}\)

Input:

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

Output:

(a + b*ArcSinh[1/(c*x)])/(4*e*(Sqrt[-d]*Sqrt[e] - d/x)) - (a + b*ArcSinh[1 
/(c*x)])/(4*e*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcTanh[(c^2*d - (Sqrt[-d]*Sq 
rt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[d]*S 
qrt[c^2*d - e]*e) - (b*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*S 
qrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4*Sqrt[d]*Sqrt[c^2*d - e]*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*Sqrt[-d]*e^(3/2)) - ((a + b*ArcSinh[1/(c*x)])*Lo 
g[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4* 
Sqrt[-d]*e^(3/2)) + ((a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSin 
h[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)) - ((a + 
b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqr 
t[-(c^2*d) + e])])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^Ar 
cSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(4*Sqrt[-d]*e^(3/2)) + ( 
b*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e] 
)])/(4*Sqrt[-d]*e^(3/2)) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)]) 
/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(4*Sqrt[-d]*e^(3/2)) + (b*PolyLog[2, (c 
*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*Sqrt[-d] 
*e^(3/2))
 

Defintions of rubi rules used

rule 2009

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

rule 6208

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

rule 6858

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^2*(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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