\(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x^2)} \, dx\) [103]

Optimal result

Integrand size = 21, antiderivative size = 518 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {b c \sqrt {1+\frac {1}{c^2 x^2}}}{d}-\frac {a}{d x}-\frac {b \text {csch}^{-1}(c x)}{d x}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 (-d)^{3/2}} \] Output:

b*c*(1+1/c^2/x^2)^(1/2)/d-a/d/x-b*arccsch(c*x)/d/x+1/2*e^(1/2)*(a+b*arccsc 
h(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)-1/2*e^(1/2)*(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)+1/2*e^(1/2)*( 
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)-1/2*e^(1/2)*(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)-1/2 
*b*e^(1/2)*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)+1/2*b*e^(1/2)*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)-1/2*b*e^(1/2)*po 
lylog(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)+1/2*b*e^(1/2)*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)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.66 (sec) , antiderivative size = 1211, normalized size of antiderivative = 2.34 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*((c*Sqrt[ 
1 + 1/(c^2*x^2)] - ArcCsch[c*x]/x)/d - ((I/16)*Sqrt[e]*(Pi^2 - (4*I)*Pi*Ar 
cCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*ArcSin[Sqrt[1 + Sqrt[e]/(c*Sqrt[d])]/Sq 
rt[2]]*ArcTan[((c*Sqrt[d] - Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqr 
t[-(c^2*d) + e]] - 8*ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi* 
Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 
8*ArcCsch[c*x]*Log[1 - (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/ 
(c*Sqrt[d])] + (16*I)*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])] + (4*I)* 
Pi*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] 
+ 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x]) 
/(c*Sqrt[d])] - (16*I)*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])] - (4*I)* 
Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*PolyLog[2, E^(-2*ArcCsch[c*x])] + 8*Po 
lyLog[2, (I*(-Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt[d])] + 
 8*PolyLog[2, ((-I)*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsch[c*x])/(c*Sqrt 
[d])]))/d^(3/2) + ((I/16)*Sqrt[e]*(Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsc 
h[c*x]^2 - 32*ArcSin[Sqrt[1 - Sqrt[e]/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((c*Sqr 
t[d] + Sqrt[e])*Cot[(Pi + (2*I)*ArcCsch[c*x])/4])/Sqrt[-(c^2*d) + e]] - 8* 
ArcCsch[c*x]*Log[1 - E^(-2*ArcCsch[c*x])] + (4*I)*Pi*Log[1 + (I*(-Sqrt[...
 

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6858, 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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6238

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b*c*Sqrt[1 + 1/(c^2*x^2)])/d - a/(d*x) - (b*ArcSinh[1/(c*x)])/(d*x) + (Sq 
rt[e]*(a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sq 
rt[e] - Sqrt[-(c^2*d) + e])])/(2*(-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])])/(2*(-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])])/(2*(-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])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -(( 
c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(2*(-d)^( 
3/2)) + (b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - S 
qrt[-(c^2*d) + e])])/(2*(-d)^(3/2)) - (b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]* 
E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e]))])/(2*(-d)^(3/2)) + (b* 
Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d 
) + e])])/(2*(-d)^(3/2))
 

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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