3.2.14 \(\int \frac {a+b \text {csch}^{-1}(c x)}{x (d+e x^2)^3} \, dx\) [114]

3.2.14.1 Optimal result

Integrand size = 21, antiderivative size = 657 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e \sqrt {1+\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d-e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b d^3}-\frac {b \left (c^2 d-2 e\right ) \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d-e\right )^{3/2}}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} \sqrt {1+\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d-e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^3}-\frac {b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 d^3} \]

output

1/4*e^2*(a+b*arccsch(c*x))/d^3/(e+d/x^2)^2-e*(a+b*arccsch(c*x))/d^3/(e+d/x 
^2)+1/2*(a+b*arccsch(c*x))^2/b/d^3-1/2*(a+b*arccsch(c*x))*ln(1-c*(1/c/x+(1 
+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(-c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arc 
csch(c*x))*ln(1+c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(-c^2*d+ 
e)^(1/2)))/d^3-1/2*(a+b*arccsch(c*x))*ln(1-c*(1/c/x+(1+1/c^2/x^2)^(1/2))*( 
-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arccsch(c*x))*ln(1+c*(1 
/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))/d^3-1/2*b 
*polylog(2,-c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(-c^2*d+e)^( 
1/2)))/d^3-1/2*b*polylog(2,c*(1/c/x+(1+1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/ 
2)-(-c^2*d+e)^(1/2)))/d^3-1/2*b*polylog(2,-c*(1/c/x+(1+1/c^2/x^2)^(1/2))*( 
-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))/d^3-1/2*b*polylog(2,c*(1/c/x+(1+1/c^ 
2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(-c^2*d+e)^(1/2)))/d^3-1/8*b*(c^2*d-2*e) 
*arctan((c^2*d-e)^(1/2)/c/x/e^(1/2)/(1+1/c^2/x^2)^(1/2))*e^(1/2)/d^3/(c^2* 
d-e)^(3/2)+b*arctan((c^2*d-e)^(1/2)/c/x/e^(1/2)/(1+1/c^2/x^2)^(1/2))*e^(1/ 
2)/d^3/(c^2*d-e)^(1/2)-1/8*b*c*e*(1+1/c^2/x^2)^(1/2)/d^2/(c^2*d-e)/(e+d/x^ 
2)/x
 

3.2.14.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.08 (sec) , antiderivative size = 2081, normalized size of antiderivative = 3.17 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]

input

Integrate[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^3),x]
 

output

a/(4*d*(d + e*x^2)^2) + a/(2*d^2*(d + e*x^2)) + (a*Log[x])/d^3 - (a*Log[d 
+ e*x^2])/(2*d^3) + b*((Sqrt[e]*((I*c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)/(Sq 
rt[d]*(c^2*d - e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/(Sqrt[e]*((-I 
)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d - e 
)*Log[(4*d*Sqrt[c^2*d - e]*Sqrt[e]*(Sqrt[e] + I*c*(c*Sqrt[d] - Sqrt[c^2*d 
- e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] + I*Sqrt[e]*x))])/ 
(d*(c^2*d - e)^(3/2))))/(16*d^2) + (Sqrt[e]*(((-I)*c*Sqrt[e]*Sqrt[1 + 1/(c 
^2*x^2)]*x)/(Sqrt[d]*(c^2*d - e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsch[c*x]/( 
Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSinh[1/(c*x)]/(d*Sqrt[e]) + (I*(2* 
c^2*d - e)*Log[((4*I)*d*Sqrt[c^2*d - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] 
+ Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])*x))/((2*c^2*d - e)*(Sqrt[d] - I*S 
qrt[e]*x))])/(d*(c^2*d - e)^(3/2))))/(16*d^2) - (((5*I)/16)*Sqrt[e]*(-(Arc 
Csch[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[ 
(2*Sqrt[d]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] + I*Sqrt[-(c^2*d) + e]*Sqrt[1 
 + 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) + e]*(I*Sqrt[d] + Sqrt[e]*x))]/Sqrt[-( 
c^2*d) + e]))/Sqrt[d]))/d^(5/2) + (((5*I)/16)*Sqrt[e]*(-(ArcCsch[c*x]/((-I 
)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSinh[1/(c*x)]/Sqrt[e] - Log[(-2*Sqrt[d] 
*Sqrt[e]*(Sqrt[e] + c*(I*c*Sqrt[d] + Sqrt[-(c^2*d) + e]*Sqrt[1 + 1/(c^2*x^ 
2)])*x))/(Sqrt[-(c^2*d) + e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) + e]) 
)/Sqrt[d]))/d^(5/2) - (Pi^2 - (4*I)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 ...
 

3.2.14.3 Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.09, 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.

\(\displaystyle \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6858

\(\displaystyle -\int \frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^5}d\frac {1}{x}\)

\(\Big \downarrow \) 6238

\(\displaystyle -\int \left (\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^3 x}-\frac {2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )^2 x}+\frac {a+b \text {arcsinh}\left (\frac {1}{c x}\right )}{d^2 \left (\frac {d}{x^2}+e\right ) x}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 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 )}{2 d^3}-\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 )}{2 d^3}-\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 )}{2 d^3}-\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 )}{2 d^3}+\frac {e^2 \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 d^3 \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{d^3 \left (\frac {d}{x^2}+e\right )}+\frac {\left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )^2}{2 b d^3}-\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 )}{2 d^3}-\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 )}{2 d^3}-\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 )}{2 d^3}-\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 )}{2 d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}+1}}\right )}{d^3 \sqrt {c^2 d-e}}-\frac {b \sqrt {e} \left (c^2 d-2 e\right ) \arctan \left (\frac {\sqrt {c^2 d-e}}{c \sqrt {e} x \sqrt {\frac {1}{c^2 x^2}+1}}\right )}{8 d^3 \left (c^2 d-e\right )^{3/2}}-\frac {b c e \sqrt {\frac {1}{c^2 x^2}+1}}{8 d^2 x \left (c^2 d-e\right ) \left (\frac {d}{x^2}+e\right )}\)

input

Int[(a + b*ArcCsch[c*x])/(x*(d + e*x^2)^3),x]
 

output

-1/8*(b*c*e*Sqrt[1 + 1/(c^2*x^2)])/(d^2*(c^2*d - e)*(e + d/x^2)*x) + (e^2* 
(a + b*ArcSinh[1/(c*x)]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSinh[1/(c*x 
)]))/(d^3*(e + d/x^2)) + (a + b*ArcSinh[1/(c*x)])^2/(2*b*d^3) - (b*(c^2*d 
- 2*e)*Sqrt[e]*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e]*Sqrt[1 + 1/(c^2*x^2)]*x)] 
)/(8*d^3*(c^2*d - e)^(3/2)) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d - e]/(c*Sqrt[e] 
*Sqrt[1 + 1/(c^2*x^2)]*x)])/(d^3*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]) 
])/(2*d^3) - ((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) - ((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) - ((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) - (b*PolyLog[2, -((c*Sqrt[-d]*E^ 
ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e]))])/(2*d^3) - (b*PolyLog[2 
, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(2*d^3) 
 - (b*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d 
) + e]))])/(2*d^3) - (b*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e 
] + Sqrt[-(c^2*d) + e])])/(2*d^3)
 

3.2.14.3.1 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]
 

3.2.14.4 Maple [F]

\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{x \left (e \,x^{2}+d \right )^{3}}d x\]

input

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

output

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

3.2.14.5 Fricas [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]

input

integrate((a+b*arccsch(c*x))/x/(e*x^2+d)^3,x, algorithm="fricas")
 

output

integral((b*arccsch(c*x) + a)/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x 
), x)
 

3.2.14.6 Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

input

integrate((a+b*acsch(c*x))/x/(e*x**2+d)**3,x)
 

output

Timed out
 

3.2.14.7 Maxima [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]

input

integrate((a+b*arccsch(c*x))/x/(e*x^2+d)^3,x, algorithm="maxima")
 

output

1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d 
)/d^3 + 4*log(x)/d^3) + b*integrate(log(sqrt(1/(c^2*x^2) + 1) + 1/(c*x))/( 
e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)
 

3.2.14.8 Giac [F]

\[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]

input

integrate((a+b*arccsch(c*x))/x/(e*x^2+d)^3,x, algorithm="giac")
 

output

integrate((b*arccsch(c*x) + a)/((e*x^2 + d)^3*x), x)
 

3.2.14.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]

input

int((a + b*asinh(1/(c*x)))/(x*(d + e*x^2)^3),x)
 

output

int((a + b*asinh(1/(c*x)))/(x*(d + e*x^2)^3), x)