3.2.7 \(\int \frac {x^4 (a+b \text {csch}^{-1}(c x))}{(d+e x^2)^2} \, dx\) [107]

3.2.7.1 Optimal result

Integrand size = 21, antiderivative size = 756 \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \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 {c^2 d-e} e^2}+\frac {b \sqrt {d} \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 {c^2 d-e} e^2}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}+\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 \sqrt {-d} \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 e^{5/2}}-\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}-\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}}+\frac {3 b \sqrt {-d} \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{4 e^{5/2}} \]

output

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

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

Result contains complex when optimal does not.

Time = 6.07 (sec) , antiderivative size = 1593, normalized size of antiderivative = 2.11 \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx =\text {Too large to display} \]

input

Integrate[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]
 

output

(a*x)/e^2 + (a*d*x)/(2*e^2*(d + e*x^2)) - (3*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/ 
Sqrt[d]])/(2*e^(5/2)) + b*(-1/4*(d*(-(ArcCsch[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]))/e^2 - (d* 
(-(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]))/(4*e^2) - (((3*I)/32)*Sqrt[d]*(Pi^2 - (4*I 
)*Pi*ArcCsch[c*x] - 8*ArcCsch[c*x]^2 + 32*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]] - 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*Sqr 
t[d])] + 8*ArcCsch[c*x]*Log[1 + (I*(Sqrt[e] + Sqrt[-(c^2*d) + e])*E^ArcCsc 
h[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])...
 

3.2.7.3 Rubi [A] (verified)

Time = 2.65 (sec) , antiderivative size = 816, 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, 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.

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

\(\Big \downarrow \) 6858

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

\(\Big \downarrow \) 6238

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{e^2}+\frac {3 \sqrt {-d} \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {e-c^2 d}}\right ) \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (\frac {\sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right ) \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}+\frac {3 \sqrt {-d} \log \left (1-\frac {c \sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {e-c^2 d}}\right ) \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}-\frac {3 \sqrt {-d} \log \left (\frac {\sqrt {-d} e^{\text {arcsinh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {e-c^2 d}}+1\right ) \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^{5/2}}-\frac {d \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {d \left (a+b \text {arcsinh}\left (\frac {1}{c x}\right )\right )}{4 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {b \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{c e^2}+\frac {b \sqrt {d} \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 {c^2 d-e} e^2}+\frac {b \sqrt {d} \text {arctanh}\left (\frac {d c^2+\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 {c^2 d-e} e^2}-\frac {3 b \sqrt {-d} \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 e^{5/2}}+\frac {3 b \sqrt {-d} \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 e^{5/2}}-\frac {3 b \sqrt {-d} \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 e^{5/2}}+\frac {3 b \sqrt {-d} \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 e^{5/2}}\)

input

Int[(x^4*(a + b*ArcCsch[c*x]))/(d + e*x^2)^2,x]
 

output

-1/4*(d*(a + b*ArcSinh[1/(c*x)]))/(e^2*(Sqrt[-d]*Sqrt[e] - d/x)) + (d*(a + 
 b*ArcSinh[1/(c*x)]))/(4*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) + (x*(a + b*ArcSinh 
[1/(c*x)]))/e^2 + (b*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]])/(c*e^2) + (b*Sqrt[d]* 
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*Sqrt[c^2*d - e]*e^2) + (b*Sqrt[d]*ArcTanh[(c^2*d + (Sq 
rt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d - e]*Sqrt[1 + 1/(c^2*x^2)])])/(4* 
Sqrt[c^2*d - e]*e^2) + (3*Sqrt[-d]*(a + b*ArcSinh[1/(c*x)])*Log[1 - (c*Sqr 
t[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4*e^(5/2)) - ( 
3*Sqrt[-d]*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcSinh[1/(c*x)] 
)/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4*e^(5/2)) + (3*Sqrt[-d]*(a + b*ArcSin 
h[1/(c*x)])*Log[1 - (c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2* 
d) + e])])/(4*e^(5/2)) - (3*Sqrt[-d]*(a + b*ArcSinh[1/(c*x)])*Log[1 + (c*S 
qrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*e^(5/2)) - 
 (3*b*Sqrt[-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] - Sqr 
t[-(c^2*d) + e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^A 
rcSinh[1/(c*x)])/(Sqrt[e] - Sqrt[-(c^2*d) + e])])/(4*e^(5/2)) - (3*b*Sqrt[ 
-d]*PolyLog[2, -((c*Sqrt[-d]*E^ArcSinh[1/(c*x)])/(Sqrt[e] + Sqrt[-(c^2*d) 
+ e]))])/(4*e^(5/2)) + (3*b*Sqrt[-d]*PolyLog[2, (c*Sqrt[-d]*E^ArcSinh[1/(c 
*x)])/(Sqrt[e] + Sqrt[-(c^2*d) + e])])/(4*e^(5/2))
 

3.2.7.3.1 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]
 

3.2.7.4 Maple [F]

\[\int \frac {x^{4} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\left (e \,x^{2}+d \right )^{2}}d x\]

input

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

output

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

3.2.7.5 Fricas [F]

\[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

input

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
 

output

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

3.2.7.6 Sympy [F]

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

input

integrate(x**4*(a+b*acsch(c*x))/(e*x**2+d)**2,x)
 

output

Integral(x**4*(a + b*acsch(c*x))/(d + e*x**2)**2, x)
 

3.2.7.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate(x^4*(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
 

3.2.7.8 Giac [F]

\[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

input

integrate(x^4*(a+b*arccsch(c*x))/(e*x^2+d)^2,x, algorithm="giac")
 

output

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

3.2.7.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \text {csch}^{-1}(c x)\right )}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

input

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^2,x)
 

output

int((x^4*(a + b*asinh(1/(c*x))))/(d + e*x^2)^2, x)