3.2.27 \(\int \frac {x^4 (a+b \text {sech}^{-1}(c x))}{(d+e x^2)^3} \, dx\) [127]

3.2.27.1 Optimal result

Integrand size = 21, antiderivative size = 1272 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {-d} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 e^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b c \sqrt {-d} \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{16 e^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}+\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}+\frac {3 \left (a+b \text {sech}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {\sqrt {-d} \left (a+b \text {sech}^{-1}(c x)\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {sech}^{-1}(c x)\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}+\frac {d}{x}\right )}-\frac {3 b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}-\frac {b d \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2} e}-\frac {3 b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}-\frac {b d \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {-1+\frac {1}{c x}}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2} e}+\frac {3 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {sech}^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{16 \sqrt {-d} e^{5/2}} \]

output

3/16*(a+b*arcsech(c*x))*ln(1-c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(- 
d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)/(-d)^(1/2)-3/16*(a+b*arcsech(c 
*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-( 
c^2*d+e)^(1/2)))/e^(5/2)/(-d)^(1/2)+3/16*(a+b*arcsech(c*x))*ln(1-c*(1/c/x+ 
(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^ 
(5/2)/(-d)^(1/2)-3/16*(a+b*arcsech(c*x))*ln(1+c*(1/c/x+(-1+1/c/x)^(1/2)*(1 
+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)/(-d)^(1/2)-3/ 
16*b*polylog(2,-c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^( 
1/2)-(c^2*d+e)^(1/2)))/e^(5/2)/(-d)^(1/2)+3/16*b*polylog(2,c*(1/c/x+(-1+1/ 
c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/e^(5/2)/ 
(-d)^(1/2)-3/16*b*polylog(2,-c*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(- 
d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/e^(5/2)/(-d)^(1/2)+3/16*b*polylog(2,c* 
(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/ 
2)))/e^(5/2)/(-d)^(1/2)-1/8*b*d*arctan((1+1/c/x)^(1/2)*(c*d-(-d)^(1/2)*e^( 
1/2))^(1/2)/(-1+1/c/x)^(1/2)/(c*d+(-d)^(1/2)*e^(1/2))^(1/2))/e/(c*d-(-d)^( 
1/2)*e^(1/2))^(3/2)/(c*d+(-d)^(1/2)*e^(1/2))^(3/2)-1/8*b*d*arctan((1+1/c/x 
)^(1/2)*(c*d+(-d)^(1/2)*e^(1/2))^(1/2)/(-1+1/c/x)^(1/2)/(c*d-(-d)^(1/2)*e^ 
(1/2))^(1/2))/e/(c*d-(-d)^(1/2)*e^(1/2))^(3/2)/(c*d+(-d)^(1/2)*e^(1/2))^(3 
/2)+1/16*(a+b*arcsech(c*x))*(-d)^(1/2)/e^(3/2)/(-d/x+(-d)^(1/2)*e^(1/2))^2 
+3/16*(a+b*arcsech(c*x))/e^2/(-d/x+(-d)^(1/2)*e^(1/2))-1/16*(a+b*arcsec...
 

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

Result contains complex when optimal does not.

Time = 6.11 (sec) , antiderivative size = 2022, normalized size of antiderivative = 1.59 \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]

input

Integrate[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
 

output

(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[ 
(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*(((I/16)*Sqrt[d]*(((-I)*Sqrt 
[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d 
] + Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + Lo 
g[x]/(d*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/ 
(1 + c*x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e] 
*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + 
c*Sqrt[c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*((-I)*Sqrt[ 
d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*((I*Sqrt[ 
e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + 
Sqrt[e]*x)) - ArcSech[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + Log[x]/(d 
*Sqrt[e]) - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c* 
x)]]/(d*Sqrt[e]) + ((2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*(Sqrt[ 
e] + I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[(1 - c*x)/(1 + c*x)] + c*Sqrt[ 
c^2*d + e]*x*Sqrt[(1 - c*x)/(1 + c*x)]))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[ 
e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 + (5*(-(ArcSech[c*x]/(I*Sqrt[d]*Sqrt[ 
e] + e*x)) + (I*(Log[x]/Sqrt[e] - Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x* 
Sqrt[(1 - c*x)/(1 + c*x)]]/Sqrt[e] + Log[((2*I)*Sqrt[e]*(Sqrt[d]*Sqrt[(1 - 
 c*x)/(1 + c*x)]*(1 + c*x) + (Sqrt[d]*Sqrt[e] + I*c^2*d*x)/Sqrt[c^2*d + e] 
))/(I*Sqrt[d] + Sqrt[e]*x)]/Sqrt[c^2*d + e]))/Sqrt[d]))/(16*e^2) + (5*(...
 

3.2.27.3 Rubi [A] (verified)

Time = 2.52 (sec) , antiderivative size = 1336, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6857, 6324, 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.

\(\displaystyle \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\)

\(\Big \downarrow \) 6857

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

\(\Big \downarrow \) 6324

\(\displaystyle -\int \left (-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^3}-\frac {\left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d^3}{8 (-d)^{3/2} e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^3}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{8 e^2 \left (-\frac {d^2}{x^2}-e d\right )}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) d}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \sqrt {-d} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}+\frac {b \sqrt {-d} \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} c}{16 e^{3/2} \left (d c^2+e\right ) \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^2 \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )}+\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\sqrt {-d} \sqrt {e}-\frac {d}{x}\right )^2}-\frac {\sqrt {-d} \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{16 e^{3/2} \left (\frac {d}{x}+\sqrt {-d} \sqrt {e}\right )^2}-\frac {b d \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2} e}-\frac {3 b \arctan \left (\frac {\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}-\frac {b d \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \left (c d-\sqrt {-d} \sqrt {e}\right )^{3/2} \left (c d+\sqrt {-d} \sqrt {e}\right )^{3/2} e}-\frac {3 b \arctan \left (\frac {\sqrt {c d+\sqrt {-d} \sqrt {e}} \sqrt {1+\frac {1}{c x}}}{\sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {\frac {1}{c x}-1}}\right )}{8 \sqrt {c d-\sqrt {-d} \sqrt {e}} \sqrt {c d+\sqrt {-d} \sqrt {e}} e^2}+\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right ) \log \left (\frac {\sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {arccosh}\left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{16 \sqrt {-d} e^{5/2}}\)

input

Int[(x^4*(a + b*ArcSech[c*x]))/(d + e*x^2)^3,x]
 

output

(b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(16*e^(3/2)*(c^2*d + e 
)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*Sqrt[-d]*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/ 
(c*x)])/(16*e^(3/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) + (Sqrt[-d]*(a + 
 b*ArcCosh[1/(c*x)]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (3*(a + b* 
ArcCosh[1/(c*x)]))/(16*e^2*(Sqrt[-d]*Sqrt[e] - d/x)) - (Sqrt[-d]*(a + b*Ar 
cCosh[1/(c*x)]))/(16*e^(3/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (3*(a + b*ArcCo 
sh[1/(c*x)]))/(16*e^2*(Sqrt[-d]*Sqrt[e] + d/x)) - (3*b*ArcTan[(Sqrt[c*d - 
Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 
 + 1/(c*x)])])/(8*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e] 
]*e^2) - (b*d*ArcTan[(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqr 
t[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e] 
)^(3/2)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)*e) - (3*b*ArcTan[(Sqrt[c*d + Sqrt[- 
d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/( 
c*x)])])/(8*Sqrt[c*d - Sqrt[-d]*Sqrt[e]]*Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*e^2) 
 - (b*d*ArcTan[(Sqrt[c*d + Sqrt[-d]*Sqrt[e]]*Sqrt[1 + 1/(c*x)])/(Sqrt[c*d 
- Sqrt[-d]*Sqrt[e]]*Sqrt[-1 + 1/(c*x)])])/(8*(c*d - Sqrt[-d]*Sqrt[e])^(3/2 
)*(c*d + Sqrt[-d]*Sqrt[e])^(3/2)*e) + (3*(a + b*ArcCosh[1/(c*x)])*Log[1 - 
(c*Sqrt[-d]*E^ArcCosh[1/(c*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d] 
*e^(5/2)) - (3*(a + b*ArcCosh[1/(c*x)])*Log[1 + (c*Sqrt[-d]*E^ArcCosh[1/(c 
*x)])/(Sqrt[e] - Sqrt[c^2*d + e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*A...
 

3.2.27.3.1 Defintions of rubi rules used

rule 2009

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

rule 6324

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

rule 6857

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[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]
 

3.2.27.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 110.96 (sec) , antiderivative size = 1960, normalized size of antiderivative = 1.54

method	result	size
parts	\(\text {Expression too large to display}\)	\(1960\)
derivativedivides	\(\text {Expression too large to display}\)	\(1983\)
default	\(\text {Expression too large to display}\)	\(1983\)

input

int(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x,method=_RETURNVERBOSE)
 

output

a*((-5/8/e*x^3-3/8*d/e^2*x)/(e*x^2+d)^2+3/8/e^2/(d*e)^(1/2)*arctan(e*x/(d* 
e)^(1/2)))+b/c^5*(-1/8*x*c^7*(-(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*c^ 
3*d*e*x-(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)*e^2*c^3*x^3+3*d^2*c^4*arc 
sech(c*x)+5*c^4*d*e*arcsech(c*x)*x^2+3*c^2*d*e*arcsech(c*x)+5*e^2*arcsech( 
c*x)*c^2*x^2)/e^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2-3/8*(-(c^2*d-2*(e*(c^2*d+e 
))^(1/2)+2*e)*d)^(1/2)*(c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2*e^2+2*(e*(c^ 
2*d+e))^(1/2)*e)*c^3*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/ 
((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)^2/e^2/d^2-3/8*((c^ 
2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(-c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2* 
d*e+2*e^2-2*(e*(c^2*d+e))^(1/2)*e)*c^3*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)* 
(1+1/c/x)^(1/2))/((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)^2/ 
e^2/d^2+1/2*(-(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d+2*(e*(c^2* 
d+e))^(1/2)+2*e)*c*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/(( 
-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)/e/d^3-1/2*(-(c^2*d-2 
*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d*(e*(c^2*d+e))^(1/2)+2*c^2*d*e+2* 
e^2+2*(e*(c^2*d+e))^(1/2)*e)*c*arctanh(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/ 
x)^(1/2))/((-c^2*d+2*(e*(c^2*d+e))^(1/2)-2*e)*d)^(1/2))/(c^2*d+e)^2/e/d^3+ 
1/2*((c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*d)^(1/2)*(c^2*d-2*(e*(c^2*d+e))^(1/ 
2)+2*e)*c*arctan(c*d*(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))/((c^2*d+2*(e 
*(c^2*d+e))^(1/2)+2*e)*d)^(1/2))/(c^2*d+e)/e/d^3-1/2*((c^2*d+2*(e*(c^2*...
 

3.2.27.5 Fricas [F]

\[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

input

integrate(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="fricas")
 

output

integral((b*x^4*arcsech(c*x) + a*x^4)/(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 
 + d^3), x)
 

3.2.27.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

input

integrate(x**4*(a+b*asech(c*x))/(e*x**2+d)**3,x)
 

output

Timed out
 

3.2.27.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]

input

integrate(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,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
 

3.2.27.8 Giac [F]

\[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]

input

integrate(x^4*(a+b*arcsech(c*x))/(e*x^2+d)^3,x, algorithm="giac")
 

output

integrate((b*arcsech(c*x) + a)*x^4/(e*x^2 + d)^3, x)
 

3.2.27.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \text {sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]

input

int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3,x)
 

output

int((x^4*(a + b*acosh(1/(c*x))))/(d + e*x^2)^3, x)