\(\int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x^2)^3} \, dx\) [128]

Optimal result

Integrand size = 18, antiderivative size = 1272 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.05 (sec) , antiderivative size = 1813, normalized size of antiderivative = 1.43 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((b*Sqrt[d]*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/((c^2*d + e)*((-I 
)*Sqrt[d] + Sqrt[e]*x)) + (b*Sqrt[d]*Sqrt[e]*Sqrt[(1 - c*x)/(1 + c*x)]*(1 
+ c*x))/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) + (4*a*d^(3/2)*x)/(d + e*x^2 
)^2 + (6*a*Sqrt[d]*x)/(d + e*x^2) + (I*b*d*ArcSech[c*x])/(Sqrt[e]*(Sqrt[d] 
 + I*Sqrt[e]*x)^2) + (I*b*d*ArcSech[c*x])/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x) 
^2) + (3*b*Sqrt[d]*ArcSech[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (3*b*Sqrt[ 
d]*ArcSech[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + (6*a*ArcTan[(Sqrt[e]*x)/Sqrt[ 
d]])/Sqrt[e] + (I*b*(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] + Sq 
rt[e]*x))])/(c^2*d + e)^(3/2) - (I*b*(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))])/(c^2*d + e)^(3/2) - (3*I)*b*(Log[x]/Sqrt[e] - Lo 
g[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]) + (3*I)*b*(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*Sqrt[e]*(I*Sqrt[d]*Sqrt[(1 
- c*x)/(1 + c*x)]*(1 + c*x) + (I*Sqrt[d]*Sqrt[e] + c^2*d*x)/Sqrt[c^2*d ...
 

Rubi [A] (verified)

Time = 4.26 (sec) , antiderivative size = 1336, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6847, 6374, 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.

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

\(\Big \downarrow \) 6847

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

\(\Big \downarrow \) 6374

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 6374

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

rule 6847

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

Maple [C] (warning: unable to verify)

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

Time = 94.87 (sec) , antiderivative size = 1950, normalized size of antiderivative = 1.53

method	result	size
parts	\(\text {Expression too large to display}\)	\(1950\)
derivativedivides	\(\text {Expression too large to display}\)	\(1975\)
default	\(\text {Expression too large to display}\)	\(1975\)

Input:

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

Output:

1/4*a*x/d/(e*x^2+d)^2+3/8*a/d^2*x/(e*x^2+d)+3/8*a/d^2/(d*e)^(1/2)*arctan(x 
*e/(d*e)^(1/2))+b/c*(1/8*c^3*x*((-(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+5*d^2*c^4*ar 
csech(c*x)+3*c^4*d*e*arcsech(c*x)*x^2+5*c^2*d*e*arcsech(c*x)+3*e^2*arcsech 
(c*x)*c^2*x^2)/d^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)+5/8*(-(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)*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))/d^4/(c^2*d+e)^2/c+5/8*((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)*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))/d^4/(c^2*d+e)^2/c-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)*e*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))/d^5/(c^2*d+e)/c^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)*e*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))/d^5/(c^2*d+e)^2/c^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) 
*e*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))/d^5/(c^2*d+e)/c^3+1/2*((c^2*d+2*(e*(c^2*d+e))...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^3} \, dx=\frac {3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,d^{2}+6 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d e \,x^{2}+3 \sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a \,e^{2} x^{4}+8 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{5} e +16 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{4} e^{2} x^{2}+8 \left (\int \frac {\mathit {asech} \left (c x \right )}{e^{3} x^{6}+3 d \,e^{2} x^{4}+3 d^{2} e \,x^{2}+d^{3}}d x \right ) b \,d^{3} e^{3} x^{4}+5 a \,d^{2} e x +3 a d \,e^{2} x^{3}}{8 d^{3} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(3*sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d**2 + 6*sqrt(e)*sqrt(d 
)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d*e*x**2 + 3*sqrt(e)*sqrt(d)*atan((e*x)/ 
(sqrt(e)*sqrt(d)))*a*e**2*x**4 + 8*int(asech(c*x)/(d**3 + 3*d**2*e*x**2 + 
3*d*e**2*x**4 + e**3*x**6),x)*b*d**5*e + 16*int(asech(c*x)/(d**3 + 3*d**2* 
e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**4*e**2*x**2 + 8*int(asech(c*x) 
/(d**3 + 3*d**2*e*x**2 + 3*d*e**2*x**4 + e**3*x**6),x)*b*d**3*e**3*x**4 + 
5*a*d**2*e*x + 3*a*d*e**2*x**3)/(8*d**3*e*(d**2 + 2*d*e*x**2 + e**2*x**4))