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

Optimal result

Integrand size = 21, antiderivative size = 704 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3} \] Output:

-1/8*b*c*e*(1-1/c^2/x^2)^(1/2)/d^2/(c^2*d+e)/(e+d/x^2)/x+1/4*e^2*(a+b*arcc 
sc(c*x))/d^3/(e+d/x^2)^2-e*(a+b*arccsc(c*x))/d^3/(e+d/x^2)+1/2*I*(a+b*arcc 
sc(c*x))^2/b/d^3+b*e^(1/2)*arctan((c^2*d+e)^(1/2)/c/e^(1/2)/(1-1/c^2/x^2)^ 
(1/2)/x)/d^3/(c^2*d+e)^(1/2)-1/8*b*e^(1/2)*(c^2*d+2*e)*arctan((c^2*d+e)^(1 
/2)/c/e^(1/2)/(1-1/c^2/x^2)^(1/2)/x)/d^3/(c^2*d+e)^(3/2)-1/2*(a+b*arccsc(c 
*x))*ln(1-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-(c^2*d+e)^(1 
/2)))/d^3-1/2*(a+b*arccsc(c*x))*ln(1+I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^( 
1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arccsc(c*x))*ln(1-I*c*(-d)^( 
1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3-1/2*(a+b*a 
rccsc(c*x))*ln(1+I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2* 
d+e)^(1/2)))/d^3+1/2*I*b*polylog(2,-I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1 
/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3+1/2*I*b*polylog(2,I*c*(-d)^(1/2)*(I/c/ 
x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3+1/2*I*b*polylog(2,-I 
*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3+1 
/2*I*b*polylog(2,I*c*(-d)^(1/2)*(I/c/x+(1-1/c^2/x^2)^(1/2))/(e^(1/2)+(c^2* 
d+e)^(1/2)))/d^3
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2114\) vs. \(2(704)=1408\).

Time = 6.06 (sec) , antiderivative size = 2114, normalized size of antiderivative = 3.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \] Input:

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

Rubi [A] (verified)

Time = 1.84 (sec) , antiderivative size = 764, 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 = {5764, 5232, 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.

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

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}+\frac {e^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 d^3 \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{d^3 \left (\frac {d}{x^2}+e\right )}+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\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} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 x \left (c^2 d+e\right ) \left (\frac {d}{x^2}+e\right )}\)

Input:

Int[(a + b*ArcCsc[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*ArcSin[1/(c*x)]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSin[1/(c*x)] 
))/(d^3*(e + d/x^2)) + ((I/2)*(a + b*ArcSin[1/(c*x)])^2)/(b*d^3) + (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]) - (b*Sqrt[e]*(c^2*d + 2*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)) - ((a + b*ArcSin[1/(c 
*x)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + 
 e])])/(2*d^3) - ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSi 
n[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[1/(c*x 
)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e 
])])/(2*d^3) - ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[ 
1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) + ((I/2)*b*PolyLog[2, ((- 
I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^3 + ( 
(I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^ 
2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x) 
]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]* 
E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3
 

Defintions of rubi rules used

rule 2009

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

rule 5232

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[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 5764

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

Maple [C] (warning: unable to verify)

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

Time = 3.51 (sec) , antiderivative size = 3479, normalized size of antiderivative = 4.94

method	result	size
parts	\(\text {Expression too large to display}\)	\(3479\)
derivativedivides	\(\text {Expression too large to display}\)	\(3553\)
default	\(\text {Expression too large to display}\)	\(3553\)

Input:

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

Output:

a/d^3*ln(x)+1/4*a/d/(e*x^2+d)^2-1/2*a/d^3*ln(e*x^2+d)+1/2*a/d^2/(e*x^2+d)+ 
b*(-(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*e^2*ln(1-d*c^2*(I/c/x+(1-1/c^2/x^2)^ 
(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))*arccsc(c*x)/c^4/d^5/(c^2*d+e)- 
I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*e^2* 
arccsc(c*x)^2/(c^4*d^2+2*c^2*d*e+e^2)/d^5/c^4+1/4*I*(e*(c^2*d+e))^(1/2)/(c 
^2*d+e)^2/e/d*arccsc(c*x)^2*c^4-I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*( 
e*(c^2*d+e))^(1/2)*e+2*e^2)*e*polylog(2,d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^ 
2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))/d^4/(c^4*d^2+2*c^2*d*e+e^2)/c^2+3/2*I 
*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*arccsc(c*x)^2*e/c^2/d^4/(c^2*d+e)+1/4*( 
(e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*c^2*ln( 
1-d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))*a 
rccsc(c*x)/d^2/e/(c^4*d^2+2*c^2*d*e+e^2)+1/8*I*(e*(c^2*d+e))^(1/2)/(c^2*d+ 
e)^2/e/d*polylog(2,d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d+2*(e*(c^2*d+ 
e))^(1/2)+2*e))*c^4-1/4*I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d 
+e))^(1/2)*e+2*e^2)*c^2*arccsc(c*x)^2/d^2/e/(c^4*d^2+2*c^2*d*e+e^2)+((e*(c 
^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*e^2*ln(1-d*c 
^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))*arccsc 
(c*x)/(c^4*d^2+2*c^2*d*e+e^2)/d^5/c^4-2*I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2 
*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e^2)*e*arccsc(c*x)^2/d^4/(c^4*d^2+2*c^2*d*e 
+e^2)/c^2+3/4*I*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*polylog(2,d*c^2*(I/c/...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*arccsc(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(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1) 
)/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)
 

Giac [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{5}+8 \left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{4} e \,x^{2}+4 \left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{3} x^{7}+3 d \,e^{2} x^{5}+3 d^{2} e \,x^{3}+d^{3} x}d x \right ) b \,d^{3} e^{2} x^{4}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,d^{2}-4 \,\mathrm {log}\left (e \,x^{2}+d \right ) a d e \,x^{2}-2 \,\mathrm {log}\left (e \,x^{2}+d \right ) a \,e^{2} x^{4}+4 \,\mathrm {log}\left (x \right ) a \,d^{2}+8 \,\mathrm {log}\left (x \right ) a d e \,x^{2}+4 \,\mathrm {log}\left (x \right ) a \,e^{2} x^{4}+2 a \,d^{2}-a \,e^{2} x^{4}}{4 d^{3} \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(4*int(acsc(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3*x**7),x)*b 
*d**5 + 8*int(acsc(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x**5 + e**3*x** 
7),x)*b*d**4*e*x**2 + 4*int(acsc(c*x)/(d**3*x + 3*d**2*e*x**3 + 3*d*e**2*x 
**5 + e**3*x**7),x)*b*d**3*e**2*x**4 - 2*log(d + e*x**2)*a*d**2 - 4*log(d 
+ e*x**2)*a*d*e*x**2 - 2*log(d + e*x**2)*a*e**2*x**4 + 4*log(x)*a*d**2 + 8 
*log(x)*a*d*e*x**2 + 4*log(x)*a*e**2*x**4 + 2*a*d**2 - a*e**2*x**4)/(4*d** 
3*(d**2 + 2*d*e*x**2 + e**2*x**4))