\(\int \frac {a+b \csc ^{-1}(c x)}{x^2 (d+e x^2)} \, dx\) [102]

Optimal result

Integrand size = 21, antiderivative size = 572 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{d}-\frac {a}{d x}-\frac {b \csc ^{-1}(c x)}{d x}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {\sqrt {e} \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}-\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}}+\frac {i b \sqrt {e} \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{3/2}} \] Output:

-b*c*(1-1/c^2/x^2)^(1/2)/d-a/d/x-b*arccsc(c*x)/d/x-1/2*e^(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)^(3/2)+1/2*e^(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)^(3/2)-1/2*e^(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)^(3/2)+1/2*e^(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)^(3/2)-1/2 
*I*b*e^(1/2)*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)^(3/2)+1/2*I*b*e^(1/2)*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)^(3/2)-1/2*I*b*e^ 
(1/2)*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)^(3/2)+1/2*I*b*e^(1/2)*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)^(3/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1241\) vs. \(2(572)=1144\).

Time = 1.72 (sec) , antiderivative size = 1241, normalized size of antiderivative = 2.17 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*(-((c*Sqr 
t[1 - 1/(c^2*x^2)]*x + ArcCsc[c*x])/(d*x)) + (Sqrt[e]*(Pi^2 - 4*Pi*ArcCsc[ 
c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[ 
2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c 
^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I* 
ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c* 
Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d 
])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c* 
x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCs 
c[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[ 
d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/S 
qrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] 
 + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] 
 + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E 
^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d] 
*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(16*d^(3/2)) 
 - (Sqrt[e]*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 
+ (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(P 
i + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqr 
t[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 ...
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 624, 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.

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

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((b*c*Sqrt[1 - 1/(c^2*x^2)])/d) - a/(d*x) - (b*ArcSin[1/(c*x)])/(d*x) - ( 
Sqrt[e]*(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)^(3/2)) + (Sqrt[e]*(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)^(3/2)) - (Sqrt[e]*(a + b*ArcSin[1/(c*x)])*Log[1 - (I*c*S 
qrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*(-d)^(3/2) 
) + (Sqrt[e]*(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)^(3/2)) - ((I/2)*b*Sqrt[e]*Po 
lyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e 
])])/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/ 
(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(-d)^(3/2) - ((I/2)*b*Sqrt[e]*PolyL 
og[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])] 
)/(-d)^(3/2) + ((I/2)*b*Sqrt[e]*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c* 
x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(-d)^(3/2)
 

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] (verified)

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

Time = 61.64 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.58

method	result	size
parts	\(-\frac {a}{d x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d \sqrt {d e}}-\frac {b c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{d x}+\frac {b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2 d}+\frac {b c e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 d}\)	\(332\)
derivativedivides	\(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2 d}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 d}\right )\)	\(340\)
default	\(c \left (-\frac {a}{d c x}-\frac {a e \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c d \sqrt {d e}}-\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{d}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{d c x}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e \right )}\right )}{2 d}+\frac {b e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c^{2} d \,\textit {\_Z}^{4}+\left (-2 c^{2} d -4 e \right ) \textit {\_Z}^{2}+c^{2} d \right )}{\sum }\frac {\textit {\_R1} \left (i \operatorname {arccsc}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} c^{2} d -c^{2} d -2 e}\right )}{2 d}\right )\)	\(340\)

Input:

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

Output:

-a/d/x-a*e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2))-b*c/d*((c^2*x^2-1)/c^2/x^ 
2)^(1/2)-b*arccsc(c*x)/d/x+1/2*b*c*e/d*sum(1/_R1/(_R1^2*c^2*d-c^2*d-2*e)*( 
I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1- 
1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))+ 
1/2*b*c*e/d*sum(_R1/(_R1^2*c^2*d-c^2*d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-( 
1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=R 
ootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2+c^2*d))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((a+b*arccsc(c*x))/x^2/(e*x^2+d),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(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \left (d+e x^2\right )} \, dx=\frac {-\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a x +\left (\int \frac {\mathit {acsc} \left (c x \right )}{e \,x^{4}+d \,x^{2}}d x \right ) b \,d^{2} x -a d}{d^{2} x} \] Input:

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

Output:

( - sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*x + int(acsc(c*x)/(d*x 
**2 + e*x**4),x)*b*d**2*x - a*d)/(d**2*x)