\(\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x^2)^2} \, dx\) [106]

Optimal result

Integrand size = 21, antiderivative size = 566 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx=-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{2 d^2 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^2}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 d^2 \sqrt {c^2 d+e}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^2} \] Output:

-1/2*e*(a+b*arccsc(c*x))/d^2/(e+d/x^2)+1/2*I*(a+b*arccsc(c*x))^2/b/d^2+1/2 
*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^2/(c^ 
2*d+e)^(1/2)-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^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^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^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^2+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^2+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^2+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^2+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^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. \(1408\) vs. \(2(566)=1132\).

Time = 1.14 (sec) , antiderivative size = 1408, normalized size of antiderivative = 2.49 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.10, 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.

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

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/2*(e*(a + b*ArcSin[1/(c*x)]))/(d^2*(e + d/x^2)) + ((I/2)*(a + b*ArcSin[ 
1/(c*x)])^2)/(b*d^2) + (b*Sqrt[e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 
 - 1/(c^2*x^2)]*x)])/(2*d^2*Sqrt[c^2*d + e]) - ((a + b*ArcSin[1/(c*x)])*Lo 
g[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/( 
2*d^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^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^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^2) + ((I/2)*b*PolyLog[2, ((-I)*c*Sqr 
t[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^2 + ((I/2)*b* 
PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e] 
)])/d^2 + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqr 
t[e] + Sqrt[c^2*d + e])])/d^2 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*Arc 
Sin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^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] (warning: unable to verify)

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

Time = 3.54 (sec) , antiderivative size = 2071, normalized size of antiderivative = 3.66

method	result	size
parts	\(\text {Expression too large to display}\)	\(2071\)
derivativedivides	\(\text {Expression too large to display}\)	\(2120\)
default	\(\text {Expression too large to display}\)	\(2120\)

Input:

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

Output:

a/d^2*ln(x)-1/2*a/d^2*ln(e*x^2+d)+1/2*a/d/(e*x^2+d)+b*(-(c^2*d+2*(e*(c^2*d 
+e))^(1/2)+2*e)*e*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)/d^4/c^4+1/2*I*(c^2*d+2*(e*(c^2*d+e))^(1/2) 
+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))*e/d^4/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)*arccsc(c*x)^2/c^2/d^3/(c^2*d+e)+((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)*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^2/d^3/(c^2*d+ 
e)-1/8*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)*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^2*d+e)/d^2/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)*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^2*d+e)/d^2/e+I*(c^2*d+2*(e*(c^2*d 
+e))^(1/2)+2*e)*arccsc(c*x)^2*e/d^4/c^4-1/2*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)*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^2/d^3/(c^2*d+e)-1/2*x^2*c 
^2*arccsc(c*x)*e/(c^2*e*x^2+c^2*d)/d^2-1/4*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)*arccsc(c*x)^2/(c^2*d+e)/d^2/e-1/2*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*pol 
ylog(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)...
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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