\(\int \frac {a+b \csc ^{-1}(c x)}{x (d+e x^2)} \, dx\) [101]

Optimal result

Integrand size = 21, antiderivative size = 479 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1089\) vs. \(2(479)=958\).

Time = 0.35 (sec) , antiderivative size = 1089, normalized size of antiderivative = 2.27 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*(I*b*Pi^2 - (4*I)*b*Pi*ArcCsc[c*x] + (4*I)*b*ArcCsc[c*x]^2 - (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]] - (16*I)*b*ArcSi 
n[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 
[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]))] - 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*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 
*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*ArcCsc[c...
 

Rubi [A] (verified)

Time = 1.41 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.11, 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.

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

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5232

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((I/2)*(a + b*ArcSin[1/(c*x)])^2)/(b*d) - ((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) 
- ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(S 
qrt[e] - Sqrt[c^2*d + e])])/(2*d) - ((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) - ((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) + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*Ar 
cSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d + ((I/2)*b*PolyLog[2, (I*c 
*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d + ((I/2)* 
b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d 
 + e])])/d + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqr 
t[e] + Sqrt[c^2*d + e])])/d
 

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 = 2.69 (sec) , antiderivative size = 1934, normalized size of antiderivative = 4.04

method	result	size
parts	\(\text {Expression too large to display}\)	\(1934\)
derivativedivides	\(\text {Expression too large to display}\)	\(1961\)
default	\(\text {Expression too large to display}\)	\(1961\)

Input:

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

Output:

a/d*ln(x)-1/2*a/d*ln(e*x^2+d)+b*(-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/(c^2*d+e)/d^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/ 
c^4/(c^2*d+e)/d^3+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^3/c^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))*arc 
csc(c*x)/c^2/(c^2*d+e)/d^2+1/2*I*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*arccsc( 
c*x)^2/c^2/d^2+1/2*I*(e*(c^2*d+e))^(1/2)/d/(c^2*d+e)*arccsc(c*x)^2-1/4*(e* 
(c^2*d+e))^(1/2)/e/(c^2*d+e)*c^2*arccsc(c*x)*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))+1/2*I/d*arccsc(c*x)^2-1/2*( 
c^2*d+2*(e*(c^2*d+e))^(1/2)+2*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)/c^2/d^2+I*(c^2*d+2*(e*(c^2* 
d+e))^(1/2)+2*e)*arccsc(c*x)^2*e/d^3/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/(c^2*d+e)/d^2+1/2*I/d* 
sum((_R1^2*c^2*d-2*c^2*d-4*e)/(_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))+1/8*I*(e*(c^2*d+e)) 
^(1/2)/e/(c^2*d+e)*polylog(2,d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d...
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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