\(\int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx\) [32]

Optimal result

Integrand size = 12, antiderivative size = 254 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)^2}{a}-\frac {\csc ^{-1}(a+b x)^2}{x}-\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 i b \csc ^{-1}(a+b x) \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \] Output:

-b*arccsc(b*x+a)^2/a-arccsc(b*x+a)^2/x-2*I*b*arccsc(b*x+a)*ln(1+I*a*(I/(b* 
x+a)+(1-1/(b*x+a)^2)^(1/2))/(1-(-a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)+2*I*b*arc 
csc(b*x+a)*ln(1+I*a*(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/(1+(-a^2+1)^(1/2)))/ 
a/(-a^2+1)^(1/2)-2*b*polylog(2,-I*a*(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/(1-( 
-a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)+2*b*polylog(2,-I*a*(I/(b*x+a)+(1-1/(b*x+a 
)^2)^(1/2))/(1+(-a^2+1)^(1/2)))/a/(-a^2+1)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(802\) vs. \(2(254)=508\).

Time = 2.18 (sec) , antiderivative size = 802, normalized size of antiderivative = 3.16 \[ \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx =\text {Too large to display} \] Input:

Integrate[ArcCsc[a + b*x]^2/x^2,x]
 

Output:

-((b*(((a + b*x)*ArcCsc[a + b*x]^2)/(b*x) + (2*Pi*ArcTan[(a - Tan[ArcCsc[a 
 + b*x]/2])/Sqrt[1 - a^2]])/Sqrt[1 - a^2] + (2*(-2*ArcCos[a^(-1)]*ArcTanh[ 
((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] + (Pi - 2*ArcCsc 
[a + b*x])*ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^ 
2]] + (ArcCos[a^(-1)] + (2*I)*(-ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b* 
x])/4])/Sqrt[-1 + a^2]] + ArcTanh[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4 
])/Sqrt[-1 + a^2]]))*Log[((1/2 + I/2)*Sqrt[-1 + a^2])/(Sqrt[a]*E^((I/2)*Ar 
cCsc[a + b*x])*Sqrt[-((b*x)/(a + b*x))])] + (ArcCos[a^(-1)] + (2*I)*ArcTan 
h[((1 + a)*Cot[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]] - (2*I)*ArcTan 
h[((-1 + a)*Tan[(Pi + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((1/2 - 
I/2)*Sqrt[-1 + a^2]*E^((I/2)*ArcCsc[a + b*x]))/(Sqrt[a]*Sqrt[-((b*x)/(a + 
b*x))])] - (ArcCos[a^(-1)] - (2*I)*ArcTanh[((1 + a)*Cot[(Pi + 2*ArcCsc[a + 
 b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(I + I*a + Sqrt[-1 + a^2])*(-I + 
 Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a^2]*Cot[(Pi + 2 
*ArcCsc[a + b*x])/4]))] - (ArcCos[a^(-1)] + (2*I)*ArcTanh[((1 + a)*Cot[(Pi 
 + 2*ArcCsc[a + b*x])/4])/Sqrt[-1 + a^2]])*Log[((-1 + a)*(-I - I*a + Sqrt[ 
-1 + a^2])*(I + Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a + Sqrt[-1 + a 
^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + I*(-PolyLog[2, ((1 - I*Sqrt[-1 + 
a^2])*(1 - a + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))/(a*(-1 + a 
 + Sqrt[-1 + a^2]*Cot[(Pi + 2*ArcCsc[a + b*x])/4]))] + PolyLog[2, ((1 +...
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5782, 4927, 3042, 4679, 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.

\(\displaystyle \int \frac {\csc ^{-1}(a+b x)^2}{x^2} \, dx\)

\(\Big \downarrow \) 5782

\(\displaystyle -b \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^2 x^2}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 4927

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

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

Input:

Int[ArcCsc[a + b*x]^2/x^2,x]
 

Output:

-(b*(ArcCsc[a + b*x]^2/(b*x) + 2*(ArcCsc[a + b*x]^2/(2*a) + (I*ArcCsc[a + 
b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - 
 a^2]) - (I*ArcCsc[a + b*x]*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[ 
1 - a^2])])/(a*Sqrt[1 - a^2]) + PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/ 
(1 - Sqrt[1 - a^2])]/(a*Sqrt[1 - a^2]) - PolyLog[2, ((-I)*a*E^(I*ArcCsc[a 
+ b*x]))/(1 + Sqrt[1 - a^2])]/(a*Sqrt[1 - a^2]))))
 

Defintions of rubi rules used

rule 2009

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 4927

Int[Cot[(c_.) + (d_.)*(x_)]*Csc[(c_.) + (d_.)*(x_)]*(Csc[(c_.) + (d_.)*(x_) 
]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(e + 
f*x)^m)*((a + b*Csc[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b*d*( 
n + 1)))   Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

rule 5782

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Csc[x]*Cot 
[x]*(d*e - c*f + f*Csc[x])^m, x], x, ArcCsc[c + d*x]], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.24 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i+\sqrt {a^{2}-1}}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i+\sqrt {a^{2}-1}}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\)	\(302\)
default	\(b \left (-\frac {\left (b x +a \right ) \operatorname {arccsc}\left (b x +a \right )^{2}}{a b x}-\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i+\sqrt {a^{2}-1}}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}+\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i+\sqrt {a^{2}-1}}{i+\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}-\frac {2 i \operatorname {dilog}\left (\frac {-\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +i-\sqrt {a^{2}-1}}{i-\sqrt {a^{2}-1}}\right )}{a \sqrt {a^{2}-1}}\right )\)	\(302\)

Input:

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

Output:

b*(-(b*x+a)*arccsc(b*x+a)^2/a/b/x-2/a*arccsc(b*x+a)/(a^2-1)^(1/2)*ln((-(I/ 
(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+I+(a^2-1)^(1/2))/(I+(a^2-1)^(1/2)))+2/a*a 
rccsc(b*x+a)/(a^2-1)^(1/2)*ln((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+I-(a^2 
-1)^(1/2))/(I-(a^2-1)^(1/2)))+2*I/a/(a^2-1)^(1/2)*dilog((-(I/(b*x+a)+(1-1/ 
(b*x+a)^2)^(1/2))*a+I+(a^2-1)^(1/2))/(I+(a^2-1)^(1/2)))-2*I/a/(a^2-1)^(1/2 
)*dilog((-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+I-(a^2-1)^(1/2))/(I-(a^2-1)^ 
(1/2))))
 

Fricas [F]

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

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

Output:

integral(arccsc(b*x + a)^2/x^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/4*(4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 + 4*x*integrate( 
(2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x*arctan2(1, sqrt(b*x + a + 1)*sq 
rt(b*x + a - 1)) + (b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a)*log 
(b*x + a)^2 + (b^3*x^3 + 2*a*b^2*x^2 + (a^2 - 1)*b*x - (b^3*x^3 + 3*a*b^2* 
x^2 + a^3 + (3*a^2 - 1)*b*x - a)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2 
))/(b^3*x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*x^2), x) - log(b 
^2*x^2 + 2*a*b*x + a^2)^2)/x
 

Giac [F]

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

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

Output:

integrate(arccsc(b*x + a)^2/x^2, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(acsc(a + b*x)**2/x**2,x)