3.1.37 \(\int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx\) [37]

3.1.37.1 Optimal result

Integrand size = 12, antiderivative size = 378 \[ \int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx=-\frac {b \csc ^{-1}(a+b x)^3}{a}-\frac {\csc ^{-1}(a+b x)^3}{x}-\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {3 i b \csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 b \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 b \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}-\frac {6 i b \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}}+\frac {6 i b \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )}{a \sqrt {1-a^2}} \]

output

-b*arccsc(b*x+a)^3/a-arccsc(b*x+a)^3/x-3*I*b*arccsc(b*x+a)^2*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)+3*I*b*a 
rccsc(b*x+a)^2*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)-6*b*arccsc(b*x+a)*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)+6*b*arccsc(b*x+a)*polyl 
og(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)-6*I*b*polylog(3,-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)+6*I*b*polylog(3,-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)
 

3.1.37.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.76 \[ \int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx=-\frac {\frac {(a+b x) \csc ^{-1}(a+b x)^3}{x}+\frac {3 i b \left (\csc ^{-1}(a+b x)^2 \log \left (1-\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-\csc ^{-1}(a+b x)^2 \log \left (1+\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )+2 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+2 \operatorname {PolyLog}\left (3,\frac {i a e^{i \csc ^{-1}(a+b x)}}{-1+\sqrt {1-a^2}}\right )-2 \operatorname {PolyLog}\left (3,-\frac {i a e^{i \csc ^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )\right )}{\sqrt {1-a^2}}}{a} \]

input

Integrate[ArcCsc[a + b*x]^3/x^2,x]
 

output

-((((a + b*x)*ArcCsc[a + b*x]^3)/x + ((3*I)*b*(ArcCsc[a + b*x]^2*Log[1 - ( 
I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - ArcCsc[a + b*x]^2*Log[1 
 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])] - (2*I)*ArcCsc[a + b*x 
]*PolyLog[2, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] + (2*I)*Arc 
Csc[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2]) 
] + 2*PolyLog[3, (I*a*E^(I*ArcCsc[a + b*x]))/(-1 + Sqrt[1 - a^2])] - 2*Pol 
yLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])]))/Sqrt[1 - a^2 
])/a)
 

3.1.37.3 Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.01, 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.

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

\(\Big \downarrow \) 5782

\(\displaystyle -b \int \frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^3}{b^2 x^2}d\csc ^{-1}(a+b x)\)

\(\Big \downarrow \) 4927

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

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

input

Int[ArcCsc[a + b*x]^3/x^2,x]
 

output

-(b*(ArcCsc[a + b*x]^3/(b*x) + 3*(ArcCsc[a + b*x]^3/(3*a) + (I*ArcCsc[a + 
b*x]^2*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]^2*Log[1 + (I*a*E^(I*ArcCsc[a + b*x]))/(1 + S 
qrt[1 - a^2])])/(a*Sqrt[1 - a^2]) + (2*ArcCsc[a + b*x]*PolyLog[2, ((-I)*a* 
E^(I*ArcCsc[a + b*x]))/(1 - Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - (2*ArcCsc 
[a + b*x]*PolyLog[2, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 + Sqrt[1 - a^2])])/ 
(a*Sqrt[1 - a^2]) + ((2*I)*PolyLog[3, ((-I)*a*E^(I*ArcCsc[a + b*x]))/(1 - 
Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]) - ((2*I)*PolyLog[3, ((-I)*a*E^(I*ArcCsc 
[a + b*x]))/(1 + Sqrt[1 - a^2])])/(a*Sqrt[1 - a^2]))))
 

3.1.37.3.1 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]
 

3.1.37.4 Maple [F]

\[\int \frac {\operatorname {arccsc}\left (b x +a \right )^{3}}{x^{2}}d x\]

input

int(arccsc(b*x+a)^3/x^2,x)
 

output

int(arccsc(b*x+a)^3/x^2,x)
 

3.1.37.5 Fricas [F]

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

input

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

output

integral(arccsc(b*x + a)^3/x^2, x)
 

3.1.37.6 Sympy [F]

\[ \int \frac {\csc ^{-1}(a+b x)^3}{x^2} \, dx=\int \frac {\operatorname {acsc}^{3}{\left (a + b x \right )}}{x^{2}}\, dx \]

input

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

output

Integral(acsc(a + b*x)**3/x**2, x)
 

3.1.37.7 Maxima [F]

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

input

integrate(arccsc(b*x+a)^3/x^2,x, algorithm="maxima")
 

output

-1/4*(4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3*arctan2(1, s 
qrt(b*x + a + 1)*sqrt(b*x + a - 1))*log(b^2*x^2 + 2*a*b*x + a^2)^2 - 4*x*i 
ntegrate(-3/4*(4*(b^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) 
+ 3*a*b^2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + a^3*arctan 
2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a 
 + 1)*sqrt(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) 
)*b*x - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*log(b*x + a)^2 
+ (4*b*x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b*x*log(b^2*x 
^2 + 2*a*b*x + a^2)^2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 4*(b^3*x^3*ar 
ctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^2*arctan2(1, sqr 
t(b*x + a + 1)*sqrt(b*x + a - 1)) + (a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt 
(b*x + a - 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x - (b 
^3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^2*arcta 
n2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 
 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a 
- 1)) - arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*b*x - a*arctan2(1 
, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*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)) 
/x
 

3.1.37.8 Giac [F]

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

input

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

output

integrate(arccsc(b*x + a)^3/x^2, x)
 

3.1.37.9 Mupad [F(-1)]

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

input

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

output

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