\(\int x^2 \csc ^{-1}(a+b x)^3 \, dx\) [33]

Optimal result

Integrand size = 12, antiderivative size = 464 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \] Output:

(b*x+a)*arccsc(b*x+a)/b^3-3*I*a*arccsc(b*x+a)^2/b^3-3*a*(b*x+a)*(1-1/(b*x+ 
a)^2)^(1/2)*arccsc(b*x+a)^2/b^3+1/2*(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)*arccsc 
(b*x+a)^2/b^3+1/3*a^3*arccsc(b*x+a)^3/b^3+1/3*x^3*arccsc(b*x+a)^3+arccsc(b 
*x+a)^2*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3+6*a^2*arccsc(b*x+a)^2 
*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3+arctanh((1-1/(b*x+a)^2)^(1/2 
))/b^3+6*a*arccsc(b*x+a)*ln(1-(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))^2)/b^3+I*a 
rccsc(b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3+6*I*a^2*arccsc 
(b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3-6*I*a^2*arccsc(b*x+ 
a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3-I*arccsc(b*x+a)*polylog 
(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3-3*I*a*polylog(2,(I/(b*x+a)+(1-1/( 
b*x+a)^2)^(1/2))^2)/b^3+polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3+6* 
a^2*polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3-polylog(3,I/(b*x+a)+(1 
-1/(b*x+a)^2)^(1/2))/b^3-6*a^2*polylog(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/ 
b^3
 

Mathematica [A] (warning: unable to verify)

Time = 6.42 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.41 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/24*((72*I)*a*ArcCsc[a + b*x]^2 - 12*ArcCsc[a + b*x]*Cot[ArcCsc[a + b*x] 
/2] + 36*a*ArcCsc[a + b*x]^2*Cot[ArcCsc[a + b*x]/2] - 2*ArcCsc[a + b*x]^3* 
Cot[ArcCsc[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Cot[ArcCsc[a + b*x]/2] - 
 3*ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^2 + 6*a*ArcCsc[a + b*x]^3*Csc[ 
ArcCsc[a + b*x]/2]^2 - (ArcCsc[a + b*x]^3*Csc[ArcCsc[a + b*x]/2]^4)/(2*(a 
+ b*x)) + 12*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] + 72*a^2*Arc 
Csc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 12*ArcCsc[a + b*x]^2*Log[1 
 + E^(I*ArcCsc[a + b*x])] - 72*a^2*ArcCsc[a + b*x]^2*Log[1 + E^(I*ArcCsc[a 
 + b*x])] - 144*a*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + 24* 
Log[Tan[ArcCsc[a + b*x]/2]] + (24*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2 
, -E^(I*ArcCsc[a + b*x])] - (24*I)*(1 + 6*a^2)*ArcCsc[a + b*x]*PolyLog[2, 
E^(I*ArcCsc[a + b*x])] + (72*I)*a*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] - 
24*PolyLog[3, -E^(I*ArcCsc[a + b*x])] - 144*a^2*PolyLog[3, -E^(I*ArcCsc[a 
+ b*x])] + 24*PolyLog[3, E^(I*ArcCsc[a + b*x])] + 144*a^2*PolyLog[3, E^(I* 
ArcCsc[a + b*x])] + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a + b*x]/2]^2 + 6*a*Arc 
Csc[a + b*x]^3*Sec[ArcCsc[a + b*x]/2]^2 - 8*(a + b*x)^3*ArcCsc[a + b*x]^3* 
Sin[ArcCsc[a + b*x]/2]^4 - 12*ArcCsc[a + b*x]*Tan[ArcCsc[a + b*x]/2] - 36* 
a*ArcCsc[a + b*x]^2*Tan[ArcCsc[a + b*x]/2] - 2*ArcCsc[a + b*x]^3*Tan[ArcCs 
c[a + b*x]/2] - 12*a^2*ArcCsc[a + b*x]^3*Tan[ArcCsc[a + b*x]/2])/b^3
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 420, normalized size of antiderivative = 0.91, 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, 4678, 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.

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

\(\Big \downarrow \) 5782

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

\(\Big \downarrow \) 4927

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4678

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {1}{3} a^3 \csc ^{-1}(a+b x)^3-6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )+6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )-6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )-\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )-\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )-\frac {1}{3} b^3 x^3 \csc ^{-1}(a+b x)^3+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+3 i a \csc ^{-1}(a+b x)^2-\frac {1}{2} (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2+3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-(a+b x) \csc ^{-1}(a+b x)-6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}\)

Input:

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

Output:

-((-((a + b*x)*ArcCsc[a + b*x]) + (3*I)*a*ArcCsc[a + b*x]^2 + 3*a*(a + b*x 
)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x]^2 - ((a + b*x)^2*Sqrt[1 - (a + 
b*x)^(-2)]*ArcCsc[a + b*x]^2)/2 - (a^3*ArcCsc[a + b*x]^3)/3 - (b^3*x^3*Arc 
Csc[a + b*x]^3)/3 - ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])] - 6*a 
^2*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])] - ArcTanh[Sqrt[1 - (a 
+ b*x)^(-2)]] - 6*a*ArcCsc[a + b*x]*Log[1 - E^((2*I)*ArcCsc[a + b*x])] + I 
*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + (6*I)*a^2*ArcCsc[a + 
 b*x]*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - I*ArcCsc[a + b*x]*PolyLog[2, E^ 
(I*ArcCsc[a + b*x])] - (6*I)*a^2*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a 
+ b*x])] + (3*I)*a*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] - PolyLog[3, -E^( 
I*ArcCsc[a + b*x])] - 6*a^2*PolyLog[3, -E^(I*ArcCsc[a + b*x])] + PolyLog[3 
, E^(I*ArcCsc[a + b*x])] + 6*a^2*PolyLog[3, E^(I*ArcCsc[a + b*x])])/b^3)
 

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 4678

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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 = 1.11 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.61

method	result	size
derivativedivides	\(\frac {-6 i a \operatorname {arccsc}\left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2}+6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )+\operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (6 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}-18 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i a \,\operatorname {arccsc}\left (b x +a \right )+6 b x +6 a \right )}{6}-\operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+6 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}-3 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}}{b^{3}}\)	\(749\)
default	\(\frac {-6 i a \operatorname {arccsc}\left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2}+6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )+\operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{2}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (6 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-6 \operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+2 \operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}-18 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}+18 i a \,\operatorname {arccsc}\left (b x +a \right )+6 b x +6 a \right )}{6}-\operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )+6 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a \,\operatorname {arccsc}\left (b x +a \right )-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}-3 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )^{2}+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}}{b^{3}}\)	\(749\)

Input:

int(x^2*arccsc(b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(-6*I*a*arccsc(b*x+a)^2+1/2*arccsc(b*x+a)^2*ln(1+I/(b*x+a)+(1-1/(b*x 
+a)^2)^(1/2))+6*I*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a^2*arccsc(b* 
x+a)+polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-1/2*arccsc(b*x+a)^2*ln(1- 
I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+1/6*arccsc(b*x+a)*(6*arccsc(b*x+a)^2*a^2* 
(b*x+a)-6*arccsc(b*x+a)^2*a*(b*x+a)^2+2*arccsc(b*x+a)^2*(b*x+a)^3-18*arccs 
c(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*a*(b*x+a)+3*arccsc(b*x+a)*(((b*x+ 
a)^2-1)/(b*x+a)^2)^(1/2)*(b*x+a)^2+18*I*a*arccsc(b*x+a)+6*b*x+6*a)-polylog 
(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+2*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1 
/2))+6*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a*arccsc(b*x+a)+6*ln(1-I/(b*x 
+a)-(1-1/(b*x+a)^2)^(1/2))*a*arccsc(b*x+a)-6*I*polylog(2,-I/(b*x+a)-(1-1/( 
b*x+a)^2)^(1/2))*a^2*arccsc(b*x+a)-6*I*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2 
)^(1/2))*a-I*arccsc(b*x+a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+I*a 
rccsc(b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-6*I*polylog(2,I/(b 
*x+a)+(1-1/(b*x+a)^2)^(1/2))*a+3*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a^2 
*arccsc(b*x+a)^2-3*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))*a^2*arccsc(b*x+a) 
^2+6*polylog(3,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))*a^2-6*polylog(3,I/(b*x+a) 
+(1-1/(b*x+a)^2)^(1/2))*a^2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 1/4*x^3*arctan 
2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))*log(b^2*x^2 + 2*a*b*x + a^2)^2 - 
 integrate(1/4*(12*(b^3*x^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1) 
) + 3*a*b^2*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*a 
rctan2(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^3 + (a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x 
 + a - 1)) - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*x^2)*log(b 
*x + a)^2 - (4*b*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b 
*x^3*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^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^4* 
arctan2(1, sqrt(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^3 + 3*(b^3*x^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 
 3*a*b^2*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arct 
an2(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^3 + (a^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + 
a - 1)) - a*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)))*x^2)*log(b*x 
+ a))*log(b^2*x^2 + 2*a*b*x + a^2))/(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 
- 1)*b*x - a), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(acsc(a + b*x)**3*x**2,x)