\(\int x \csc ^{-1}(a+b x)^3 \, dx\) [34]

Optimal result

Integrand size = 10, antiderivative size = 264 \[ \int x \csc ^{-1}(a+b x)^3 \, dx=\frac {3 i \csc ^{-1}(a+b x)^2}{2 b^2}+\frac {3 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^2}-\frac {a^2 \csc ^{-1}(a+b x)^3}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)^3-\frac {6 a \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {3 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {6 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}-\frac {6 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {3 i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{2 b^2}-\frac {6 a \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^2}+\frac {6 a \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.19 \[ \int x \csc ^{-1}(a+b x)^3 \, dx=\frac {3 i \csc ^{-1}(a+b x)^2+3 a \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \csc ^{-1}(a+b x)^2+3 b x \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-a^2 \csc ^{-1}(a+b x)^3+b^2 x^2 \csc ^{-1}(a+b x)^3+6 a \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-6 a \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-6 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+12 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-12 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+3 i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-12 a \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+12 a \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{2 b^2} \] Input:

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

Output:

((3*I)*ArcCsc[a + b*x]^2 + 3*a*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b* 
x)^2]*ArcCsc[a + b*x]^2 + 3*b*x*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b 
*x)^2]*ArcCsc[a + b*x]^2 - a^2*ArcCsc[a + b*x]^3 + b^2*x^2*ArcCsc[a + b*x] 
^3 + 6*a*ArcCsc[a + b*x]^2*Log[1 - E^(I*ArcCsc[a + b*x])] - 6*a*ArcCsc[a + 
 b*x]^2*Log[1 + E^(I*ArcCsc[a + b*x])] - 6*ArcCsc[a + b*x]*Log[1 - E^((2*I 
)*ArcCsc[a + b*x])] + (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, -E^(I*ArcCsc[a + 
 b*x])] - (12*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (3* 
I)*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] - 12*a*PolyLog[3, -E^(I*ArcCsc[a 
+ b*x])] + 12*a*PolyLog[3, E^(I*ArcCsc[a + b*x])])/(2*b^2)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5782, 25, 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.

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

\(\Big \downarrow \) 5782

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 4927

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4678

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)^3+\frac {3}{2} \left (\frac {1}{3} a^2 \csc ^{-1}(a+b x)^3+4 a \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )-4 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+4 i a \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )-i \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )+4 a \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-4 a \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-i \csc ^{-1}(a+b x)^2+2 \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )\right )}{b^2}\)

Input:

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

Output:

-((-1/2*(b^2*x^2*ArcCsc[a + b*x]^3) + (3*((-I)*ArcCsc[a + b*x]^2 - (a + b* 
x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x]^2 + (a^2*ArcCsc[a + b*x]^3)/3 
+ 4*a*ArcCsc[a + b*x]^2*ArcTanh[E^(I*ArcCsc[a + b*x])] + 2*ArcCsc[a + b*x] 
*Log[1 - E^((2*I)*ArcCsc[a + b*x])] - (4*I)*a*ArcCsc[a + b*x]*PolyLog[2, - 
E^(I*ArcCsc[a + b*x])] + (4*I)*a*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a 
+ b*x])] - I*PolyLog[2, E^((2*I)*ArcCsc[a + b*x])] + 4*a*PolyLog[3, -E^(I* 
ArcCsc[a + b*x])] - 4*a*PolyLog[3, E^(I*ArcCsc[a + b*x])]))/2)/b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 = 0.91 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.61

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (2 \,\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )-\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}-3 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )+3 i\right )}{2}-3 a \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 a \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 a \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 a \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 i a \,\operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i a \,\operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 i \operatorname {arccsc}\left (b x +a \right )^{2}+3 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}\)	\(425\)
default	\(\frac {-\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (2 \,\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )-\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}-3 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )+3 i\right )}{2}-3 a \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 a \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 a \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \,\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 a \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 i a \,\operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i a \,\operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 i \operatorname {arccsc}\left (b x +a \right )^{2}+3 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+3 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{2}}\)	\(425\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3/8*x^2*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(3/8*(8*(b^3*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) 
 + 3*a*b^2*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*ar 
ctan2(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^2 + (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)*log(b*x 
+ a)^2 - (4*b*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - b*x^ 
2*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^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^3*arc 
tan2(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^2 + 2*(b^3*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3* 
a*b^2*x^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)*sq 
rt(b*x + a - 1)))*b*x^2 + (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)*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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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