\(\int \csc ^{-1}(a+b x)^3 \, dx\) [35]

Optimal result

Integrand size = 8, antiderivative size = 140 \[ \int \csc ^{-1}(a+b x)^3 \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)^3}{b}+\frac {6 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac {6 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {6 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {6 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac {6 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.16 \[ \int \csc ^{-1}(a+b x)^3 \, dx=\frac {a \csc ^{-1}(a+b x)^3+b x \csc ^{-1}(a+b x)^3-3 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+3 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-6 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+6 i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-6 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5776, 5740, 4245, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5776

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

\(\Big \downarrow \) 5740

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

\(\Big \downarrow \) 4245

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((-((a + b*x)*ArcCsc[a + b*x]^3) + 3*(-2*ArcCsc[a + b*x]^2*ArcTanh[E^(I*A 
rcCsc[a + b*x])] - 2*(I*ArcCsc[a + b*x]*PolyLog[2, E^(I*ArcCsc[a + b*x])] 
- PolyLog[3, E^(I*ArcCsc[a + b*x])]) + 2*(I*ArcCsc[a + b*x]*PolyLog[2, -E^ 
(I*ArcCsc[a + b*x])] - PolyLog[3, -a - b*x])))/b)
 

Defintions of rubi rules used

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4245

Int[Cot[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csc[(a_.) + (b_.)*(x_)^(n_.)]^(p_.) 
*(x_)^(m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csc[a + b*x^n]^p/(b*n*p)), 
 x] + Simp[(m - n + 1)/(b*n*p)   Int[x^(m - n)*Csc[a + b*x^n]^p, x], x] /; 
FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m, n] && EqQ[q, 1]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5740

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-c^(-1)   Su 
bst[Int[(a + b*x)^n*Csc[x]*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, 
c, n}, x] && IGtQ[n, 0]
 

rule 5776

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d 
  Subst[Int[(a + b*ArcCsc[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d}, 
 x] && IGtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.60

method	result	size
derivativedivides	\(\frac {\operatorname {arccsc}\left (b x +a \right )^{3} \left (b x +a \right )+3 \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\)	\(224\)
default	\(\frac {\operatorname {arccsc}\left (b x +a \right )^{3} \left (b x +a \right )+3 \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 \operatorname {polylog}\left (3, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-3 \operatorname {arccsc}\left (b x +a \right )^{2} \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )+6 i \operatorname {arccsc}\left (b x +a \right ) \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )-6 \operatorname {polylog}\left (3, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}\)	\(224\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

x*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^3 - 3/4*x*arctan2(1, sqr 
t(b*x + a + 1)*sqrt(b*x + a - 1))*log(b^2*x^2 + 2*a*b*x + a^2)^2 - integra 
te(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*arctan2(1, sq 
rt(b*x + a + 1)*sqrt(b*x + a - 1)) + (3*a^2*arctan2(1, sqrt(b*x + a + 1)*s 
qrt(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*arctan2(1 
, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 2*a*b^2*x^2*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 + (b^3*x^3* 
arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + 3*a*b^2*x^2*arctan2(1, s 
qrt(b*x + a + 1)*sqrt(b*x + a - 1)) + a^3*arctan2(1, sqrt(b*x + a + 1)*sqr 
t(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^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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