3.1.24 \(\int x^3 (a+b \csc ^{-1}(c x))^3 \, dx\) [24]

3.1.24.1 Optimal result

Integrand size = 14, antiderivative size = 207 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {b^3 \sqrt {1-\frac {1}{c^2 x^2}} x}{4 c^3}+\frac {b^2 x^2 \left (a+b \csc ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^4}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )^2}{2 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^3-\frac {b^2 \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{c^4}+\frac {i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 c^4} \]

output

1/4*b^2*x^2*(a+b*arccsc(c*x))/c^2+1/2*I*b*(a+b*arccsc(c*x))^2/c^4+1/4*x^4* 
(a+b*arccsc(c*x))^3-b^2*(a+b*arccsc(c*x))*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2)) 
^2)/c^4+1/2*I*b^3*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)/c^4+1/4*b^3*x*( 
1-1/c^2/x^2)^(1/2)/c^3+1/2*b*x*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/c^3 
+1/4*b*x^3*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/c
 

3.1.24.2 Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.38 \[ \int x^3 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {2 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x+b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x+a b^2 c^2 x^2+a^2 b c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3+a^3 c^4 x^4+b^2 \left (3 a c^4 x^4+b \left (2 i+2 c \sqrt {1-\frac {1}{c^2 x^2}} x+c^3 \sqrt {1-\frac {1}{c^2 x^2}} x^3\right )\right ) \csc ^{-1}(c x)^2+b^3 c^4 x^4 \csc ^{-1}(c x)^3+b \csc ^{-1}(c x) \left (c x \left (b^2 c x+3 a^2 c^3 x^3+2 a b \sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )\right )-4 b^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )-4 a b^2 \log \left (\frac {1}{c x}\right )+2 i b^3 \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{4 c^4} \]

input

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

output

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

3.1.24.3 Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5746, 4910, 3042, 4674, 3042, 4254, 24, 4672, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5746

\(\displaystyle -\frac {\int c^5 \sqrt {1-\frac {1}{c^2 x^2}} x^5 \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)}{c^4}\)

\(\Big \downarrow \) 4910

\(\displaystyle -\frac {\frac {3}{4} b \int c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{4} b \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )^4d\csc ^{-1}(c x)-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 4674

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \int c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)+\frac {1}{3} b^2 \int c^2 x^2d\csc ^{-1}(c x)-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)+\frac {1}{3} b^2 \int \csc \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{3} b^2 \int 1d\left (c \sqrt {1-\frac {1}{c^2 x^2}} x\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \left (2 b \int c \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \left (2 b \int -\left (\left (a+b \csc ^{-1}(c x)\right ) \tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )\right )d\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {3}{4} b \left (\frac {2}{3} \left (-2 b \int \left (a+b \csc ^{-1}(c x)\right ) \tan \left (\csc ^{-1}(c x)+\frac {\pi }{2}\right )d\csc ^{-1}(c x)-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3}{c^4}\)

\(\Big \downarrow \) 4200

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {3}{4} b \left (\frac {2}{3} \left (-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-2 b \left (\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-2 i \int -\frac {e^{2 i \csc ^{-1}(c x)} \left (a+b \csc ^{-1}(c x)\right )}{1-e^{2 i \csc ^{-1}(c x)}}d\csc ^{-1}(c x)\right )\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {3}{4} b \left (\frac {2}{3} \left (-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-2 b \left (2 i \int \frac {e^{2 i \csc ^{-1}(c x)} \left (a+b \csc ^{-1}(c x)\right )}{1-e^{2 i \csc ^{-1}(c x)}}d\csc ^{-1}(c x)+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}\right )\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^4}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {3}{4} b \left (\frac {2}{3} \left (-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-2 b \left (2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{2} i b \int \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}\right )\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^4}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {3}{4} b \left (\frac {2}{3} \left (-c x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-2 b \left (2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{4} b \int e^{-2 i \csc ^{-1}(c x)} \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )de^{2 i \csc ^{-1}(c x)}\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}\right )\right )-\frac {1}{3} b c^2 x^2 \left (a+b \csc ^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{3} b^2 c x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^4}\)

\(\Big \downarrow \) 2838

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

input

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

output

-((-1/4*(c^4*x^4*(a + b*ArcCsc[c*x])^3) + (3*b*(-1/3*(b^2*c*Sqrt[1 - 1/(c^ 
2*x^2)]*x) - (b*c^2*x^2*(a + b*ArcCsc[c*x]))/3 - (c^3*Sqrt[1 - 1/(c^2*x^2) 
]*x^3*(a + b*ArcCsc[c*x])^2)/3 + (2*(-(c*Sqrt[1 - 1/(c^2*x^2)]*x*(a + b*Ar 
cCsc[c*x])^2) - 2*b*(((I/2)*(a + b*ArcCsc[c*x])^2)/b + (2*I)*((I/2)*(a + b 
*ArcCsc[c*x])*Log[1 - E^((2*I)*ArcCsc[c*x])] + (b*PolyLog[2, E^((2*I)*ArcC 
sc[c*x])])/4))))/3))/4)/c^4)
 

3.1.24.3.1 Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 2620

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

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4672

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

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 4910

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x 
] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free 
Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 5746

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- 
(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC 
sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n 
, 0] || LtQ[m, -1])
 

3.1.24.4 Maple [A] (verified)

Time = 1.56 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(417\)
default	\(\frac {\frac {a^{3} c^{4} x^{4}}{4}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(417\)
parts	\(\frac {a^{3} x^{4}}{4}+\frac {b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right )^{3} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{4}+\frac {\operatorname {arccsc}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}{2}+\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {c^{2} x^{2} \operatorname {arccsc}\left (c x \right )}{4}+\frac {x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{4}-\frac {i}{4}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{4}}+\frac {3 a \,b^{2} \left (\frac {\operatorname {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {\operatorname {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {c^{2} x^{2}}{12}+\frac {\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {\ln \left (\frac {1}{c x}\right )}{3}\right )}{c^{4}}+\frac {3 a^{2} b \left (\frac {c^{4} x^{4} \operatorname {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(419\)

input

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

output

1/c^4*(1/4*a^3*c^4*x^4+b^3*(1/4*arccsc(c*x)^3*c^4*x^4+1/4*arccsc(c*x)^2*(( 
c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*x^3+1/2*arccsc(c*x)^2*((c^2*x^2-1)/c^2/x^2)^ 
(1/2)*c*x+1/2*I*arccsc(c*x)^2+1/4*c^2*x^2*arccsc(c*x)+1/4*x*c*((c^2*x^2-1) 
/c^2/x^2)^(1/2)-1/4*I-arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))-arccsc(c 
*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))+I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2)) 
+I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))+3*a*b^2*(1/4*arccsc(c*x)^2*c^4*x 
^4+1/6*arccsc(c*x)*((c^2*x^2-1)/c^2/x^2)^(1/2)*c^3*x^3+1/12*c^2*x^2+1/3*ar 
ccsc(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)^(1/2)-1/3*ln(1/c/x))+3*a^2*b*(1/4*c^4* 
x^4*arccsc(c*x)+1/12*(c^2*x^2-1)*(c^2*x^2+2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c 
/x))
 

3.1.24.5 Fricas [F]

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

input

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

output

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

3.1.24.6 Sympy [F]

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

input

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

output

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

3.1.24.7 Maxima [F]

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

input

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

output

3/4*a*b^2*x^4*arccsc(c*x)^2 + 1/4*a^3*x^4 + 1/4*(3*x^4*arccsc(c*x) + (c^2* 
x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*a^2*b + 1/ 
16*(4*x^4*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*x^4*arctan2(1, sqr 
t(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 - 16*integrate(3/16*(16*c^2*x^5*a 
rctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c)^2 - 16*x^3*arctan2(1, sqrt(c 
*x + 1)*sqrt(c*x - 1))*log(c)^2 + 16*(c^2*x^5*arctan2(1, sqrt(c*x + 1)*sqr 
t(c*x - 1)) - x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2 - (4*x 
^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 - x^3*log(c^2*x^2)^2)*sqrt(c* 
x + 1)*sqrt(c*x - 1) - 4*((4*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*l 
og(c) + c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*x^5 - (4*arctan2(1, s 
qrt(c*x + 1)*sqrt(c*x - 1))*log(c) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1 
)))*x^3 + 4*(c^2*x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - x^3*arctan2 
(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x))*log(c^2*x^2) + 32*(c^2*x^5*arcta 
n2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c) - x^3*arctan2(1, sqrt(c*x + 1)*s 
qrt(c*x - 1))*log(c))*log(x))/(c^2*x^2 - 1), x))*b^3 + 1/4*(2*c^4*x^4*arct 
an2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*c^2*x^2*arctan2(1, sqrt(c*x + 1)*s 
qrt(c*x - 1)) + (c^2*x^2 + 2*log(x^2))*sqrt(c*x + 1)*sqrt(c*x - 1) - 4*arc 
tan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*a*b^2/(sqrt(c*x + 1)*sqrt(c*x - 1)*c 
^4)
 

3.1.24.8 Giac [F]

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

input

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

output

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

3.1.24.9 Mupad [F(-1)]

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

input

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

output

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