Integrand size = 12, antiderivative size = 464 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\frac {(a+b x) \csc ^{-1}(a+b x)}{b^3}-\frac {3 i a \csc ^{-1}(a+b x)^2}{b^3}-\frac {3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2}{2 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^3+\frac {\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^3}+\frac {6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \]
(b*x+a)*arccsc(b*x+a)/b^3-3*I*a*polylog(2,(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2) )^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*arct anh(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*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^3-3*I*a*arccsc(b*x+a)^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+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*polyl og(3,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3-3*a*(b*x+a)*arccsc(b*x+a)^2*(1-1 /(b*x+a)^2)^(1/2)/b^3+1/2*(b*x+a)^2*arccsc(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)/ b^3
Time = 7.18 (sec) , antiderivative size = 656, normalized size of antiderivative = 1.41 \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=-\frac {72 i a \csc ^{-1}(a+b x)^2-12 \csc ^{-1}(a+b x) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+36 a \csc ^{-1}(a+b x)^2 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 a^2 \csc ^{-1}(a+b x)^3 \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-3 \csc ^{-1}(a+b x)^2 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {\csc ^{-1}(a+b x)^3 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}+12 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-12 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-72 a^2 \csc ^{-1}(a+b x)^2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )-144 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )\right )+24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-24 i \left (1+6 a^2\right ) \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+72 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )-144 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+24 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+144 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+3 \csc ^{-1}(a+b x)^2 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+6 a \csc ^{-1}(a+b x)^3 \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^3 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 \csc ^{-1}(a+b x) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-36 a \csc ^{-1}(a+b x)^2 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-12 a^2 \csc ^{-1}(a+b x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \]
-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
Time = 0.72 (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 |
| \(\displaystyle -\frac {-\frac {1}{3} a^3 \csc ^{-1}(a+b x)^3-6 a^2 \csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )+6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-6 i a^2 \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )-6 a^2 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+6 a^2 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )-\text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )-\csc ^{-1}(a+b x)^2 \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )-\frac {1}{3} b^3 x^3 \csc ^{-1}(a+b x)^3+i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-i \csc ^{-1}(a+b x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+3 i a \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(a+b x)}\right )+3 i a \csc ^{-1}(a+b x)^2-\frac {1}{2} (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2+3 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2-(a+b x) \csc ^{-1}(a+b x)-6 a \csc ^{-1}(a+b x) \log \left (1-e^{2 i \csc ^{-1}(a+b x)}\right )}{b^3}\) |
-((-((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)
3.1.33.3.1 Defintions of rubi rules used
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]
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]
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]
Time = 1.49 (sec) , antiderivative size = 749, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {-6 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\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}-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 )+\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 )+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 a \operatorname {arccsc}\left (b x +a \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 )-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}+\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}+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 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -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 \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\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}-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 )+\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 )+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 a \operatorname {arccsc}\left (b x +a \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 )-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}+\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}+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 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -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}}\) | \(749\) |
1/b^3*(-6*I*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))*a-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))-I*arccsc(b*x+ a)*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))+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)+I*arccsc(b*x+a)*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2 )^(1/2))-6*I*a*arccsc(b*x+a)^2+6*I*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/ 2))*a^2*arccsc(b*x+a)-3*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))*a^2*arccsc(b *x+a)^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*arccsc(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)+3*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2) ^(1/2))*a^2*arccsc(b*x+a)^2-6*I*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2)) *a-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)
\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int x^{2} \operatorname {acsc}^{3}{\left (a + b x \right )}\, dx \]
\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
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)
\[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x^2 \csc ^{-1}(a+b x)^3 \, dx=\int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^3 \,d x \]