Integrand size = 12, antiderivative size = 366 \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {2 a \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {\log (a+b x)}{3 b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {i a \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {2 i a^3 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4} \]
-a*x/b^3+1/12*(b*x+a)^2/b^4-1/4*a^4*arccsc(b*x+a)^2/b^4+1/4*x^4*arccsc(b*x +a)^2-2*a*arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4-4*a^3 *arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+1/3*ln(b*x+a)/ b^4+3*a^2*ln(b*x+a)/b^4-I*a*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4 -2*I*a^3*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+2*I*a^3*polylog(2, -I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^4+I*a*polylog(2,-I/(b*x+a)-(1-1/(b*x+a )^2)^(1/2))/b^4+1/3*(b*x+a)*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4+3*a^2* (b*x+a)*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4-a*(b*x+a)^2*arccsc(b*x+a)* (1-1/(b*x+a)^2)^(1/2)/b^4+1/6*(b*x+a)^3*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2 )/b^4
Time = 3.95 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.24 \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\frac {-16 \left (6 a-2 \left (1+9 a^2\right ) \csc ^{-1}(a+b x)+3 \left (a+2 a^3\right ) \csc ^{-1}(a+b x)^2\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \left (2-24 a \csc ^{-1}(a+b x)+\left (3+36 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {2 \csc ^{-1}(a+b x) \left (-1+6 a \csc ^{-1}(a+b x)\right ) \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a+b x}-64 \left (1+9 a^2\right ) \left (\log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+\log \left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )\right )+192 \left (a+2 a^3\right ) \left (\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )\right )+2 \left (2+24 a \csc ^{-1}(a+b x)+\left (3+36 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \sec ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-32 (a+b x)^3 \csc ^{-1}(a+b x) \left (1+6 a \csc ^{-1}(a+b x)\right ) \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-16 \left (6 a+2 \left (1+9 a^2\right ) \csc ^{-1}(a+b x)+3 \left (a+2 a^3\right ) \csc ^{-1}(a+b x)^2\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{192 b^4} \]
(-16*(6*a - 2*(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^ 2)*Cot[ArcCsc[a + b*x]/2] + 2*(2 - 24*a*ArcCsc[a + b*x] + (3 + 36*a^2)*Arc Csc[a + b*x]^2)*Csc[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Csc[ArcCsc[ a + b*x]/2]^4 - (2*ArcCsc[a + b*x]*(-1 + 6*a*ArcCsc[a + b*x])*Csc[ArcCsc[a + b*x]/2]^4)/(a + b*x) - 64*(1 + 9*a^2)*(Log[1/((a + b*x)*Sqrt[1 - (a + b *x)^(-2)])] + Log[Sqrt[1 - (a + b*x)^(-2)]]) + 192*(a + 2*a^3)*(ArcCsc[a + b*x]*(Log[1 - E^(I*ArcCsc[a + b*x])] - Log[1 + E^(I*ArcCsc[a + b*x])]) + I*(PolyLog[2, -E^(I*ArcCsc[a + b*x])] - PolyLog[2, E^(I*ArcCsc[a + b*x])]) ) + 2*(2 + 24*a*ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Sec[ArcC sc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a + b*x]/2]^4 - 32*(a + b*x)^3*ArcCsc[a + b*x]*(1 + 6*a*ArcCsc[a + b*x])*Sin[ArcCsc[a + b*x]/2]^4 - 16*(6*a + 2*(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^ 2)*Tan[ArcCsc[a + b*x]/2])/(192*b^4)
Time = 0.59 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
| \(\displaystyle \int x^3 \csc ^{-1}(a+b x)^2 \, dx\) |
| \(\Big \downarrow \) 5782 |
| \(\displaystyle -\frac {\int b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2d\csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)^2d\csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4927 |
| \(\displaystyle -\frac {\frac {1}{2} \int b^4 x^4 \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc ^{-1}(a+b x) \left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^4d\csc ^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 4678 |
| \(\displaystyle -\frac {\frac {1}{2} \int \left (\csc ^{-1}(a+b x) a^4-4 (a+b x) \csc ^{-1}(a+b x) a^3+6 (a+b x)^2 \csc ^{-1}(a+b x) a^2-4 (a+b x)^3 \csc ^{-1}(a+b x) a+(a+b x)^4 \csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)^2}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)^2+\frac {1}{2} \left (\frac {1}{2} a^4 \csc ^{-1}(a+b x)^2+8 a^3 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )-4 i a^3 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+4 i a^3 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+6 a^2 \log \left (\frac {1}{a+b x}\right )-6 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)+4 a \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )-2 i a \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )+2 i a \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )+2 a (a+b x)-\frac {1}{6} (a+b x)^2+\frac {2}{3} \log \left (\frac {1}{a+b x}\right )+2 a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)-\frac {1}{3} (a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)-\frac {2}{3} (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)\right )}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCsc[a + b*x]^2) + (2*a*(a + b*x) - (a + b*x)^2/6 - (2* (a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/3 - 6*a^2*(a + b*x)*Sq rt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x] + 2*a*(a + b*x)^2*Sqrt[1 - (a + b*x )^(-2)]*ArcCsc[a + b*x] - ((a + b*x)^3*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/3 + (a^4*ArcCsc[a + b*x]^2)/2 + 4*a*ArcCsc[a + b*x]*ArcTanh[E^(I*Ar cCsc[a + b*x])] + 8*a^3*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])] + ( 2*Log[(a + b*x)^(-1)])/3 + 6*a^2*Log[(a + b*x)^(-1)] - (2*I)*a*PolyLog[2, -E^(I*ArcCsc[a + b*x])] - (4*I)*a^3*PolyLog[2, -E^(I*ArcCsc[a + b*x])] + ( 2*I)*a*PolyLog[2, E^(I*ArcCsc[a + b*x])] + (4*I)*a^3*PolyLog[2, E^(I*ArcCs c[a + b*x])])/2)/b^4)
3.1.27.3.1 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 = 1.53 (sec) , antiderivative size = 703, normalized size of antiderivative = 1.92
| method | result | size |
| derivativedivides | \(\frac {\frac {2 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-\frac {\ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )}{3}+\frac {\left (b x +a \right )^{2}}{12}-\left (b x +a \right ) a -\operatorname {arccsc}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}+\frac {\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 )^{3}}{6}+\frac {\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 )}{3}-3 i a^{2} \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 ) a^{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 ) a^{3} \operatorname {arccsc}\left (b x +a \right )+2 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{3}-2 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 \,\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 ) a \,\operatorname {arccsc}\left (b x +a \right )+i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\frac {\ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a^{2} \left (b x +a \right )-\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 )^{2}+6 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-3 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right ) a^{2}-\frac {i \operatorname {arccsc}\left (b x +a \right )}{3}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}}{b^{4}}\) | \(703\) |
| default | \(\frac {\frac {2 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-\frac {\ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )}{3}+\frac {\left (b x +a \right )^{2}}{12}-\left (b x +a \right ) a -\operatorname {arccsc}\left (b x +a \right )^{2} a^{3} \left (b x +a \right )+\frac {3 \operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{3}+\frac {\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 )^{3}}{6}+\frac {\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 )}{3}-3 i a^{2} \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 ) a^{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 ) a^{3} \operatorname {arccsc}\left (b x +a \right )+2 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{3}-2 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 \,\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 ) a \,\operatorname {arccsc}\left (b x +a \right )+i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -\frac {\ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+3 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a^{2} \left (b x +a \right )-\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 )^{2}+6 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-3 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-3 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right ) a^{2}-\frac {i \operatorname {arccsc}\left (b x +a \right )}{3}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{4}}{4}}{b^{4}}\) | \(703\) |
1/b^4*(-1/3*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+1/12*(b*x+a)^2-(b*x+a)*a +2/3*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-1/3*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^ (1/2)-1)+2*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))*a^3*arccsc(b*x+a)-2*ln(1+ I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a^3*arccsc(b*x+a)+2*I*polylog(2,-I/(b*x+a )-(1-1/(b*x+a)^2)^(1/2))*a^3-2*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*arccsc(b*x+a)-ln(1+I/(b*x+a) +(1-1/(b*x+a)^2)^(1/2))*a*arccsc(b*x+a)+I*polylog(2,-I/(b*x+a)-(1-1/(b*x+a )^2)^(1/2))*a-I*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a-arccsc(b*x+a) ^2*a^3*(b*x+a)+3/2*arccsc(b*x+a)^2*a^2*(b*x+a)^2-arccsc(b*x+a)^2*a*(b*x+a) ^3+1/6*arccsc(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*(b*x+a)^3-3*I*a^2*arc csc(b*x+a)+1/3*arccsc(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*(b*x+a)+3*arc csc(b*x+a)*(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*a^2*(b*x+a)-arccsc(b*x+a)*(((b* x+a)^2-1)/(b*x+a)^2)^(1/2)*a*(b*x+a)^2+6*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2 ))*a^2-3*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))*a^2-3*ln(I/(b*x+a)+(1-1/(b* x+a)^2)^(1/2)-1)*a^2+1/4*arccsc(b*x+a)^2*(b*x+a)^4-1/3*I*arccsc(b*x+a))
\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]
\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int x^{3} \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]
\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]
1/4*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 1/16*x^4*log(b ^2*x^2 + 2*a*b*x + a^2)^2 + integrate(1/4*(2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - 4*(b^3*x^6 + 3*a*b^2*x^5 + (3*a^2 - 1)*b*x^4 + (a^3 - a)*x^3)*log(b*x + a)^2 + (b^3*x ^6 + 2*a*b^2*x^5 + (a^2 - 1)*b*x^4 + 4*(b^3*x^6 + 3*a*b^2*x^5 + (3*a^2 - 1 )*b*x^4 + (a^3 - a)*x^3)*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)
Exception generated. \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]