\(\int x^2 \csc ^{-1}(a+b x)^2 \, dx\) [28]

Optimal result

Integrand size = 12, antiderivative size = 272 \[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \]

[Out]

1/3*x/b^2+1/3*a^3*arccsc(b*x+a)^2/b^3+1/3*x^3*arccsc(b*x+a)^2+2/3*arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)
^2)^(1/2))/b^3+4*a^2*arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3-2*a*ln(b*x+a)/b^3-1/3*I*polylo
g(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3-2*I*a^2*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^3+1/3*I*polylo
g(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3+2*I*a^2*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^3-2*a*(b*x+a)*ar
ccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^3+1/3*(b*x+a)^2*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^3

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5367, 4512, 4275, 4268, 2317, 2438, 4269, 3556, 4270} \[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 i a^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {x}{3 b^2} \]

[In]

Int[x^2*ArcCsc[a + b*x]^2,x]

[Out]

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4268

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] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[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 4270

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

Rule 4275

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 4512

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] + Dist[f*(m/(
b*d*(n + 1))), Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& IGtQ[m, 0] && NeQ[n, -1]

Rule 5367

Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int x^2 \cot (x) \csc (x) (-a+\csc (x))^2 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3} \\ & = \frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int x (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3} \\ & = \frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int \left (-a^3 x+3 a^2 x \csc (x)-3 a x \csc ^2(x)+x \csc ^3(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3} \\ & = \frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2-\frac {2 \text {Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3} \\ & = \frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {\text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}+\frac {(2 a) \text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^3} \\ & = \frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}+\frac {\text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^3}-\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {\left (2 i a^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \\ & = \frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {2 i a^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {2 i a^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3} \\ & = \frac {x}{3 b^2}-\frac {2 a (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^3}+\frac {(a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^3}+\frac {a^3 \csc ^{-1}(a+b x)^2}{3 b^3}+\frac {1}{3} x^3 \csc ^{-1}(a+b x)^2+\frac {2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {4 a^2 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^3}-\frac {2 a \log (a+b x)}{b^3}-\frac {i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}-\frac {2 i a^2 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^3}+\frac {i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{3 b^3}+\frac {2 i a^2 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 4.46 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.28 \[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=-\frac {-2 \left (2-12 a \csc ^{-1}(a+b x)+\left (1+6 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \csc ^{-1}(a+b x) \left (-1+3 a \csc ^{-1}(a+b x)\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {\csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{2 (a+b x)}-48 a \left (\log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+\log \left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )\right )+8 \left (1+6 a^2\right ) \left (\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )\right )+2 \csc ^{-1}(a+b x) \left (1+3 a \csc ^{-1}(a+b x)\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-8 (a+b x)^3 \csc ^{-1}(a+b x)^2 \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-2 \left (2+12 a \csc ^{-1}(a+b x)+\left (1+6 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{24 b^3} \]

[In]

Integrate[x^2*ArcCsc[a + b*x]^2,x]

[Out]

-1/24*(-2*(2 - 12*a*ArcCsc[a + b*x] + (1 + 6*a^2)*ArcCsc[a + b*x]^2)*Cot[ArcCsc[a + b*x]/2] + 2*ArcCsc[a + b*x
]*(-1 + 3*a*ArcCsc[a + b*x])*Csc[ArcCsc[a + b*x]/2]^2 - (ArcCsc[a + b*x]^2*Csc[ArcCsc[a + b*x]/2]^4)/(2*(a + b
*x)) - 48*a*(Log[1/((a + b*x)*Sqrt[1 - (a + b*x)^(-2)])] + Log[Sqrt[1 - (a + b*x)^(-2)]]) + 8*(1 + 6*a^2)*(Arc
Csc[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*ArcCsc[a + b*x]*(1 + 3*a*ArcCsc[a + b*x])*Sec[ArcCsc[a + b*
x]/2]^2 - 8*(a + b*x)^3*ArcCsc[a + b*x]^2*Sin[ArcCsc[a + b*x]/2]^4 - 2*(2 + 12*a*ArcCsc[a + b*x] + (1 + 6*a^2)
*ArcCsc[a + b*x]^2)*Tan[ArcCsc[a + b*x]/2])/b^3

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.84

method	result	size
derivativedivides	\(\frac {\operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+\frac {\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {b x}{3}+\frac {a}{3}+\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+2 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-2 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}+2 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -4 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +2 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right ) a +2 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-2 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )+2 i a \,\operatorname {arccsc}\left (b x +a \right )+\frac {i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}}{b^{3}}\)	\(500\)
default	\(\frac {\operatorname {arccsc}\left (b x +a \right )^{2} a^{2} \left (b x +a \right )-\operatorname {arccsc}\left (b x +a \right )^{2} a \left (b x +a \right )^{2}+\frac {\operatorname {arccsc}\left (b x +a \right )^{2} \left (b x +a \right )^{3}}{3}-2 \,\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, a \left (b x +a \right )+\frac {\operatorname {arccsc}\left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )^{2}}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+\frac {b x}{3}+\frac {a}{3}+\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}+2 i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}-\frac {\operatorname {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}-2 i \operatorname {polylog}\left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2}+2 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a -4 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a +2 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right ) a +2 \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )-2 \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right ) a^{2} \operatorname {arccsc}\left (b x +a \right )+2 i a \,\operatorname {arccsc}\left (b x +a \right )+\frac {i \operatorname {polylog}\left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3}}{b^{3}}\)	\(500\)

[In]

int(x^2*arccsc(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^2*arccsc(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^2*arccsc(b*x + a)^2, x)

Sympy [F]

\[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\int x^{2} \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate(x**2*acsc(b*x+a)**2,x)

[Out]

Integral(x**2*acsc(a + b*x)**2, x)

Maxima [F]

\[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^2*arccsc(b*x+a)^2,x, algorithm="maxima")

[Out]

1/3*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 1/12*x^3*log(b^2*x^2 + 2*a*b*x + a^2)^2 + integrat
e(1/3*(2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x^3*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - 3*(b^3*x^
5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*x^2)*log(b*x + a)^2 + (b^3*x^5 + 2*a*b^2*x^4 + (a^2 - 1)*b*x^3
+ 3*(b^3*x^5 + 3*a*b^2*x^4 + (3*a^2 - 1)*b*x^3 + (a^3 - a)*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)

Giac [F]

\[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{2} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^2*arccsc(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(x^2*arccsc(b*x + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \csc ^{-1}(a+b x)^2 \, dx=\int x^2\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]

[In]

int(x^2*asin(1/(a + b*x))^2,x)

[Out]

int(x^2*asin(1/(a + b*x))^2, x)