\(\int x^3 \csc ^{-1}(a+b x)^2 \, dx\) [27]

Optimal result

Integrand size = 12, antiderivative size = 366 \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {2 a \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {\log (a+b x)}{3 b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {i a \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {2 i a^3 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 20, 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} \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {2 i a^3 \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {2 a \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {i a \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {(a+b x)^2}{12 b^4}+\frac {\log (a+b x)}{3 b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac {a x}{b^3}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2 \]

[In]

Int[x^3*ArcCsc[a + b*x]^2,x]

[Out]

-((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/(3*b^4) + (3*a^2*(a
+ b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/b^4 - (a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x
])/b^4 + ((a + b*x)^3*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/(6*b^4) - (a^4*ArcCsc[a + b*x]^2)/(4*b^4) + (x
^4*ArcCsc[a + b*x]^2)/4 - (2*a*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^4 - (4*a^3*ArcCsc[a + b*x]*Ar
cTanh[E^(I*ArcCsc[a + b*x])])/b^4 + Log[a + b*x]/(3*b^4) + (3*a^2*Log[a + b*x])/b^4 + (I*a*PolyLog[2, -E^(I*Ar
cCsc[a + b*x])])/b^4 + ((2*I)*a^3*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^4 - (I*a*PolyLog[2, E^(I*ArcCsc[a + b*
x])])/b^4 - ((2*I)*a^3*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^4

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))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4} \\ & = \frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int x (-a+\csc (x))^4 \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4} \\ & = \frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {\text {Subst}\left (\int \left (a^4 x-4 a^3 x \csc (x)+6 a^2 x \csc ^2(x)-4 a x \csc ^3(x)+x \csc ^4(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 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 {\text {Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}+\frac {(2 a) \text {Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4} \\ & = -\frac {a x}{b^3}+\frac {(a+b x)^2}{12 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 {4 a^3 \csc ^{-1}(a+b x) \text {arctanh}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {\text {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}+\frac {a \text {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 a^3\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4} \\ & = -\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 {3 a^2 \log (a+b x)}{b^4}-\frac {\text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}-\frac {a \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {a \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 i a^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {\left (2 i a^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4} \\ & = -\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 {2 i a^3 \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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {(i a) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4} \\ & = -\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} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 3.95 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.24 \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\frac {-16 \left (6 a-2 \left (1+9 a^2\right ) \csc ^{-1}(a+b x)+3 \left (a+2 a^3\right ) \csc ^{-1}(a+b x)^2\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \left (2-24 a \csc ^{-1}(a+b x)+\left (3+36 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {2 \csc ^{-1}(a+b x) \left (-1+6 a \csc ^{-1}(a+b x)\right ) \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a+b x}-64 \left (1+9 a^2\right ) \left (\log \left (\frac {1}{(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}\right )+\log \left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )\right )+192 \left (a+2 a^3\right ) \left (\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )\right )+2 \left (2+24 a \csc ^{-1}(a+b x)+\left (3+36 a^2\right ) \csc ^{-1}(a+b x)^2\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \sec ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-32 (a+b x)^3 \csc ^{-1}(a+b x) \left (1+6 a \csc ^{-1}(a+b x)\right ) \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-16 \left (6 a+2 \left (1+9 a^2\right ) \csc ^{-1}(a+b x)+3 \left (a+2 a^3\right ) \csc ^{-1}(a+b x)^2\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{192 b^4} \]

[In]

Integrate[x^3*ArcCsc[a + b*x]^2,x]

[Out]

(-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 - 2
4*a*ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Csc[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Csc[ArcCs
c[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)*Se
c[ArcCsc[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)

Maple [A] (verified)

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}}\)	\(703\)

[In]

int(x^3*arccsc(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

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*p
olylog(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*arccsc(b*x+a)+1/3*arccsc(b*x+a)*(
((b*x+a)^2-1)/(b*x+a)^2)^(1/2)*(b*x+a)+3*arccsc(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))

Fricas [F]

\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^3*arccsc(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(x^3*arccsc(b*x + a)^2, x)

Sympy [F]

\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int x^{3} \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]

[In]

integrate(x**3*acsc(b*x+a)**2,x)

[Out]

Integral(x**3*acsc(a + b*x)**2, x)

Maxima [F]

\[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int { x^{3} \operatorname {arccsc}\left (b x + a\right )^{2} \,d x } \]

[In]

integrate(x^3*arccsc(b*x+a)^2,x, algorithm="maxima")

[Out]

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 + integrat
e(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)

Giac [F(-2)]

Exception generated. \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^3*arccsc(b*x+a)^2,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int x^3 \csc ^{-1}(a+b x)^2 \, dx=\int x^3\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]

[In]

int(x^3*asin(1/(a + b*x))^2,x)

[Out]

int(x^3*asin(1/(a + b*x))^2, x)