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

Optimal. Leaf size=366 \[ \frac{2 i a^3 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{2 i a^3 \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{i a \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{i a \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{3 a^2 \log (a+b x)}{b^4}-\frac{a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\log (a+b x)}{3 b^4}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac{2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)^2 \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.305289, antiderivative size = 366, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5259, 4427, 4190, 4183, 2279, 2391, 4184, 3475, 4185} \[ \frac{2 i a^3 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{2 i a^3 \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{i a \text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{i a \text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{3 a^2 \log (a+b x)}{b^4}-\frac{a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{a x}{b^3}+\frac{(a+b x)^2}{12 b^4}+\frac{\log (a+b x)}{3 b^4}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac{(a+b x)^3 \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac{2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)^2 \]

Antiderivative was successfully verified.

[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 5259

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]

Rule 4427

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 4190

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 4183

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 2279

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 2391

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 4184

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 3475

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

Rule 4185

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] && G
tQ[n, 1] && NeQ[n, 2]

Rubi steps

\begin{align*} \int x^3 \csc ^{-1}(a+b x)^2 \, dx &=-\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}+\frac{(2 a) \operatorname{Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac{\left (2 a^3\right ) \operatorname{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) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{\operatorname{Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}+\frac{a \operatorname{Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac{\left (2 a^3\right ) \operatorname{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 ) \operatorname{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) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{3 a^2 \log (a+b x)}{b^4}-\frac{\operatorname{Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}-\frac{a \operatorname{Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac{a \operatorname{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 ) \operatorname{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 ) \operatorname{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) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{2 i a^3 \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{(i a) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{(i a) \operatorname{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) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\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 \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac{2 i a^3 \text{Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{i a \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac{2 i a^3 \text{Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}\\ \end{align*}

Mathematica [A]  time = 5.16831, size = 420, normalized size = 1.15 \[ \frac{192 \left (2 a^3+a\right ) \left (i \left (\text{PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )-\text{PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )\right )+\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )\right )-64 \left (9 a^2+1\right ) \log \left (\frac{1}{a+b x}\right )+2 \left (\left (36 a^2+3\right ) \csc ^{-1}(a+b x)^2-24 a \csc ^{-1}(a+b x)+2\right ) \csc ^2\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-16 \left (3 \left (2 a^3+a\right ) \csc ^{-1}(a+b x)^2-2 \left (9 a^2+1\right ) \csc ^{-1}(a+b x)+6 a\right ) \cot \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-16 \left (3 \left (2 a^3+a\right ) \csc ^{-1}(a+b x)^2+2 \left (9 a^2+1\right ) \csc ^{-1}(a+b x)+6 a\right ) \tan \left (\frac{1}{2} \csc ^{-1}(a+b x)\right )+2 \left (\left (36 a^2+3\right ) \csc ^{-1}(a+b x)^2+24 a \csc ^{-1}(a+b x)+2\right ) \sec ^2\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \csc ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )-\frac{2 \csc ^{-1}(a+b x) \left (6 a \csc ^{-1}(a+b x)-1\right ) \csc ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{a+b x}-32 (a+b x)^3 \csc ^{-1}(a+b x) \left (6 a \csc ^{-1}(a+b x)+1\right ) \sin ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \sec ^4\left (\frac{1}{2} \csc ^{-1}(a+b x)\right )}{192 b^4} \]

Warning: Unable to verify antiderivative.

[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[(a + b*x)^(-1)] + 192*(a + 2*a^3)*(ArcCsc[a + b*x]*(Log[1 - E^(I*ArcCsc[a + b*x])] - Log[1 + E^(I*A
rcCsc[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[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)

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Maple [A]  time = 0.81, size = 769, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^4*arccsc(b*x+a)^2/b^4-5/6*a*x/b^3-1/2/b^2*arccsc(b*x+a)*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*x^2*a-3/b^4*a^
2*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2)-1)-3/b^4*a^2*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+6/b^4*a^2*ln(I/(b*x+a)
+(1-1/(b*x+a)^2)^(1/2))-1/3*I/b^4*arccsc(b*x+a)+1/6/b*arccsc(b*x+a)*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*x^3+1/3/b
^3*arccsc(b*x+a)*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*x-3*I/b^4*a^2*arccsc(b*x+a)+1/b^4*a*arccsc(b*x+a)*ln(1-I/(b*
x+a)-(1-1/(b*x+a)^2)^(1/2))-1/b^4*a*arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+2/b^4*a^3*arccsc(b*x+a
)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-2/b^4*a^3*arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+13/6/b^4
*arccsc(b*x+a)*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*a^3+1/3/b^4*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*arccsc(b*x+a)*a+3
/2/b^3*arccsc(b*x+a)*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*x*a^2+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^4-2*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^4-11/12/b^4*a^2+1/12/b^2*x^2+2/3/b^4*ln(I/(b*x+a)+(1-1/(b*
x+a)^2)^(1/2))-1/3/b^4*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-1/3/b^4*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2)-1)+1/4
*x^4*arccsc(b*x+a)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right )^{2} - \frac{1}{16} \, x^{4} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \int \frac{2 \, \sqrt{b x + a + 1} \sqrt{b x + a - 1} b x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) - 4 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} - 1\right )} b x^{4} +{\left (a^{3} - a\right )} x^{3}\right )} \log \left (b x + a\right )^{2} +{\left (b^{3} x^{6} + 2 \, a b^{2} x^{5} +{\left (a^{2} - 1\right )} b x^{4} + 4 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} - 1\right )} b x^{4} +{\left (a^{3} - a\right )} x^{3}\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{4 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{3} \operatorname{arccsc}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arccsc(b*x + a)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acsc}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acsc(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arccsc}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccsc(b*x + a)^2, x)