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

Optimal. Leaf size=220 \[ -\frac{i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac{i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3} \]

[Out]

(b^2*x*(a + b*ArcCsc[c*x]))/c^2 + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x])^2)/(2*c) + (x^3*(a + b*ArcC
sc[c*x])^3)/3 + (b*(a + b*ArcCsc[c*x])^2*ArcTanh[E^(I*ArcCsc[c*x])])/c^3 + (b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]]
)/c^3 - (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2,
E^(I*ArcCsc[c*x])])/c^3 + (b^3*PolyLog[3, -E^(I*ArcCsc[c*x])])/c^3 - (b^3*PolyLog[3, E^(I*ArcCsc[c*x])])/c^3

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Rubi [A]  time = 0.188112, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5223, 4410, 4186, 3770, 4183, 2531, 2282, 6589} \[ -\frac{i b^2 \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac{i b^2 \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{c^3}+\frac{b^3 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{b \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2}{c^3}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsc[c*x])^3,x]

[Out]

(b^2*x*(a + b*ArcCsc[c*x]))/c^2 + (b*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x])^2)/(2*c) + (x^3*(a + b*ArcC
sc[c*x])^3)/3 + (b*(a + b*ArcCsc[c*x])^2*ArcTanh[E^(I*ArcCsc[c*x])])/c^3 + (b^3*ArcTanh[Sqrt[1 - 1/(c^2*x^2)]]
)/c^3 - (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (I*b^2*(a + b*ArcCsc[c*x])*PolyLog[2,
E^(I*ArcCsc[c*x])])/c^3 + (b^3*PolyLog[3, -E^(I*ArcCsc[c*x])])/c^3 - (b^3*PolyLog[3, E^(I*ArcCsc[c*x])])/c^3

Rule 5223

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rule 4410

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4186

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

Rule 3770

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

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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \cot (x) \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \csc ^3(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3-\frac{b \operatorname{Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^3}-\frac{b^3 \operatorname{Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3}+\frac{b^2 \operatorname{Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac{b^2 \operatorname{Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}-\frac{\left (i b^3\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac{b^3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ &=\frac{b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac{1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac{b \left (a+b \csc ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^2 x^2}}\right )}{c^3}-\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{i b^2 \left (a+b \csc ^{-1}(c x)\right ) \text{Li}_2\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac{b^3 \text{Li}_3\left (-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac{b^3 \text{Li}_3\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}\\ \end{align*}

Mathematica [B]  time = 7.75087, size = 580, normalized size = 2.64 \[ \frac{a b^2 \left (2 c^3 x^3 \left (\frac{4 i \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3 x^3}+4 \csc ^{-1}(c x)^2-2 \cos \left (2 \csc ^{-1}(c x)\right )-\frac{3 \csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right )}{c x}+\frac{3 \csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right )}{c x}+2 \csc ^{-1}(c x) \sin \left (2 \csc ^{-1}(c x)\right )+\csc ^{-1}(c x) \log \left (1-e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )-\csc ^{-1}(c x) \log \left (1+e^{i \csc ^{-1}(c x)}\right ) \sin \left (3 \csc ^{-1}(c x)\right )+2\right )-8 i \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )\right )}{8 c^3}+\frac{b^3 \left (-48 i \csc ^{-1}(c x) \text{PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+48 i \csc ^{-1}(c x) \text{PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+48 \text{PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-48 \text{PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )+16 c^3 x^3 \csc ^{-1}(c x)^3 \sin ^4\left (\frac{1}{2} \csc ^{-1}(c x)\right )+\frac{\csc ^{-1}(c x)^3 \csc ^4\left (\frac{1}{2} \csc ^{-1}(c x)\right )}{c x}+6 \csc ^{-1}(c x)^2 \csc ^2\left (\frac{1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \cot \left (\frac{1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \cot \left (\frac{1}{2} \csc ^{-1}(c x)\right )-24 \csc ^{-1}(c x)^2 \log \left (1-e^{i \csc ^{-1}(c x)}\right )+24 \csc ^{-1}(c x)^2 \log \left (1+e^{i \csc ^{-1}(c x)}\right )+4 \csc ^{-1}(c x)^3 \tan \left (\frac{1}{2} \csc ^{-1}(c x)\right )+24 \csc ^{-1}(c x) \tan \left (\frac{1}{2} \csc ^{-1}(c x)\right )-6 \csc ^{-1}(c x)^2 \sec ^2\left (\frac{1}{2} \csc ^{-1}(c x)\right )-48 \log \left (\tan \left (\frac{1}{2} \csc ^{-1}(c x)\right )\right )\right )}{48 c^3}+\frac{a^2 b x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{a^2 b \log \left (x \left (\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right )\right )}{2 c^3}+a^2 b x^3 \csc ^{-1}(c x)+\frac{a^3 x^3}{3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2*(a + b*ArcCsc[c*x])^3,x]

[Out]

(a^3*x^3)/3 + (a^2*b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + a^2*b*x^3*ArcCsc[c*x] + (a^2*b*Log[x*(1 + Sqr
t[(-1 + c^2*x^2)/(c^2*x^2)])])/(2*c^3) + (a*b^2*((-8*I)*PolyLog[2, -E^(I*ArcCsc[c*x])] + 2*c^3*x^3*(2 + 4*ArcC
sc[c*x]^2 - 2*Cos[2*ArcCsc[c*x]] - (3*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])])/(c*x) + (3*ArcCsc[c*x]*Log[1 + E
^(I*ArcCsc[c*x])])/(c*x) + ((4*I)*PolyLog[2, E^(I*ArcCsc[c*x])])/(c^3*x^3) + 2*ArcCsc[c*x]*Sin[2*ArcCsc[c*x]]
+ ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])]*Sin[3*ArcCsc[c*x]] - ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])]*Sin[3*Arc
Csc[c*x]])))/(8*c^3) + (b^3*(24*ArcCsc[c*x]*Cot[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Cot[ArcCsc[c*x]/2] + 6*ArcCsc
[c*x]^2*Csc[ArcCsc[c*x]/2]^2 + (ArcCsc[c*x]^3*Csc[ArcCsc[c*x]/2]^4)/(c*x) - 24*ArcCsc[c*x]^2*Log[1 - E^(I*ArcC
sc[c*x])] + 24*ArcCsc[c*x]^2*Log[1 + E^(I*ArcCsc[c*x])] - 48*Log[Tan[ArcCsc[c*x]/2]] - (48*I)*ArcCsc[c*x]*Poly
Log[2, -E^(I*ArcCsc[c*x])] + (48*I)*ArcCsc[c*x]*PolyLog[2, E^(I*ArcCsc[c*x])] + 48*PolyLog[3, -E^(I*ArcCsc[c*x
])] - 48*PolyLog[3, E^(I*ArcCsc[c*x])] - 6*ArcCsc[c*x]^2*Sec[ArcCsc[c*x]/2]^2 + 16*c^3*x^3*ArcCsc[c*x]^3*Sin[A
rcCsc[c*x]/2]^4 + 24*ArcCsc[c*x]*Tan[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Tan[ArcCsc[c*x]/2]))/(48*c^3)

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Maple [B]  time = 0.446, size = 648, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^3*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 1/4*b^3*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*lo
g(c^2*x^2)^2 - 1/2*a*b^2*c^2*(2*(c^2*x^3 + 3*x)/c^4 - 3*log(c*x + 1)/c^5 + 3*log(c*x - 1)/c^5)*log(c)^2 - 12*b
^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c)^2 + 12*b^3*c^2*integ
rate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integr
ate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) + 12*a*b^2*c^2*integrate(1
/4*x^4*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*a*b^2*c^2*integrate(1/4*x^4*log(x)/(c^2*x^2 - 1), x)*log(c)
+ 1/3*a^3*x^3 + 12*b^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)*log(x)/(c^2*
x^2 - 1), x) - 12*b^3*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)^2/(c^2*x^2 - 1), x)
+ 12*a*b^2*c^2*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) + 4*b^3*c^2*inte
grate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) - 3*a*b^2*c^2*integrate(1
/4*x^4*log(c^2*x^2)^2/(c^2*x^2 - 1), x) + 12*a*b^2*c^2*integrate(1/4*x^4*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x)
- 12*a*b^2*c^2*integrate(1/4*x^4*log(x)^2/(c^2*x^2 - 1), x) + 3/2*a*b^2*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x
- 1)/c^3)*log(c)^2 + 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^2 - 1), x)*log(c
)^2 - 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) +
24*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*log(c) - 12*a*b^2*i
ntegrate(1/4*x^2*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) + 24*a*b^2*integrate(1/4*x^2*log(x)/(c^2*x^2 - 1), x)*l
og(c) + 1/4*(4*x^3*arccsc(c*x) + (2*sqrt(-1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + log(sqrt(-1/(c^2*x^
2) + 1) + 1)/c^2 - log(sqrt(-1/(c^2*x^2) + 1) - 1)/c^2)/c)*a^2*b + 4*b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*x
- 1)*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x) - b^3*integrate(1/4*sqrt(c*x + 1)*sqrt(c*
x - 1)*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))
*log(c^2*x^2)*log(x)/(c^2*x^2 - 1), x) + 12*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(
x)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))^2/(c^2*x^2 - 1), x
) - 4*b^3*integrate(1/4*x^2*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x) + 3*a*b^2*i
ntegrate(1/4*x^2*log(c^2*x^2)^2/(c^2*x^2 - 1), x) - 12*a*b^2*integrate(1/4*x^2*log(c^2*x^2)*log(x)/(c^2*x^2 -
1), x) + 12*a*b^2*integrate(1/4*x^2*log(x)^2/(c^2*x^2 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*acsc(c*x))**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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