Integrand size = 14, antiderivative size = 220 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {b^2 x \left (a+b \csc ^{-1}(c x)\right )}{c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \csc ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \csc ^{-1}(c x)\right )^3+\frac {b \left (a+b \csc ^{-1}(c x)\right )^2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \csc ^{-1}(c x)\right ) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )}{c^3} \]
b^2*x*(a+b*arccsc(c*x))/c^2+1/3*x^3*(a+b*arccsc(c*x))^3+b*(a+b*arccsc(c*x) )^2*arctanh(I/c/x+(1-1/c^2/x^2)^(1/2))/c^3+b^3*arctanh((1-1/c^2/x^2)^(1/2) )/c^3-I*b^2*(a+b*arccsc(c*x))*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))/c^3+I* b^2*(a+b*arccsc(c*x))*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))/c^3+b^3*polylog (3,-I/c/x-(1-1/c^2/x^2)^(1/2))/c^3-b^3*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2) )/c^3+1/2*b*x^2*(a+b*arccsc(c*x))^2*(1-1/c^2/x^2)^(1/2)/c
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(580\) vs. \(2(220)=440\).
Time = 7.39 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.64 \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\frac {a^3 x^3}{3}+\frac {a^2 b x^2 \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+a^2 b x^3 \csc ^{-1}(c x)+\frac {a^2 b \log \left (x \left (1+\sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}\right )\right )}{2 c^3}+\frac {a b^2 \left (-8 i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+2 c^3 x^3 \left (2+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}+\frac {4 i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )}{c^3 x^3}+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 )\right )\right )}{8 c^3}+\frac {b^3 \left (24 \csc ^{-1}(c x) \cot \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 )+6 \csc ^{-1}(c x)^2 \csc ^2\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}-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 )-48 \log \left (\tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )-48 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )+48 i \csc ^{-1}(c x) \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )+48 \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )-48 \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )-6 \csc ^{-1}(c x)^2 \sec ^2\left (\frac {1}{2} \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 )+24 \csc ^{-1}(c x) \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )+4 \csc ^{-1}(c x)^3 \tan \left (\frac {1}{2} \csc ^{-1}(c x)\right )\right )}{48 c^3} \]
(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 + Sqrt[(-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*ArcCsc [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)*PolyLo g[2, E^(I*ArcCsc[c*x])])/(c^3*x^3) + 2*ArcCsc[c*x]*Sin[2*ArcCsc[c*x]] + Ar cCsc[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*ArcCsc[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*A rcCsc[c*x]^2*Log[1 - E^(I*ArcCsc[c*x])] + 24*ArcCsc[c*x]^2*Log[1 + E^(I*Ar cCsc[c*x])] - 48*Log[Tan[ArcCsc[c*x]/2]] - (48*I)*ArcCsc[c*x]*PolyLog[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*Arc Csc[c*x]^2*Sec[ArcCsc[c*x]/2]^2 + 16*c^3*x^3*ArcCsc[c*x]^3*Sin[ArcCsc[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)
Time = 0.82 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.97, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5746, 4910, 3042, 4674, 3042, 4257, 4671, 3011, 2720, 7143}
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.
| \(\displaystyle \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 5746 |
| \(\displaystyle -\frac {\int c^4 \sqrt {1-\frac {1}{c^2 x^2}} x^4 \left (a+b \csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)}{c^3}\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle -\frac {b \int c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3}{c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )^3d\csc ^{-1}(c x)-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3}{c^3}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {b \left (\frac {1}{2} \int c x \left (a+b \csc ^{-1}(c x)\right )^2d\csc ^{-1}(c x)+b^2 \int c xd\csc ^{-1}(c x)-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3}{c^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {1}{2} \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)+b^2 \int \csc \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )\right )-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3}{c^3}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {b \left (\frac {1}{2} \int \left (a+b \csc ^{-1}(c x)\right )^2 \csc \left (\csc ^{-1}(c x)\right )d\csc ^{-1}(c x)-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )+b^2 \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )\right )-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3}{c^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3+b \left (\frac {1}{2} \left (-2 b \int \left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)+2 b \int \left (a+b \csc ^{-1}(c x)\right ) \log \left (1+e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )+b^2 \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3+b \left (\frac {1}{2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-i b \int \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-i b \int \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )d\csc ^{-1}(c x)\right )-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )+b^2 \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3+b \left (\frac {1}{2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \int e^{-i \csc ^{-1}(c x)} \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \int e^{-i \csc ^{-1}(c x)} \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right )de^{i \csc ^{-1}(c x)}\right )-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )+b^2 \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )\right )}{c^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {-\frac {1}{3} c^3 x^3 \left (a+b \csc ^{-1}(c x)\right )^3+b \left (\frac {1}{2} \left (-2 \text {arctanh}\left (e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2+2 b \left (i \operatorname {PolyLog}\left (2,-e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \operatorname {PolyLog}\left (3,-e^{i \csc ^{-1}(c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,e^{i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )-b \operatorname {PolyLog}\left (3,e^{i \csc ^{-1}(c x)}\right )\right )\right )-\frac {1}{2} c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-b c x \left (a+b \csc ^{-1}(c x)\right )+b^2 \left (-\text {arctanh}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )\right )\right )}{c^3}\) |
-((-1/3*(c^3*x^3*(a + b*ArcCsc[c*x])^3) + b*(-(b*c*x*(a + b*ArcCsc[c*x])) - (c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*(a + b*ArcCsc[c*x])^2)/2 - b^2*ArcTanh[Sq rt[1 - 1/(c^2*x^2)]] + (-2*(a + b*ArcCsc[c*x])^2*ArcTanh[E^(I*ArcCsc[c*x]) ] + 2*b*(I*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])] - b*PolyLog[ 3, -E^(I*ArcCsc[c*x])]) - 2*b*(I*(a + b*ArcCsc[c*x])*PolyLog[2, E^(I*ArcCs c[c*x])] - b*PolyLog[3, E^(I*ArcCsc[c*x])]))/2))/c^3)
3.1.25.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), 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] + Simp[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] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
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 ] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[- (c^(m + 1))^(-1) Subst[Int[(a + b*x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcC sc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (GtQ[n , 0] || LtQ[m, -1])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> 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]
Time = 1.90 (sec) , antiderivative size = 535, normalized size of antiderivative = 2.43
| method | result | size |
| derivativedivides | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(535\) |
| default | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )+3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(535\) |
| parts | \(\frac {a^{3} x^{3}}{3}+\frac {b^{3} \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (2 c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+3 \,\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+6\right ) c x}{6}-\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}+i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {polylog}\left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {\operatorname {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{2}-i \operatorname {arccsc}\left (c x \right ) \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\operatorname {polylog}\left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 \,\operatorname {arctanh}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{c^{3}}+\frac {3 a \,b^{2} \left (\frac {\left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )^{2}+\operatorname {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}+1\right ) c x}{3}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}-\frac {i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{3}\right )}{c^{3}}+\frac {3 a^{2} b \left (\frac {c^{3} x^{3} \operatorname {arccsc}\left (c x \right )}{3}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (c x \sqrt {c^{2} x^{2}-1}+\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )\right )}{6 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}\) | \(537\) |
1/c^3*(1/3*c^3*x^3*a^3+b^3*(1/6*arccsc(c*x)*(2*c^2*x^2*arccsc(c*x)^2+3*arc csc(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)^(1/2)+6)*c*x-1/2*arccsc(c*x)^2*ln(1-I/c /x-(1-1/c^2/x^2)^(1/2))+I*arccsc(c*x)*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2)) -polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))+1/2*arccsc(c*x)^2*ln(1+I/c/x+(1-1/c^ 2/x^2)^(1/2))-I*arccsc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))+polylog( 3,-I/c/x-(1-1/c^2/x^2)^(1/2))+2*arctanh(I/c/x+(1-1/c^2/x^2)^(1/2)))+3*a*b^ 2*(1/3*(c^2*x^2*arccsc(c*x)^2+arccsc(c*x)*c*x*((c^2*x^2-1)/c^2/x^2)^(1/2)+ 1)*c*x-1/3*arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+1/3*I*polylog(2,I/c /x+(1-1/c^2/x^2)^(1/2))+1/3*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/ 3*I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))+3*a^2*b*(1/3*c^3*x^3*arccsc(c*x )+1/6*(c^2*x^2-1)^(1/2)*(c*x*(c^2*x^2-1)^(1/2)+ln(c*x+(c^2*x^2-1)^(1/2)))/ ((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))
\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]
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*a rccsc(c*x) + a^3*x^2, x)
\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int x^{2} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \]
\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]
1/3*b^3*x^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 1/4*b^3*x^3*arctan 2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(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*integrate(1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqr t(c*x - 1)))*log(c^2*x^2)/(c^2*x^2 - 1), x)*log(c) - 24*b^3*c^2*integrate( 1/4*x^4*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^2 - 1), x)*l og(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*integrate(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)*l og(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...
\[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}^{3} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \csc ^{-1}(c x)\right )^3 \, dx=\int x^2\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]